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Angles Complete Lessons for KS2 to GCSE Maths

Angles Complete Lessons for KS2 to GCSE Maths

Subject: Mathematics

Age range: 11-14

Resource type: Lesson (complete)

Empower Learning

Last updated

8 February 2024

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angles in triangles problem solving tes

Complete set of comprehensive angle lessons covering angle basics, angle rules, angles in triangles and angles in quadrilaterals. They cover everything you need to know about these topics, from KS2 level through to GCSE foundation level. These are ideal to use within the classroom, for home educating families, or for tutors to use in lessons!

This pack includes: A PPT file with 5 individual lessons and A PDF file with 5 printable question worksheets (one to go with each lesson)

Each lesson includes: a starter activity, teaching content, worked examples, questions for students to complete (also included on a printable worksheet), quiz questions, an exit ticket/plenary question, and answer slides for all questions.

Lessons are designed to last approximately 50-60 minutes, but can be tailored to individual preference.

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angles in triangles problem solving tes

  • Time  - Boss Maths
  • Converting time odd one out  ( answers ) - Dr Austin Maths
  • KS2 Time  - Maths4Everyone
  • Time worksheets  - teachingimage.com 
  • Converting decimals to hours and minutes  - @mathsmrgordon via variationtheory.com
  • Reading scales  - SMILE
  • Reading and interpreting scales  - fionajones88 on TES
  • Using scale diagrams and maps  - Boss Maths
  • Scale drawings  - Boss Maths
  • Interpreting Map Scales Practice Grid ( Answers ) - Dr Austin Maths
  • Using Map Scales Practice Grid ( Answers ) - Dr Austin Maths
  • Scale Drawing and Maps - dannytheref on TES
  • Scale Drawing - CIMT

Scale Drawings Practice Grid ( answers ) - Dr Austin Maths

  • Scale drawings - Dr Austin Maths
  • DIY Units  - Andy Lutwyche on TES
  • Identifying measurements - TeachIt Maths
  • Metric/imperial conversion bursts - TeachIt Maths
  • Metric units table  - The Chalk Face
  • Metric units activity  - The Chalk Face
  • Metric and imperial units:  activity 1  &  activity 2  - The Chalk Face
  • Convert lengths bingo cards/card match  - Nuffield Foundation
  • Units and large numbers  - Teachit Maths
  • Units of Measurement - CIMT
  • History of the metric system - Dan Walker on TES
  • Why the metric system matters - Matt Anticole on Youtube
  • Converting between metric units of measures of length and mass  - Boss Maths
  • Metric unit conversions: lengths - Josh Cutts via variationtheory.com
  • Metric unit conversions quiz - by me!
  • Converting between metric units of measures of area and volume  - Boss Maths
  • Money  - Boss Maths
  • Universcale  (magnitude exploration tool) - Nikon
[ back to top ]
  • Loci and regions - Median Don Steward
  • Loci challenges booklet - The Chalk Face
  • Loci investigations - The Chalk Face
  • Loci matching exercise - @GiftedBA
  • Mixed loci problems lesson - Boss Maths
  • Grid Loci - Median Don Steward
  • Loci presentation and worksheets -  @DJUdall 
  • Festival Safety - idearefresh.co.uk
  • Loci practice grid  ( answers ) - Dr Austin Maths
  • Loci exam questions
  • Constructing perpendicular bisectors, angle bisectors, and perpendiculars to or from a point lesson - Boss Maths
  • Constructing triangles lesson - Boss Maths
  • Constructing triangles lesson - cparkinson3 on TES
  • Constructing perpendicular and angle bisectors lesson - cparkinson3 on TES
  • Construct a scenario - Teachit Maths
  • Constructions tasks - Median Don Steward
  • Contructions - Dan Walker on TES
  • Online constructions demonstrations and tools
  • Construction animated gifs  (to paste into PowerPoints)
  • Interactive Construction Tool - MathsPad.co.uk
  • Euclid the Game
  • Speed, distance, time task - Segar Rogers
  • Speed activity - The Chalk Face
  • Using compound units - Boss Maths
  • Finding speed  - @fortyninecubed via variationtheory.com 
  • Speed, distance, time questions  ( answers ) - @taylorda01
  • Speed, distance, time questions II  ( answers ) - @taylorda01
  • Speeding - Median Don Steward
  • Speed calculations fill in the blanks  ( answers ) - Dr Austin Maths
  • Harder speed calculations fill in the blanks  ( answers ) - Dr Austin Maths
  • Distance-Time Graphs lesson - cparkinson3 on TES
  • Distance-time graphs step-by-step worksheet - Labrown20 on TES
  • Distance-time graphs and average speed  - Median Don Steward
  • Interpreting Distance-Time Graphs  - Maths Assessment Project (activities at the back)
  • Distance Time Graphs - mathshko
  • Journey - Maths Assessment Project (activities at the back)
  • Hurdles Race - Transum
  • Super hero activity  - @jase_wanner 
  • Compound Measures - Median Don Steward
  • Density, mass, volume questions  ( answers ) - @taylorda01
  • Exam question practice - density - Maths4Everyone on TES
  • Density questions from Linked Pair GCSE - compiled by me!
  • Identifying similar triangles  - Mathematics Assessment Project. 
  • Similar shapes booklet  - tumshy on TES.
  • Similarity  - CIMT
  • Similar triangles - steele1989 on TES
  • Similar Triangles notes and exercises - martahidegkuti.com
  • Similar shapes odd one out ( answers ) - Dr Austin Maths
  • Similar Shapes Practice Strips ( Answers ) - Dr Austin Maths
  • Similar Shapes Crack the Code ( Answers ) - Dr Austin Maths
  • Similar triangles questions  ( Answers ) - @taylorda01
  • Similarity slides - Don Steward
  • Similar triangles exam style questions ( answers ) - 1st Class Maths
  • Similar shapes exam style questions  ( answers ) - 1st Class Maths
  • Model Boat  - The Chalk Face 
  • Similar shapes and equations - Mr Thompson on TES
  • Evaluating Statements about Enlargements  - Mathematics Assessment Project
  • Length, Area and Volume Factors - Median Don Steward
  • Lengths, areas and volumes in similar shapes - Boss Maths
  • Similar Areas and Volumes Fill In The Blanks ( Answers ) - Dr Austin Maths
  • Similar Areas and Volumes Practice Strips ( Answers ) - Dr Austin Maths
  • Harder Similar Areas and Volumes Fill In The Blanks ( Answers ) - Dr Austin Maths
  • Harder Similar Areas and Volumes Practice Strips ( Answers ) - Dr Austin Maths
  • Similar area and volume - mathshelper.co.uk
  • Similar shapes GCSE revision - Maths4Everyone on TES
  • Baby surface area  - The Chalk Face
  • Similarity Check In - OCR
  • Similar shapes (area and volume) exam questions ( answers ) - Maths Genie
  • Prove it! - MathsPad
  • Similar shapes full coverage GCSE questions  - compiled by Dr Frost
  • Similar area and volume exam style questions ( answers ) - 1st Class Maths
  • Congruence criteria for triangles (SSS, SAS, ASA, RHS)  - Boss Maths
  • Congruent triangles worksheet - mathsmalakiss
  • Analysing congruency proofs - Mathematics Assessment Project
  • Congruent triangles
  • Complete the congruence proof  - topdrawer.aamt.edu.au
  • Constructions and congruence in triangles - Median Don Steward
  • Congruence and similarity assessment - topdrawer.aamt.edu.au
  • Congruence Check In - OCR
  • Identify and Label Angles and Lengths  - fionaryan88 on TES
  • Conventions for labelling the sides and angles of triangles  - Boss Maths
  • Lines and angles naming and vocabulary activity  - alicecreswick on TES
  • Drawing diagrams from a written description  - Boss Maths
  • Understanding Angle Labels - Dan Draper
  • Estimating and naming angles  - MathsPad
  • CIMT angles: KS3 and KS4 - CIMT via TES
  • Measuring line segments and angles in geometric figures - Boss Maths
  • Measuring angles - demo and worksheets  - GoTeachMaths
  • Introducing angles  - Gareth Evans on TES
  • Basic angle properties activity  - Mark Horley
  • Angle rules crack the code  ( answers ) - Dr Austin Maths
  • Angles on a straight line worksheets  - Maths4Everyone on TES
  • Angles around a point worksheets  - Maths4Everyone on TES
  • Angles at a point, angles at a point on a straight line, vertically opposite angles  - Boss Maths
  • Angles in triangles (worksheet bundle) - Maths4Everyone on TES
  • Angles in a triangle extra practice  - Maths4Everyone on TES
  • Isosceles triangle angles  - Median Don Steward
  • Angles practice makes perfect - Median Don Steward
  • Angles with algebra - Don Steward
  • Angle facts questions  ( answers & supporting material ) - @taylorda01
  • Angles in triangles and quadrilateral questions  ( answers ) - @taylorda01
  • Missing angles  - Median Don Steward
  • Two isosceles triangles stuck together - Median Don Steward
  • Isosceles triangle proofs  - Median Don Steward
  • Angles in triangles - equations  - MissBrookesMaths
  • Angles in quadrilaterals - equations  - MissBrookesMaths
  • Angle chases  - thanks to @MathedUp
  • Angles and ratio  - Sam Blatherwick
  • Angle problems - Nrich 
  • Mixed angle puzzles - collated by me, see sheet for original sources
  • There are loads of brilliant angles activities from  MathsPad  for a small subscription
  • Alternate and corresponding angles on parallel lines - Boss Maths
  • Spot the Angle  - MathsPad
  • Interactive Parallel Lines Tool  - MathsPad
  • Angles in parallel lines questions   ( answers  &  supporting material ) - @taylorda01
  • Angles in parallel lines tick or trash  - MissBrookesMaths
  • Unmarked angles  - SMILE
  • Missing angles  - SMILE
  • Angles in Parallel Lines (Worksheet Bundle)  - Maths4Everyone on TES
  • Angles in parallel lines practice grids ( Answers ) - Dr Austin Maths
  • Angles in Parallel Lines GCSE questions - Maths4Everyone
  • Parallel line angles  - Median Don Steward
  • Angles in parallel lines: comparing methods  - by me!
  • Angles in Parallel Lines - Solving Linear Equations  - Mr Thompson on TES
  • Solving linear equations from parallel lines - Nathan Day
  • Parallelogram mazes - mathymcmatherson
  • Parallel line angle relationships and proof  - Median Don Steward
  • Angles in parallel lines GCSE questions  - Labrown20 on TES
  • Death Star Angles  - dooranran on TES
  • Mixed puzzles  -  collated by me, see sheet for original sources
  • Maths GCSE Bearings  - collinbillet on TES
  • Bearings lesson - Boss Maths
  • Angles, bearings and maps - CIMT
  • Bearings teaching ideas and activities - Colin Foster
  • Angle properties and bearings - pas1001 on TES
  • Bearings worksheets  and Finding bearings - The Chalk Face
  • Calculating bearings - Richard Tock
  • Calculating bearings - Maths4Everyone on TES
  • Measuring bearings match up ( answers ) - Dr Austin Maths
  • Calculating bearings practice grid ( answers ) - Dr Austin Maths
  • Tour de France bearings activity  ( answers ) - Dr Austin Maths
  • Find the Treasure - MathsPad
  • Reasoning with Bearings - Dan Draper
  • Bearings - Median Don Steward
  • GCSE 9-1 Exam Practice Questions (Bearings)  - Maths4Everyone on TES

angles in triangles problem solving tes

  • Exterior angles tool - MathsPad
  • Identifying Exterior Angles  - jamesgreenland on TES
  • Angle sums investigation - MathsPad
  • Angle in polygons  - The Chalk Face
  • Applying angle theorems  - Mathematics Assessment Project
  • Angles practice makes perfect  - Median Don Steward
  • Angles homework  - collated by me! Questions from mathsmalakiss.com
  • Angles in polygons  ( solutions ) - CazoomMaths
  • Angles in Regular Polygons Practice Grid ( Answers ) - Dr Austin Maths
  • Angles in Irregular Polygons ( Answers ) - Dr Austin Maths
  • Exterior angles of convex polygons - Mr Draper Maths
  • Using ratio to find angles in polygons - Stephen Gregory
  • Angle Chasing II - Polygons and Parallel Lines - ItsMattKennedy on TES
  • Angles in Polygons Multi Step Challenges - Maths4Everyone
  • Angles in polygons GCSE revision - Maths4Everyone on TES
  • Regular polygon angle compilation  and Pentagon angle compilation  - Median Don Steward
  • Flowers - Median Don Steward
  • Polygon Pile Up - Jon Orr
  • Stained glass tessellations  - Teachit Maths
  • Surrounding a point   - Median Don Steward

angles in triangles problem solving tes

  • 2D geometry – terms and notation - Boss Maths
  • Properties of special triangles and quadrilaterals - Boss Maths
  • Shape properties logic puzzle  - Teachit Maths
  • Special Quadrilaterals - CIMT
  • Describing and defining quadrilaterals - Mathematics Assessment Project
  • Quadrilateral family tree
  • Quadrilateral flowchart puzzle  - topdrawer.aamt.edu.au
  • Quadrilateral Properties Quiz  - topdrawer.aamt.edu.au
  • Quadrilateral property match-up  - topdrawer.aamt.edu.au
  • Quadrilaterals Always Sometimes Never  - Lisa Bejarano
  • Special quadrilaterals  - Teachit Maths
  • What quadrilateral am I?   - Eddie Woo
  • Polygon Properties  - Transum
  • Shape Shoot  - Flash Maths
  • Draw the shape - Don Steward
  • Shapes and their Descriptions  - Shahira Ibrahim on TES
  • Circle theorems on Geogebra - Michael Borcherds
  • Geogebra apps and worksheet  - themathsteacher.com
  • Circle theorems - including Geogebra applets - Boss Maths
  • Introducing Circle Theorems - Nathan Day
  • Circle theorems - first steps - Maths4Everyone on TES
  • Blank circle theorems - The Chalk Face
  • Circle Theorems Practice Grids - Dr Austin Maths
  • Circle Theorems misconceptions tasks  - mathshko.com
  • Circle theorems worksheet  - MathsPad
  • Circle Theorems Reasoning Task - @rodrigotweets1 (with questions from Don Steward)
  • Great angle chase  - topdrawer.aamt.edu.au
  • Circle theorems codebreaker - alutwyche on TES
  • Circle theorems problems - Maths Malakiss
  • Solving equations with circle theorems  - Nathan Day
  • Solving further equations with circle theorems - Karen Hancock
  • Circle theorems revision exercise  - keyboardmonkey on TES
  • Circle theorems meet 0.5absinC  - Median Don Steward
  • Don Steward's circle theorems presentations  
  • Circle Theorems Exam Style Questions ( answers ) - 1st Class Maths
  • See my post  Ideas for Teaching Circle Theorems for more ideas and resources
  • Area and Perimeter Lessons and Resources - MEI via Oak National Academy
  • Right angled triangle areas - SMILE
  • Counting Squares  - MathsPad
  • Areas of Polygons  - SMILE
  • Polygon Areas - SMILE
  • Practical perimeters - TeachIt Maths
  • Perimeter Overlearning - Dan Draper
  • Rectangle areas - Don Steward
  • Area of a triangle  - Kyle Gillies vis Starting Point Maths
  • Triangle areas, various ways - Median Don Steward
  • Area tasks - @giftedHKO
  • Area of a parallelogram  - Kyle Gillies vis Starting Point Maths
  • Area of Parallelograms and Trapeziums Match-Up  ( answers ) - Dr Austin Maths
  • Area of a parallelogram and algebra - Mr Thompson on TES
  • Area of parallelogram problem solving - Mr Thompson on TES
  • Areas of trapeziums interwoven - Karen Hancock
  • Area of trapezium - finding a missing length - Mr Thompson on TES
  • Blog post: Thinking About Areas of Parallelograms - Paul Rowlandson
  • Perimeter questions  ( answers ) - @taylorda01
  • Area of 2D shapes questions ( answers ) - @taylorda01
  • Area of 2D shapes questions II  ( answers ) - @taylorda01
  • Area of 2D shapes Crack the Code  ( answers ) - Dr Austin Maths
  • Perimeter and Area Tasks - John Mason
  • Area and Perimeter Match - Don Steward
  • Area practice - TeachIt Maths
  • Examples and exercises from BossMaths: 
  • Area of a rectangle
  • Area of a triangle
  • Area of a parallelogram
  • Area of a trapezium
  • Perimeter of polygons
  • Perimeter and area of composite shapes made up of polygons .
  • Area of 2D shapes review - TeachIt Maths
  • Areas of Flags  - Owen134866 on TES
  • L shapes (slides and worksheet) - MathsPad
  • Area of L shaped diagrams - TeachIt Maths
  • L shapes with fractions - Nathan Day
  • Rectilinear areas - TeachIt Maths
  • L shaped perimeters  - Don Steward
  • Border areas - TeachIt Maths
  • Compound Shapes Matching Cards  - Nuffield Foundation
  • Compound rectangular shapes  - Median Don Steward 
  • Equable shapes - Median Don Steward
  • Area challenge - TeachIt Maths
  • Area mazes - Don Steward
  • Area puzzles - via STEM Centre

angles in triangles problem solving tes

  • Rhymes for the area and circumference of a circle
  • Circles - introducing circumference - Median Don Steward 
  • Circumference of a circle lesson - cparkinson3 on TES
  • Circles questions - Teacher Resources Online
  • Blog post: Thinking About Calculating Areas of Circles  - Paul Rowlandson
  • Pi and the circumference of circles - Maths4Everyone on TES
  • Non calculator area of a circle - Mr Thompson on TES
  • Areas of a circle lesson - cparkinson3 on TES
  • Area and circumference of a circle lesson - cparkinson3 on TES
  • Area of a circle - Maths4Everyone on TES
  • Circumference and Area - TeachIt Maths
  • Area of a circle
  • Circumference of a circle
  • Perimeter and area of composite shapes made up of polygons and sectors of circles
  • Area of a circle questions  ( answers ) - @taylorda01
  • Mensuration - Teacher Resources Online
  • Penny Farthing - The Chalk Face
  • Geogebra showing revolution is circumference   - Tim Brzezinski
  • Mixed circles questions with answers - collated by me from various sources
  • Areas of Flags with Circles  - Owen134866 on TES
  • Eight Circles - illustrativemathematics.org
  • Area and perimeter of a sector lesson - cparkinson3 on TES
  • Arc length and sector area questions  ( answers ) - @taylorda01
  • Arc length and sector area - Median Don Steward
  • Arcs and sectors worksheet - source unknown (sorry!)
  • Area and perimeter of compound shapes  - mrwhy1089 on TES 
  • Angle and radius of a sector lesson - cparkinson3 on TES
  • Sectors Fill in the Gaps - by me!
  • Circle sector problems - Access Maths
  • Paper clip - Illustrative Mathematics
  • Concentric circular rings - Median Don Steward
  • Segment areas - mathshelper.co.uk
  • Sectors GCSE Exam Practice  - Maths4Everyone on TES
  • Areas of sectors GCSE revision - Maths4Everyone on TES
  • Area of shaded regions GCSE revision - Maths4Everyone on TES
  • Sectors, arcs and perimeters GCSE revision - Maths4Everyone on TES
  • Basic cuboid volume - SMILE
  • Interactive cuboid  - The Chalk Face
  • Volume of a Cuboid Practice Grid  ( Answers ) - Dr Austin Maths
  • Volume of a Cuboid Challenge Activity ( Answers ) - Dr Austin Maths
  • Volume of Cubes and Cuboids Match-Up ( Answers ) - Dr Austin Maths
  • Cuboid surface area - Median Don Steward
  • Surface area of cuboids - problem solving - Mr Thompson on TES
  • Cuboid volume and surface area - Median Don Steward
  • Volume of prism lessons  - cparkinson3 on TES
  • Volume of a Prism Practice Grid  ( Answers ) - Dr Austin Maths
  • Volume and Surface Area of Prisms Crack the Code ( Answers ) - Dr Austin Maths
  • Surface area of prism lesson  - cparkinson3 on TES
  • Triangular prisms  - Median Don Steward
  • Cornflakes problem  - The Chalk Face
  • Fuel tank  - The Chalk Face
  • Cylinder volume questions - Median Don Steward
  • Volume of a Cylinder Practice Grid ( Answers ) - Dr Austin Maths
  • Problem solving with volumes of prisms  - Teachit Maths
  • Harder surface area questions  - Median Don Steward
  • Sphere volume - Median Don Steward
  • Golden balls  - The Chalk Face
  • Volume of a cone questions  ( answers ) - @taylorda01
  • Cone Volume - Median Don Steward
  • Investigating the surface area of a cone  - Maths Sandpit
  • Surface area of a cone questions  ( answers ) - @taylorda01
  • Cone Surface Area - Median Don Steward
  • Volume and area GCSE questions  - collated by me
  • Volume GCSE questions  - UKMaths on TES
  • Volume and Surface Area Revision Practice Grid  ( Answers ) - Dr Austin Maths
  • Backwards Volumes - Starting Point Maths
  • Volume of solids questions  ( answers ) - @taylorda01
  • Surface area of solids questions  ( answers ) - @taylorda01
  • Finding the volume of compound objects  - Shell Centre
  • Functional Volume Questions - Access Maths
  • Volume and Surface Area Revision Carousel - alisongilroy on TES
  • Spheres, Cones & Cylinders GCSE revision - Maths4Everyone on TES
  • Spheres, Cones & Cylinders, Working Backwards - Maths4Everyone on TES
  • Matching graphs and scenarios cards  ( slides ) - Nuffield Foundation
  • Conversion graphs - Nuffield Foundation
  • Conversion Graphs Practice Grid ( Answers ) - DrAustinMaths
  • Plumber's call-out - Nuffield Foundation
  • Graphical interpretation - Median Don Steward
  • Linear rules, with contexts  - Median Don Steward 
  • Bath card sort - adapted from mattsteel87 on TES
  • Archimedes' Bath - ColmanWeb
  • Desmos Waterline  
  • The Language of Functions and Graphs  - Shell Centre
  • Kinematics assorted problems - Boss Maths

angles in triangles problem solving tes

  • 3D geometry - terms and notation - Boss Maths
  • Properties of 3D Shapes Practice Strips ( Answers ) - Dr Austin Maths
  • Investigating 3D Shapes Worksheet ( Answers ) - Dr Austin Maths
  • Faces and elevations activity - Median Don Steward
  • Plans and elevations of 3D shapes - Boss Maths
  • Plan and Elevation - mathshko
  • Drawing in 2D and 3D  (slides/activities) - Dan Walker on TES
  • Drawing skills: isometric projection - source unknown, sorry
  • Isometric drawing activities - Median Don Steward
  • Isometric Drawing - Owen134866 on TES
  • Net Tasks - Median Don Steward
  • The Box Problem - 1001 Math Problems
  • Nets of a Cube - numeracycd.com
  • 3D geometry: faces, edges and vertices - Don Steward
  • Plotting Coordinates - Ashton Coward
  • Coordinate Message  - SMILE
  • Coordinates lesson - Boss Maths
  • Coordinates - Dan Walker on TES
  • Coordinate Practice - Median Don Steward
  • Quadrilateral Coordinates - Median Don Steward
  • Squares and Coordinates - Median Don Steward
  • Graphs - examples and exercises - CIMT
  • Coordinates and Shapes - TeachIt Maths
  • Coordinate CBSE Questions - Median Don Steward
  • Coordinates Problem Solving - White Rose Maths Hub
  • Rectangles on a Grid - Don Steward
  • Working with sketches - Boss Maths
  • Solve geometrical problems on coordinate axes - Boss Maths
  • 3D Coordinates examples and exercises - CIMT
  • Plotting 3D Coordinates ppt - Teachitmaths
  • Lines of symmetry - Median Don Steward
  • Reflection symmetry - Boss Maths
  • Rotation symmetry - Boss Maths
  • Rotational designs  (& interactive activity ) - MathsPad
  • Symmetry gifs
  • Making symmetrical shapes - Teachit Maths
  • Add one square - Median Don Steward
  • Symmetry challenge questions ( answers ) - Dr Austin Maths
  • Transformations tasks - Dr Austin Maths
  • Transformations activities - Teachit Maths
  • Transformations lessons - cparkinson3 on TES
  • Reflecting diagonally - Median Don Steward
  • Reflection lesson - Boss Maths
  • Grid reflections - Median Don Steward
  • Wordly reflections - Median Don Steward
  • Rotations booklet - The Chalk Face
  • Rotation questions - Median Don Steward
  • Rotations Jigsaw - Tristan Jones on TES
  • Rotation lesson - Boss Maths
  • Interactive rotation tool and worksheet - MathsPad
  • Still pools (reflection) - Median Don Steward
  • Vector messages - SMILE
  • Translations with vectors - alicescreswick on TES
  • Vectors Snakes and Ladders  - MrBartonMaths on TES
  • Interactive translation tool and worksheet - MathsPad
  • Transformers Activity - danwalker on TES
  • Transforming Words - Tristan Jones on TES
  • Transformations Challenge - G Westwater on TES
  • Transformation Station game - flashymaths.co.uk
  • Transformation workbook  - Math4Everyone on TES
  • Congruence, similarity and transformations - Boss Maths
  • Transformation activities - @DJUdall
  • Combinations of transformations - Boss Maths
  • Transforming Shapes Codebreaker - Andy Lutwyche
  • Invariance activity sheet (new GCSE) - Peter Mattock on TES
  • Invariant Points - Miss Konstantine
  • Integer enlargement practice
  • Fractional enlargement practice
  • Negative enlargement practice  
  • Enlargement lesson - cparkinson3 on TES
  • Positive integer enlargement jigsaw puzzle - Mr Thompson on TES
  • Fractional enlargement jigsaw puzzle - Mr Thompson on TES
  • Describe the enlargement  - Median Don Steward
  • Enlargement lesson - Boss Maths
  • Drawing it a bit bigger  - Median Don Steward
  • Also see resources listed under 'Similarity' above
  • Investigating the sides of right-angled triangles  - Teachit Maths 
  • Pythagoras' Theorem - complete topic booklet  - Teachit Maths
  • Pythagoras fading examples - evm86 on TES
  • Pythagoras practice questions  (includes converse) - Frank Tapson (trol)
  • Pythagorean Theorem questions  ( answers ) - @taylorda01
  • Pythagoras Puzzle - Dan Walker on TES
  • Pythagoras tasks - @giftedHKO
  • Pythagoras problems  - The Chalk Face
  • Pythagorean Stacks - equationfreak.blogspot.com
  • Pythagoras Pile-Up  - @mrandersonmaths 
  • Pythagoras' Theorem Student Sheets  (&  notes ) - Nuffield Foundation
  • KS3 SAT Pythagoras Questions - Median Don Steward
  • Pythagoras interactive activities - MathsPad
  • Applied Pythagoras - Dan Draper
  • Distance between two points - Corbett Maths
  • Pythagorean Triples Tasks  - pythagoreantriples.blogspot.co.uk
  • Triple triangle lengths - Median Don Steward
  • Pythagoras and surd form - Median Don Steward
  • Pythagoraean Theorem in 3D questions  ( answers ) - @taylorda01 
  • 3D Pythagoras - Median Don Steward
  • 3D Pythagoras Questions - collated from textbooks
  • Pythagoras Topic Review Sheet - Maths4Everyone
  • Circles and Pythagoras  - National 5 Maths
  • GCSE 9-1 Exam Question Practice (Pythagoras) - Maths4Everyone
  • Pythagoras exercises - Transum
  • Star Wars Pythagoras Questions (challenging) - Nathan Day
  • Pythagoras Problem Solving Set - see final page for sources
  • Pythagoras  Problem Solving Questions  - steele1989
  • See my blog post about interesting Pythagoras problems from the new GCSE
  • More ideas in my blog post about teaching Pythagoras' Theorem
Right angled
  • Trigonometry introduction booklet  - Teachit Maths
  • Introduction to Trigonometry  - lesson plans and activities from Project Maths
  • Sine, Cosine and Tangent Ratios Fill in the Blanks ( Answers ) - Dr Austin Maths
  • Finding Angles Using Trigonometry Fill In The Blanks ( Answers ) - Dr Austin Maths
  • Finding Angles Using Trigonometry Name the Film ( Answers ) - Dr Austin Maths
  • Finding Lengths Using Trigonometry Fill In The Blanks ( Answers )  - Dr Austin Maths
  • Finding Lengths Using Trigonometry Name the Film ( Answers ) - Dr Austin Maths
  • Trigonometric ratios questions  - @taylorda01
  • Trigonometry Pile Up 1  - Great Maths Teaching Ideas
  • Trigonometry booklet  & answers - MissBrookesMaths
  • Let's draw some diagrams  - Teachit Maths 
  • Trigonometry in Isosceles Triangles  - Mr Thompson on TES
  • Finding lengths - faded scaffolding - evm86 on TES 
  • SOHCAHTOA slides - Dan Walker
  • Right-angled trigonometry - mathshelper.co.uk
  • Trigonometry - Median Don Steward
  • Trigonometry in isosceles triangles   - Mr Thompson on TES
  • Trigonometry worksheets  - Frank Tapson
  • Trigonometry tests with answers - Frank Tapson
  • Multi-Step Trigonometry Problems Practice Strips ( Answers ) - draustinmaths
  • Pirate Trigonometry  - Matthew Kennedy on TES
  • MEP trigonometry exercises - CIMT
  • Trigonometry and Pythagoras Post It Challenge  - steele1989 on TES
  • SOHCAHTOA GCSE revision - Maths4Everyone on TES
Non right angled
  • When do we need 1/2absinc? card match  - Teachit Maths
  • Trigonometry (area) fill in the gaps - Andy Lutwyche on TES
  • Areas of triangles - Cazoom Maths
  • Area of a triangle GCSE revision  - Maths4Everyone on TES
  • Area of triangle using trig exam style questions ( answers ) - 1st Class Maths
  • Sine Rule Introduction  - SMILE
  • Sine Rule lesson  - Boss Maths
  • Sine Rule - Median Don Steward
  • Finding Lengths Using Sine Rule Fill In The Blanks ( Answers ) - Dr Austin Maths
  • Finding Lengths Using Sine Rule Practice Grid ( Answers ) - Dr Austin Maths
  • Finding Angles Using Sine Rule Fill In The Blanks ( Answers ) - Dr Austin Maths
  • Finding Angles Using Sine Rule Practice Grid ( Answers ) - Dr Austin Maths
  • Sine Rule Target Table  - SimplyEffectiveEducation on TES
  • Sine Rule exam style questions ( answers ) - 1st Class Maths
  • Cosine Rule lesson - Boss Maths
  • Cosine Rule - Median Don Steward
  • Cosine Rule exam style questions ( answers ) - 1st Class Maths
  • Sine Rule Codebreaker - MathsPad (with subscription)
  • Sine Rule (ambiguous case) - langy74 on TES
  • Finding Lengths Using Cosine Rule Fill In The Blanks ( Answers ) - Dr Austin Maths
  • Finding Lengths Using Cosine Rule Practice Grid ( Answers ) - Dr Austin Maths
  • Finding Angles Using Cosine Rule Fill In The Blanks ( Answers ) - Dr Austin Maths
  • Cosine Rule Topic Review Sheet  - Maths4Everyone on TES
  • Trigonometry collect a joke - Dan Walker on TES
  • Trigonometry and bearings  
  • Simple Bearings & Trigonometry - Scaffolded worksheet - Mr Thompson on TES
  • MEP trigonometry exercises  - CIMT
  • Trig Pile Up - MathematicQuinn on TES
  • Sine and Cosine rule trigonometry pile up - MrGrayMaths on TES
  • Sine and Cosine Rule GCSE questions - Maths4Everyone on TES
  • Non-right angled triangles full coverage GCSE questions  - compiled by Dr Frost
  • See my blog post New GCSE: Trigonometry Questions for some challenging questions
Graphs and exact values
  • Trigonometry worksheets  - Cleave Books (includes trig graph questions)
  • Symmetry in Trigonometric Graphs  - pas1001 on TES
  • Graphs of trigonometric functions - Boss Maths
  • Plotting Trigonometric Graphs Practice Grid  ( Answers ) - Dr Austin
  • Trigonometric Graphs Sort It Out ( Answers ) - Dr Austin
  • Exact trig values worksheet and codebreaker - MissBrookesMaths
  • GCSE Trigonometry - Exact Values - Mr Thompson on TES
  • Finding Exact Trig Values - Discovery Learning   - emcnicholl on TES
  • Exact Trigonometric Values - Median Don Steward
  • Exact Trigonometric Values Crack the Code ( Answers ) - draustinmaths
  • Trigonometry without a calculator - joashknight on TES
  • Exact Trigonometric Values textbook exercise  - CorbettMaths 
  • If you subscribe to MathsPad , they have great exact trig values resources.
3 Dimensional
  • 3D Trigonometry lesson - cparkinson3 on TES
  • 3D Trigonometry  - Median Don Steward
  • 3D Trigonometry - mathshelper.co.uk
  • Get 21 - 3D Trigonometry  - HSpedding on TES 
  • Trigonometry worksheets  - Cleave Books (includes 3D trig)
  • GCSE 9-1 Exam Question Practice (3D Pythagoras + Trigonometry) - Maths4Everyone
  • 3D Trig and Pythagoras Exam Style Questions ( Answers ) - 1st Class Maths
  • See my blog post on trigonometry for more ideas
  • See my post New GCSE: trigonometry questions for some challenging questions
  • Upper and lower bounds lesson - Boss Maths
  • Error intervals lesson - Boss Maths
  • Bounds lesson - Tick Tock Maths
  • Upper and lower bounds questions  - The Chalk Face
  • Painting bounds  - The Chalk Face
  • Highest and lowest bounds - Median Don Steward
  • Errors Student Sheet  (&  notes ) - Nuffield Foundation
  • Bounds and Error Intervals  - @dooranran on TES
  • Lesson plan: bounds and speed - Colin Foster
  • Bounds GCSE revision - Maths4Everyone on TES
  • Bounds full coverage GCSE questions - compiled by Dr Frost
  • Error intervals exercises  - @alcmaths
  • Upper and Lower Bounds Fill in the Blanks ( Answers ) - Dr Austin
  • More Upper and Lower Bounds Fill in the Blanks ( Answers ) - Dr Austin
  • Upper and Lower Bounds Decode the Joke ( Answers ) - Dr Austin
  • Upper and Lower Bounds Odd One Out ( Answers ) - Dr Austin
  • Upper and Lower Bounds Revision Practice Grid ( Answers ) - Dr Austin
  • Upper and lower bounds  - mathshelper.co.uk
  • Error Intervals exam style questions ( answers ) - 1st Class Maths
  • Upper and Lower Bounds exam style questions ( Answers ) - 1st Class Maths
  • See  my blog post about GCSE bounds  for teaching advice
  • Vectors resources from Dr Austin Maths
  • Adding and subtracting of column vectors - Boss Maths
  • Multiplying column vectors by a scalar  - Boss Maths
  • Vectors and quadrilaterals - MathsPad
  • Vectors problem - The Chalk Face
  • Vectors lessons - Dan Walker on TES
  • Don Steward's vectors resources
  • Vector Hunt -  Peter Mattock on TES
  • Defining Vectors - Peter Mattock on TES
  • Vector proof - Peter Mattock on TES
  • Proofs using vectors - Boss Maths
  • Vectors exam questions - bland.in
  • Vectors exam questions - teachingmaths.net
  • KS3-4 Bridging the gap Pocket 9 - Vectors - AQA All About Maths 
  • Harder GCSE vector questions  - Don Steward
  • Vectors - Harder GCSE Practice questions for Edexcel 1MA1 syllabus  - THarman on TES
  • Vectors workbook - Maths4Everyone on TES
  • Vectors exam question practice - Maths4Everyone on TES
  • Vectors full coverage GCSE questions  - compiled by Dr Frost
  • Simple geometric proofs - Boss Maths
  • Translating geometric descriptions activity  - topdrawer.aamt.edu.au
  • Geometry toolkit  - topdrawer.aamt.edu.au
  • Adding auxiliary lines  - topdrawer.aamt.edu.au
  • Mathematical Proof (includes geometric proof) & answers - CIMT
  • Prove it! (similar triangles) - MathsPad
  • Working with sketches  - Boss Maths
  • Formula sheet  and sheet showing  formula to learn for GCSE
  • Geometry revision from Dr Austin
  • Geometry rich venn tasks - mathsvenns.com
  • Circle theorem revision cards  -  teachitmaths.co.uk
  • Trigonometry revision mat - MathedUp
  • Shape and space textbook extract - Oxford University Press
  • Exam questions  (various formats, including answers) compiled by Dr Gareth Evans (WJEC)

angles in triangles problem solving tes

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Angles, Polygons and Geometrical Proof Short Problems

This is part of our collection of Short Problems . You may also be interested in our longer problems on Angles, Polygons and Geometrical Proof  Age 11-14 and Age 14-16 . Printable worksheets containing selections of these problems are available here:

angles in triangles problem solving tes

Half Past Two

Weekly Problem 21 - 2009 What is the angle between the two hands of a clock at 2.30?

Fraction of a Square

What fraction of this square is shaded?

Right-angled Request

Weekly Problem 26 - 2006 How many right angled triangles are formed by the points in this diagram?

Equilateral Pair

Weekly Problem 39 - 2016 In the diagram, VWX and XYZ are congruent equilateral triangles. What is the size of angle VWY?

Angular Reflection

Weekly Problem 28 - 2013 Two lines meet at a point. Another line through this point is reflected in both of these lines. What is the angle between the image lines?

Weekly Problem 39 - 2010 If you know three lengths and an angle in this diagram, can you find another angle by calculation?

Weekly Problem 40 - 2015 In the diagram, $PT = QT = TS$ and $QS = SR$. What is the value of $x$?

Shared Vertex

Weekly Problem 38 - 2017 In the diagram, what is the value of $x$?

Outside the Nonagon

Weekly Problem 44 - 2010 Extend two of the sides of a nonagon to form an angle. How large is this acute angle?

Parallel Base

Weekly Problem 46 - 2015 The diagram shows two parallel lines and two angles. What is the value of x?

Two Exterior Triangles

Weekly Problem 35 - 2009 Two equilateral triangles have been drawn on two adjacent sides of a square. What is the angle between the triangles?

Weekly Problem 16 - 2007 Can you figure out how far the robot has travelled by the time it is first facing due East?

Stellar Angles

Weekly Problem 30 - 2013 What is the angle $x$ in the star shape shown?

Square Bisection

Weekly Problem 8 - 2008 In how many ways can a square be cut in half using a single straight line cut?

Weekly Problem 8 - 2016 Can you work out the size of the angles in a quadrilateral?

Square Within a Square Within...

Polygon cradle.

Weekly Problem 18 - 2007 A regular pentagon together with three sides of a regular hexagon form a cradle. What is the size of one of the angles?

Isosceles Meld

Weekly Problem 9 - 2012 What is the angle QPT in this diagram?

Diagonal Division

Weekly Problem 45 - 2008 The diagram shows a regular pentagon with two of its diagonals. If all the diagonals are drawn in, into how many areas will the pentagon be divided?

Stacking Shapes

Weekly Problem 28 - 2017 The diagram on the right shows an equilateral triangle, a square and a regular pentagon. What is the sum of the interior angles of the resulting polygon?

As Long as Possible

Weekly Problem 40 - 2013 Given three sides of a quadrilateral, what is the longest that the fourth side can be?

Homely Angles

Weekly Problem 18 - 2011 Draw an equilateral triangle onto one side of a square. Can you work out one particular angle?

Angle of Overlap

Weekly Problem 26 - 2007 The diagram shows two equilateral triangles. What is the value of x?

Overlapping Beer Mats

Can you find the area of the overlap when these two beer mats are placed on top of each other?

Long Shadows

Weekly Problem 10 - 2012 If you know how long Meg's shadow is, can you work out how long the shadow is when she stands on her brother's shoulders?

Angle Please

Weekly Problem 19 - 2017 In the figure, what is the value of x?

Bishop's Paradise

Weekly Problem 37 - 2013 Which of the statements about diagonals of polygons is false?

Triangle in a Corner

The diagram shows an equilateral triangle touching two straight lines. What is the sum of the four marked angles?

Distinct Diagonals

Weekly Problem 21 - 2010 How many diagonals can you draw on this square...

Regular Vertex

A square, regular pentagon and equilateral triangle share a vertex. What is the size of the other angle?

Central Distance

Weekly Problem 1 - 2006 The diagram shows two circles enclosed in a rectangle. What is the distance between the centres of the circles?

Isometric Rhombuses

Weekly Problem 31 - 2016 The diagram shows a grid of $16$ identical equilateral triangles. How many rhombuses are there made up of two adjacent small triangles?

Extended Parallelogram

Weekly Problem 11 - 2014 The diagram shows a parallelogram and an isosceles triangle. What is the size of angle TQR?

Right Angled Octagon

Weekly Problem 38 - 2008 A quadrilateral can have four right angles. What is the largest number of right angles an octagon can have?

Hexapentagon

Weekly Problem 53 - 2007 The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of x?

Two Triangles

Prove that the angle marked $a$ is half the size of the angle marked $b$.

Square in a Triangle

Weekly Problem 33 - 2006 A square is inscribed in an isoscles right angled triangle of area $x$. What is the area of the square?

Rhombus Diagonal

Weekly Problem 19 - 2014 The diagram shows a rhombus and an isosceles triangle. Can you work out the size of the angle JFI?

Handy Angles

Weekly Problem 39 - 2008 How big is the angle between the hour hand and the minute hand of a clock at twenty to five?

Perimeter Puzzle

If four copies of this triangle are joined together to form a parallelogram, what is the largest possible perimeter of the parallelogram?

Perimeter in a Hexagon

Can you find the perimeter of this triangle inscribed in a hexagon?

Nonagon Angle

Weekly Problem 53 - 2012 ABCDEFGHI is a regular nine-sided polygon (called a 'nonagon' or 'enneagon'). What is the size of the angle FAE ?

Triangle in the Corner

A triangle is shaded within a regular hexagon. Can you find its area?

Adding Angles

Weekly Problem 47 - 2016 What is the sum of the six marked angles?

Diagonal Side

Can you work out the area of a square drawn on a diagonal?

Tricky Tessellations

Can you work out the fraction of the tiles that are painted black in this pattern?

Descending Angles

Given four of the angles in two triangles, can you find the smallest angle overall?

Outside the Boxes

Weekly Problem 13 - 2008 The diagram shows three squares drawn on the sides of a triangle. What is the sum of the three marked angles?

Integral Polygons

Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have?

Funky Hexagon

Can you find the value of x in this diagram?

Radioactive Triangle

Weekly Problem 41 - 2014 Three straight lines divide an equilateral triangle into seven regions. What is the side length of the original triangle?

Triangle Split

Weekly Problem 50 - 2008 The lengths SP, SQ and SR are equal and the angle SRQ is x degrees. What is the size of angle PQR?

Weekly Problem 7 - 2013 Three of the angles in this diagram all have size $x$. What is the value of $x$?

U in a Pentagon

Weekly Problem 18 - 2008 The diagram shows a regular pentagon. Can you work out the size of the marked angle?

Dodecagon Angles

Weekly Problem 50 - 2012 The diagram shows a regular dodecagon. What is the size of the marked angle?

Rectangle Dissection

Weekly Problem 2 - 2009 The 16 by 9 rectangle is cut as shown. Rearrange the pieces to form a square. What is the perimeter of the square?

Inscribed Hexagon

Weekly Problem 1 - 2014 The diagram shows a regular hexagon inside a rectangle. What is the sum of the four marked angles?

Six Minutes Past Eight

Weekly Problem 45 - 2007 What is the obtuse angle between the hands of a clock at 6 minutes past 8 o'clock?

Heptagon Has

Weekly Problem 15 - 2012 How many of the five properties can a heptagon have?

Overbearing

A village has a pub, church and school. What is the bearing of the school from the church?

Equal Lengths

Weekly Problem 29 - 2013 An equilateral triangle is drawn inside a rhombus, both with equal side lengths. What is one of the angles of the rhombus?

Pentagon Ring

Weekly Problem 47 - 2011 Place equal, regular pentagons together to form a ring. How many pentagons will be needed?

Inner Rectangle

If the shape on the inside is a rectangle, what can you say about the shape on the outside?

Hexagon Cut Out

Weekly Problem 52 - 2012 An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?

Add All the Angles

Find the sum of all of the angles denoted by letters in this diagram

Prove that the angle bisectors of a triangle can never meet at right angles.

Clock Face Angles

The time is 20:14. What is the smaller angle between the hour hand and the minute hand on an accurate analogue clock?

Trapezium Arch

Weekly Problem 27 - 2007 Ten stones form an arch. What is the size of the smallest angles of the trapezoidal stones?

Clock Angle

What is the angle between the the hands of a clock at 8:24?

Two Isosceles

Weekly Problem 37 - 2017 A quadrilateral is divided into two isosceles triangles. Can you work out the perimeter of the quadrilateral?

Parallelogram in the Middle

Weekly Problem 27 - 2013 The diagram shows a parallelogram inside a triangle. What is the value of $x$?

Cat on a Wall

How high is the wall that this cat is lying on?

Weekly Problem 37 - 2014 Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?

Weekly Problem 13 - 2012 The diagram shows contains some equal lengths. Can you work out one of the angles?

Inscribed Semicircle

Weekly Problem 43 - 2017 The diagram shows a semicircle inscribed in a right angled triangle. What is the radius of the semicircle?

Tunnel Vision

How wide is this tunnel?

Octagonal Ratio

Can you find the ratio of the area shaded in this regular octagon to the unshaded area?

Diagonal Touch

Weekly Problem 21 - 2012 Two rectangles are drawn in a rectangle. What fraction of the rectangle is shaded?

Circles on a Triangle

Find the missing length on this diagram.

How much of the field can the animals graze?

Shaded Square

Weekly Problem 41 - 2016 The diagram shows a square, with lines drawn from its centre. What is the shaded area?

Two Right Angles

Weekly Problem 4 - 2008 In the figure given in the problem, calculate the length of an edge.

Weekly Problem 27 - 2014 Four congruent isosceles trapezia are placed in a square. What fraction of the square is shaded?

3-4-5 Circle

Can you find the radius of the circle inscribed inside a '3-4-5 triangle'?

Incentre Angle

Weekly Problem 1 - 2011 Use facts about the angle bisectors of this triangle to work out another internal angle.

Slide Height

How high is the top of the slide?

Internal - External

Weekly Problem 12 - 2016 The diagram shows a square PQRS and two equilateral triangles RSU and PST. PQ has length 1. What is the length of TU?

Square and Triangle

Can you find the area of the yellow square?

Doubly Isosceles

Find the missing distance in this diagram with two isosceles triangles

Block Challenge

Can you work out the shaded area in this shape?

Angle to Chord

Weekly Problem 23 - 2008 A triangle has been drawn inside this circle. Can you find the length of the chord it forms?

Isosceles Reduction

Weekly Problem 29 - 2010 An isosceles triangle is drawn inside another triangle. Can you work out the length of its base?

Overlapping Semicircles

Two semicircles overlap, can you find this length?

Height and Sides

Can you find the area of the triangle from its height and two sides?

Circular Inscription

In the diagram, the radius of the circle is equal to the length AB. Can you find the size of angle ACB?

Two Equilateral Triangles

Prove that these two lengths are equal.

Griddy Region

Weekly Problem 34 - 2008 What is the area of the region common to this triangle and square?

Garden Fence

Weekly Problem 44 - 2009 A garden has the shape of a right-angled triangle. A fence goes from the corner with the right-angle to a point on the opposite side. How long is the fence?

Overlapping Annuli

Just from the diagram, can you work out the radius of the smaller circles?

Altitude Inequalities

Weekly Problem 8 - 2010 Are you able to find triangles such that these five statements are true?

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High school geometry

Course: high school geometry   >   unit 3.

  • Corresponding angles in congruent triangles

Find angles in congruent triangles

  • Isosceles & equilateral triangles problems
  • Find angles in isosceles triangles
  • Finding angles in isosceles triangles
  • Finding angles in isosceles triangles (example 2)
  • Your answer should be
  • an integer, like 6 ‍  
  • an exact decimal, like 0.75 ‍  
  • a simplified proper fraction, like 3 / 5 ‍  
  • a simplified improper fraction, like 7 / 4 ‍  
  • a mixed number, like 1   3 / 4 ‍  

One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

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This topic is relevant for:

GCSE Maths

Angles In Polygons

Here we will learn about angles in polygons including how to calculate angles in polygons using a variety of methods and an overview of interior and exterior angles

There are also angles in polygons worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are angles in polygons?

Angles in polygons relate to the interior and exterior angles of regular and irregular polygons.

Interior angles are the angles within a polygon made by two sides. We can calculate the sum of the interior angles of a polygon by subtracting 2 from the number of sides and then multiplying by 180º .

Step-by-step guide: Interior angles of a polygon

Exterior angles are the angles between a polygon and the extended line from the next side. The sum of the exterior angles of a polygon is always equal to 360º .

Step-by-step guide: Exterior angles of a polygon

A polygon is a two dimensional shape with at least three sides, where the sides are all straight lines. ‘Poly’ comes from the greek for ‘many’ whilst ‘gon’ means ‘angles’. You will be familiar with many types of polygons such as triangle, rectangle and pentagon.

Regular polygons have all angles are that are equal in size and all sides that are equal in length.

Irregular polygons have angles that are not equal in size and sides that are not equal in length.

What are angles in polygons

What are angles in polygons

Sum of the interior angles of a polygon

The ‘ sum of interior angles ’ of a polygon means finding the total of all the angles in a polygon. This is the key step in helping us solve many problems involving angles in polygons.

We know that the sum of all the angles in a triangle is equal to 180º , but what about angles in a quadrilateral? A regular pentagon? Or even an irregular octagon?

Sum of the angles in a triangle :

We know that the three angles in any triangle add up to 180º .

Mathematically we would say:

“The sum of interior angles for a triangle is 180 degrees” .

Step-by-step guide: Angles in a triangle

Sum of the angles in a quadrilateral :

A quadrilateral is a four sided shape. We can ‘split’ a quadrilateral into two triangles by drawing a line from one corner to an opposite one.

If the sum of interior angles one triangle is 180º , then the sum of the interior angles of two triangles is 180º × 2 = 360º .

So the sum of the interior angles of quadrilateral is 360º .

Step-by-step guide: Angles in a quadrilateral

Using our knowledge of triangles we can find the sum of the interior angles of any polygon by splitting it into triangles.

Step-by-step guide: Angles in a pentagon

Step-by-step guide: Angles in a hexagon

Key relationships

Interior and exterior angles add up to 180 0 .

Interior and exterior angles lie on a straight line. This means that when added together they will equal 180º .

This is useful when working on more complex questions.

The number of triangles created inside a shape is always 2 lower than the number of sides.

Here is a list of all the polygons we will work with in this lesson:

Notice how the number of triangles created is always 2 lower than the number of sides .

The sum of the interior angles of a polygon depends on how many sides it has, not what it looks like.

Despite these two decagons ( 10 sided shapes) looking very different, the sum of their interior angles is the same.

How to find the sum of the interior angles of a polygon

In order to find the sum of interior angles for any polygon you should:

Identify how many sides the polygon has.

Identify if the polygon is regular or irregular.

  • If possible work out how many triangles could be created within the polygon by drawing lines from one vertex to all the other vertices.
  • Multiply the number of triangles by 180 to calculate the sum of the interior angles.

State your findings e.g. sides, regular/irregular, the sum of interior angles.

How to find the sum of the interior angles for a polygon

How to find the sum of the interior angles for a polygon

Angles in polygons worksheet

Get your free angles in polygons worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Angles in polygons examples

Example 1: irregular quadrilateral.

Find the sum of interior angles for this polygon.

This polygon has four sides.

2   Identify if the polygon is regular or irregular.

This polygon is irregular as all the sides are not of equal length.

3 If possible work out how many triangles could be created within the polygon by drawing lines from one corner to all the other vertices.

The polygon can be broken up into two triangles.

4 Multiply the number of triangles by 180 to get the sum of the interior angles.

5 State your findings e.g. sides, regular/irregular, the sum of interior angles.

The polygon is a irregular quadrilateral (specifically called a parallelogram as both opposite sides are parallel) with a sum of interior angles of 360º .

Example 2: regular decagon

This polygon has 10 sides.

This polygon is regular as all the sides are of equal length and the angles are of equal size.

If possible work out how many triangles could be created within the polygon by drawing lines from one corner to all the other vertices.

The polygon can be broken up into eight triangles.

Multiply the number of triangles by 180 o to get the sum of the interior angles.

The polygon is a regular 10 sided polygon (decagon) with a sum of interior angles of 1440º .

Example 3: irregular pentagon with concave angle

This polygon has five sides.

This polygon is irregular as all the sides are not of equal length and the angles are not of equal size

The polygon can be broken up into three triangles.

The polygon is a irregular five sided polygon (pentagon) with a sum of interior angles of 540º .

Example 4: complex polygon

This polygon has 10 sides

This polygon is irregular as all the angles are not of equal size.

Notice how we had to do this differently to previous examples. But we have not created any new angles because our lines do not cross.

The polygon is a irregular 10 sided polygon with a sum of interior angles of 1440º .

STOP AND THINK: Notice how this is the same sum of interior angles for example 2. This is because the polygon has the same number of sides .

Example 5: finding an exterior angle

  What is the size of one of the exterior angles of an equilateral triangle?

The sum of interior angle in a triangle is 180º .

An equilateral triangle has 3 angles of equal size.

Therefore each interior angle is 60º because 180 ÷ 3 = 60 .

Therefore one exterior angle is equal to 180 – 60 because the interior angle and exterior angle of a polygon lie on a straight line.

Therefore one exterior angle is 120º .

Example 6: finding the sum of interior angles of larger polygons

What is the sum of the interior angles for a 30 sided polygon?

To work out the number of triangles a polygon can be split into we subtract 2 .

A 30 sided polygon can be split into 30 − 2 = 28 triangles.

Therefore the sum of its interior angles can be found by 28 × 180º = 5040º .

The sum of interior angles of a 30 sided polygon is 5040º .

This will be looked at in more detail in the lesson on interior angles in a polygon .

Example 7: finding the sum of interior angles of a complex polygon

This polygon has 6 sides. The sides are not all the same length so this polygon is an irregular hexagon.

To work out the number of triangles a polygon can be split into we subtract 2 from the number of sides it has.

A 6 sided polygon can be split into 6 − 2 = 4 triangles.

Therefore the sum of its interior angles can be found by 4 x 180º = 720º .

The sum of interior angles in a hexagon is 720º .

Common misconceptions

  • Miscounting the number of sides
  • Misidentifying if a polygon is regular or irregular
  • Incorrectly assuming all the angles are the same size
  • Crossing lines when drawing the triangles, this creates false interior angles

Practice angles in polygons questions

1. Is a square a regular or irregular polygon?

GCSE Quiz True

All the side lengths are the same, and all angles are right angles, hence a square is a regular polygon.

2. Is a semi circle a polygon?

A semicircle has a side which is not straight, so it is not a polygon.

3. What is the sum of interior angles for a triangle?

180^{\circ} 

360^{\circ} 

270^{\circ} 

The angles in a triangle add up to 180^{\circ}.

4. What is the sum of interior angles for a regular hexagon?

720^{\circ}

540^{\circ}

360^{\circ}

1080^{\circ}

We know the angles in a triangle add up to 180^{\circ} . A regular hexagon can be divided into 4 triangles, and four lots of 180 is 720 .

5. What is the sum of interior angles for an irregular hexagon?

We know the angles in a triangle add up to 180^{\circ} . An irregular hexagon can be divided into 4 triangles, and four lots of 180 is 720 .

6. What is the sum of interior angles for a regular 12 sided polygon?

2700^{\circ}  

2160^{\circ}

1800^{\circ}  

1080^{\circ}  

We know the angles in a triangle add up to 180^{\circ} . A 12 sided polygon can be divided into 10 triangles, and ten lots of 180 is 1800 .

7. What is the sum of interior angles for a regular 25 sided polygon?

4500^{\circ}

4140^{\circ}

9000^{\circ}

3600^{\circ}

We know the angles in a triangle add up to 180^{\circ} . A 25 sided polygon can be divided into 23 triangles, and 23 lots of 180 is 4140 .

Angles in polygons GCSE questions

1.  Each exterior angle of a regular polygon is 15^{\circ} . Work out the number of sides the polygon has.

360 \div 15

            (1)

2.  Each of the interior angles of a regular polygon is 140^{\circ} . Show that this polygon has 9 sides

Exterior angle = 40 seen or implied

3.  In a regular polygon each exterior angle is 18^{\circ} . Find the sum of interior angles for this polygon

Learning checklist

You have now learned how to:

  • Use conventional terms for geometry e.g. regular or irregular
  • Derive and apply the properties and definitions of: special types of quadrilaterals
  • Knowing names and properties of polygons
  • Calculate the sum of interior angles for a regular polygon
  • Calculate the sum of interior angles for a irregular polygon

The next lessons are

  • Angle rules
  • Angles in parallel lines
  • How to calculate volume

Still stuck?

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Triangle Solving Practice

Practice solving triangles .

You only need to know:

  • Angles Add to 180°
  • The Law of Sines
  • The Law of Cosines

Try to solve each triangle yourself first, using pen and paper.

Then use the buttons to solve it step-by-step (more Instructions below).

Instructions

  • Look at the triangle and decide whether you need to find another angle using 180°, or use the sine rule, or the cosine rule.  Click your choice .
  • The formula you chose appears, now click on the variable you want to find.
  • The calculation is done for you.
  • Click again for other rules until you have solved the triangle.

Note: answers are rounded to 1 decimal place.

What Does "AAS", "ASA" etc Mean?

It means which sides or angles we already know:

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Choose Your Test

Sat / act prep online guides and tips, triangles on act math: geometry guide and practice problems.

feature_Triangle

If you thought the ACT was a big fan of circles , then brace yourself for its absolutely shameless love of triangles. In one breath, you may be expected to find the various dimensions of an obtuse triangle, and the next, an isosceles right triangle. ACT triangle problems will be as numerous as they are varied, so make sure you familiarize yourself with all the different types before test day.

This will be your complete guide to ACT triangles --the types of triangles that will show up on the ACT, the formulas you’ll need to know to solve them, and the strategies you’ll need to apply when approaching a triangle question. We’ll also break down real ACT math problems and give you the walk-throughs on how to most efficiently and effectively tackle any and all triangle problems you come up against.

What Are Triangles?

Before we go through how to solve a triangle problem, let’s discuss the basics. A triangle is a flat figure made up of three straight lines that connect together at three angles. The sum of these angles is 180°.

Each of the three sides of a triangle is called a “leg” of the triangle, and the largest (longest) leg is called the “hypotenuse.” The angle opposite the hypotenuse will always be the largest of the three angles.

body_SAT_triangles_21.2-1

The sum of any two legs of a triangle must always be greater than the measure of the third side. Why? Because when the sum of two lines is smaller than the measure a third line, they cannot all connect to form a triangle.

Triangles that have legs which sum only slightly more than the hypotenuse are quite long and skinny, but they still make the “bump” of a triangle because they combine to be longer than the third side.

body_SAT_triangles_21-2

But if the legs are too short, they will never meet, no matter how shallow the angle.

body_SAT_triangles_21.01-1

And if the lines are the exact length of the hypotenuse, then they will flatten to a perfectly straight line, overlapping the hypotenuse precisely.

Let's look at an example ACT problem of this kind:

A triangle has side lengths of 6 inches and 9 inches. If the third side is an integer, what is the least possible perimeter, in inches, of the triangle?

We know, based on our rules for the side lengths of triangles, that the sum of two sides must be greater than the third. Because we are trying to find the smallest perimeter, we must find our missing side by taking the difference of our two leg lengths:

$9 - 6 = 3$

Considering the sum of two legs must be greater than the third side, our missing side must be greater than 3. (Why? Because $6 + 3 = 9$ and we need the sum to be larger than 9.)

If our missing side is an integer value (which we are told is true), and we are trying to find the minimum perimeter value, then our missing side must be the smallest integer greater than 3.

Which means that our missing side is 4.

To find our perimeter, then, we must add all our sides together:

$4 + 6 + 9 = 19$

Our final answer is D , 19.

(Note: always pay attention to the exact question you’re being asked and don’t get tricked by bait answers! If you were going too quickly through the test, you might have been tempted to select answer choice A, 4, which was the value of the missing side length. But, since we were asked to find the perimeter , this would have been the wrong answer.)

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Ready to enter the realm of special triangles (and become insanely awesome)?

Special Triangles

There are several different kinds of special triangles, all of which commonly appear on the ACT.

In this section, we will define and describe all the different kinds of triangles you’ll see on the test. In the next section, we will go through all the formulas you’ll need to know for your ACT triangle problems, as well as how to use them.

Equilateral Triangles

An equilateral triangle is a triangle that has three equal legs and three equal angles. Though the leg measurements can be anything (so long as they are all equal), the angle measurements must all equal 60°. Why? Because a triangle’s angles must always total 180°, and $180/3 = 60$.

body_equilateral-1

Isosceles Triangles

An isosceles triangle is a triangle in which two sides and two angles are equal.

body_isosceles-1

The sides opposite equal angles will always be equal and the angles opposite equal sides will always be equal. This knowledge will often lead you to the correct answers for many ACT questions in which it seems you are given very little information.

body_ACT_Triangles_11

(We will go through how to solve this problem later in the guide, but for now, note how it seems as if you are not given enough information. But, if you remember that angles opposite equal lines are also equal, then you’ll see that you now have exactly enough to solve the problem)

Right Triangles

A right triangle is a triangle in which one of the angles measures 90° (90° is a right angle). This means that the sum of the other two angles must be 90° as well, since a triangle’s angles always add up to 90°.

body_right_triangle-1

The leg opposite the 90° angle will always be the triangle’s hypotenuse. This is due to the fact that the 90° angle will always be the largest angle in a right triangle. (Why? Because two 90° angles would make a straight line , not a triangle.)

body_right_angles-1

Special Right Triangles

There are many different kinds of right triangle and some are considered “special.” These are triangles that have set angles or side lengths and formulas to correspond with them. Understanding these types of triangles (and their formulas) will save you a significant amount of time as you go through your test.

We will go through the formulas that correspond with these types of triangles in the next section, but for now, let’s go through their definitions.

Isosceles Right Triangle

An isosceles right triangle is just what it sounds like--a right triangle in which two sides and two angles are equal.

Though the side measurements may change, an isosceles triangle will always have one 90° angle and two 45° angles. (Why? Because a right triangle has to have one 90° angle by definition and the other two angles must add up to 90°. So $90/2 = 45$.)

body_isosceles_right-1

30-60-90 Triangles

A 30-60-90 triangle is a special right triangle defined by its angles. It is a right triangle due to its 90° angle, and the other two angles must be 30° and 60°.

body_30-60-90-1

3-4-5, and 5-12-13 Right Triangles

3-4-5 and 5-12-13 triangles are special right triangles defined by their side lengths. The numbers 3-4-5 and 5-12-13 describe the lengths of the triangle’s legs, meaning that, when you have a right triangle with two leg lengths of 4 and 5, then you automatically know that the third leg equals 3. Any consistent multiples of these numbers will also work the same way. So a right triangle could have leg lengths of:

3(1)-4(1)-5(1) => 3-4-5

3(2)-4(2)-5(2) => 6-8-10

3(3)-4(3)-5(3) => 9-12-15

These are considered special right triangles because all their sides are integers.

body_side_lengths-2

Now it's triangle formula time!

Triangle Formulas

Now that you know what all your triangles will look like, let’s go through how to find missing variables and information about them.

You will not be given any formulas on the ACT, so you must know all of these formulas by heart. (For more on the formulas you’ll need for the ACT math section, check out our guide to the 31 formulas you must know before test day .)  

But beyond memorizing your formulas, you also must take care to understand them--how they work and when. All the rote memorization in the world won’t help you if you don’t know when or how to apply them when solving your problems.

All Triangles

$a = {1/2}bh$

$b$ is the base of the triangle, which is the length of any one of the triangle’s legs.

$h$ is the height of a triangle, found by drawing a straight line (at a 90° angle) from the base of the triangle to the opposite angle from the base.

This means that, in a right triangle, the height is the length of the leg that meets at a 90° angle to the base. In a non-right triangle, you must create a new line for your height.

body_triangle_height-1

$p = l_1 + l_2 + l_3$

Just like with any other kind of plane geometry figure, the perimeter of a triangle is the sum of its outer sides (the triangle’s three legs).

body_leg_perimeter-1

Some triangle formulas apply specifically to right triangles, so let's take a look.

Pythagorean Theorem

$a^2 + b^2 = c^2$

The Pythagorean theorem allows you to find the side lengths of a right triangle by using the lengths of its other sides. $a$ and $b$ signify the shorter legs of the triangle, while $c$ is always the leg opposite the 90° angle (the hypotenuse).

body_ACT_Triangles_15

According to the Pythagorean theorem, $a^2 + b^2 = c^2$. We know that the side with $y$ meters must be our hypotenuse, as it is opposite the 90 degree angle. This means that:

$a^2 + b^2 = c^2$ 

$4^2 + x^2 = y^2$

Now, we need to find $y$ in terms of $x$, which means we need to isolate our $y$. 

$16 + x^2 = y^2$ 

$y = √{16 + x^2}$

Our final answer is E , $√{x^2 + 16}$

3-4-5 and 5-12-13 triangles (and their multiples) are special because you do not need to work through the pythagorean theorem in order to find the side measures of the third length. Remember, if two sides of a right triangle are 12 and 15, then you automatically know the third side is 9 (because $3(3)-4(3)-5(3) = 9-12-15$).

body_ACT_Triangles_14

Though we can find the length of BC using the Pythagorean theorem, we can also simply know that it is 5. (Why? Because it is the hypotenuse of a right triangle with leg lengths of 3 and 4). 

Now, we can set up a proportion to find the measure of side AE. The length of AE to its hypotenuse will be in proportion to the length of BD to its hypotenuse. 

${AE}/20 = 3/5$

Our final answer is B , 12.

$x, x, x√2$

Though you can find the missing side lengths of an isosceles triangle using the Pythagorean theorem, you can also take a shortcut and say that the equal side lengths are $x$ and the hypotenuse is $x√2$.

body_x_x_x_root_2-2

Why does this work? Let’s look at an isosceles right triangle problem.

body_ACT_Triangles_2

It is given to us that one side length equals 10, so we know the second leg must also equal 10 (because the two legs are equal in an isosceles triangle). We can also find the hypotenuse using the Pythagorean theorem because it is a right triangle. So:

$10^2 + 10^2 = c^2$

$100 + 100 = c^2$

$200 = c^2$

$c = √100 * √2$ (Why were we able to split up our root this way? Check out our guide to ACT advanced integers and its section on roots if this process is unfamiliar to you.)

So, we are left with side lengths of 10, 10, and 10√2. Or, in other words, our side lengths are $x, x$, and $x√2$.

So our final answer is E , $10√2$

30-60-90 Triangle

$x, x√3, 2x$

Just like with an isosceles right triangle, a 30-60-90 triangle has side lengths that are dictated by a set of rules. 

Again, you can find these lengths with the Pythagorean theorem, but you can also always find them using the rule: $x, x√3, 2x$, where $x$ is the side opposite 30°, $x√3$ is the side opposite 60°, and $2x$ is the side opposite 90°.

body_30-60-90_example-1

Make a note now of any formulas that are unfamiliar to you. You will need to know them by test day, so a little practice and organization now will go a long way to keeping them straight in your head.

Typical Triangle Questions

Most triangle question on the ACT will involve a diagram, though a rare few will be purely word problems. Let’s look at some of the standard types of question in each category.

Word Problems

Most triangle word problems are fairly simplistic once you draw them out. In fact, often times, the very reason why they give you the problem as a word problem instead of providing you with a diagram is because the test-makers thought the problem would be too easy to solve with a picture.

Whenever possible, draw your own diagram when you are given a triangle problem without one. It won’t take you long and it’ll be much simpler for you to visualize the question.

body_ACT_Triangles_10

This should be a simple figure, but it never hurts to quickly sketch it out in order to keep all our parts in order.

body_diagram_problem_1.1

We are told that this is a right triangle and we need to find one missing side length, so we will need to use the Pythagorean theorem.

Using our given side lengths for $a$ and $b$, we have:

$6^2 + 7^2 = c^2$

$36 + 49 = c^s$

Our final answer is G , $√85$

Diagram Problems

There are several different kinds of triangle problems that involve diagrams. Let’s break them into categories and discuss the strategies for each.

Diagram Type 1 - Finding Missing Values

Most triangle problems will fall into this category--you will be asked to find a missing angle, an area, a perimeter, or a side length (among other things) based on given information.

Some of these questions will be more complicated than others, but the ACT will always provide you will enough information to solve a problem, so it’s up to you to put the clues together.

Let’s walk through some real ACT math examples of this type:

body_ACT_Triangles_1

First, let us fill in our given information so that we don't lose track of which angles measure what.

body_diagram_problem_2

We know that the interior angles in a triangle sum up to 180 degrees, so we can find ACB by subtracting our givens from 180. 

$180 - 30 - 110$

body_diagram_problem_2.2

We also know that any straight line will measure 180 degrees. BCD are collinear, which means that they lie on a straight line. We can therefore find angle ACD by subtracting our ACB measure from 180.

Our final answer is G , 140°.

body_ACT_Triangles_13

Similar triangles are in proportion with one another, so we can find the side lengths for triangle BAC by setting up proportions with triangle LKM. 

${BA}/{AC} = {LK}/{KM}$

${BA}/3 = 12.5/7.5$

$7.5BA = 37.5$

And our second proportion will follow the same model. 

${AC}/{BC} = {KM}/{LM}$

$3/{BC} = 7.5/15$

$7.5BC = 45$

Now, we have all the side measures for triangle BAC, which means we can find its perimeter. 

body_diagram_problem_3

$5 + 3 + 6$

Our final answer is B , 14. 

Diagram Type 2 -  Ratios and (In)Equalities

These kinds of questions will generally ask you to either find the ratios between parts of different triangles or will ask you whether or not certain sides or angles of triangles are equal or unequal.

body_ACT_Triangles_8

We are told that AD is equal to BC, which means that their corresponding angles will also be equal. This means that angles CAB and DBA are equal (which consequently means that angles EAB and EBA are equal). We can therefore eliminate answer choice K. 

body_diagram_problem_4.1

Now, if angles CAB and DBA are equal, then angles CBA and DAB must ALSO be equal. Why? Well we know that each triangle has a 90 degree angle and one angle to equal to some unknown measurement (which we could call $x$). This means that the third, remaining, angle (let's call it $y$) must ALSO be the same for each triangle. 

Each triangle would then be made up of:

$180 = 90 + x + y$

This means that we can eliminate answer choice J.

By that same reckoning, if angle DAB = angle CBA, then the legs opposite those angles must also be equal. This means that AC = BD, which means that answer choice F can be eliminated.

body_diagram_problem_4.2

Because AD and CB are equal and both are part of a triangle with a hypotenuse of AB, legs CA and DB will cross in a manner that makes each half of the leg equal to the corresponding half of the leg of the other triangle. In other words, AE = EB and DE = EC. This means we can eliminate answer choice H. 

The only answer choice we are left with is G. AD CANNOT equal AE. Why? AD is the leg of triangle ADE, while AE is the hypotenuse of that same triangle. From our definitions, we know that the hypotenuse must always be the longest side of the triangle and so it cannot be equal to one of the legs. 

Our final answer is G. 

Diagram Type 3 -  Multi-Shape or Shapes Within Shapes

As you can see from earlier examples, some of the triangle problems on the ACT will involve multiple triangles (or other geometric shapes) combined together. This technique for presenting you problems is designed to challenge your understanding of lines and angles as well as triangles.

For these types of problems, you must use the information you are given and solve for more information down the line until you find exactly what you’re looking for. It’s essentially a domino effect of problem solving.

body_ACT_Triangles_7

Because this problem uses variables, the simplest way to solve it is by  plugging in our own numbers . So let us do so. 

We are told that each unshaded triangle is a congruent right triangle. Because variables can be difficult to work with, let us replace $x$ with 4. (Why 4? Why not!)

This means that each triangle has one leg that measures 4 and one leg that measures $2(4) = 8$. 

body_diagram_problem_5.1

Now, we can find the length of one side of the square ABCD by adding our values together. 

body_diagram_problem_5.2

Each side of the square ABCD is equal to 12. Now we can find the total area by squaring this side measure, so:

The total area for ABCD is 144.

Now, because each unshaded triangle is a right triangle, we can find the side measures for the shaded square using the Pythagorean theorem.

$4^2 + 8^2 = c^2$

$16 + 64 = c^2$

body_diagram_problem_5.3

Since this is the measure of one side of the shaded square, we can now find the area for the shaded square by squaring this number. So:

Now, we must simply divide our shaded square by our unshaded square, ABCD, in order to determine what fraction it is of the larger square. 

$80 ÷ 16 = 5$ and $144 ÷ 16 = 9$

Our final answer is D , $5/9$

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Life lessons and triangle strategies--win-win!

Strategies for Solving a Triangle Question

Because there are so many different kinds of triangle problems, it is difficult to break down one exact path for problem solving them.

That said, your greatest assets and strategies when solving triangle problems will be to follow these four steps:

#1: Write down your formulas

Because you are not given any formulas, you must keep them in your head and in your heart. The good news is that more you practice, the better you’ll be at rattling off triangle areas or side lengths of 30-60-90 triangles or anything else you’ll need.

But if you feel like you’ll forget your formulas as you go through your test, take a few seconds and write them down before you start solving your questions. Once you do, they will be there indelibly for you to work from for the rest of the math section, and you won’t have to worry about forgetting them.

#2: Use your formulas (and take your short-cuts)

Once you’re sure that you’ve remembered your formulas, using them is the absolute most crucial step for any triangle problem. And, considering that most of your formulas essentially act as short-cuts (why bother solving with the Pythagorean theorem when you know that the legs of a 30-60-90 triangle are $x, x√3, 2x$?), you will save yourself a great deal of time and energy when you can keep your formulas on hand and in order.

#3: When working with multi-shapes, break it into small steps

Remember that dealing with a multi-shape triangle problem is like working with dominoes. Each successive piece of information makes way for finding the next piece of information.

Don’t get intimidated that you don’t have enough information or that there are too many shapes or lines to deal with. You will always have enough data to go on--just focus on finding one shape and one piece of information at a time, and the dominoes will fall into place.

#4: Draw it out

Draw your own diagrams if you are given none. Draw on top of your diagrams when you are given pictures. Write in your givens and all the measurements you find along the way to your missing variable (or variables), mark congruent lines and angles.

The more you can clarify your diagrams, the less likely you’ll be to make careless errors in misplacing or confusing your numbers and equalities.

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Ready to put your knowledge to the test?

Test Your Knowledge

Now let's test your triangle knowledge against some more real ACT math problems.

body_ACT_Triangles_11-1

Answers: B, F, E, H

Answer Explanations:

1)  Because we are told that this is an isosceles trapezoid, we know that each non-parallel side must be equal. This means that the angles that capture these sides (angles BDC and ACD) must also be equal. 

body_diagram_problem_6.1

We also know that the interior degrees of a triangle will always sum 180 degrees, so we can find the measure of DXC by subtracting our two known angles from 180. 

$180 - 25 - 25$

body_diagram_problem_6.2

Now, DB is a straight line, which means that the angles that make the line must total 180 degrees.  This means we can find angle BXC by subtracting our known angle from 180. 

$180 - 130$

body_diagram_problem_6.3

Finally, we again know that a triangle's interior angles will sum to 180, so we can find DBC by subtracting our known angles from 180. 

$180 - 50 - 35$

Our final answer is B , 95°.

2)  We know from our triangle definitions that the larger the side opposite an angle, the larger the angle will be. (If you ever feel unsure about the relationships between angles and sides of a triangle, you can also consult your rules and definitions of trigonometry .)

So if we drew in some random side measurements for XZ and YZ (so long as they follow the rule that XZ > YZ), we can see clearly that angle Y will be greater than angle X. 

body_diagram_problem_7

Our final answer is F, angle X < angle Y. 

3)  We are told that the triangle is a hypotenuse right triangle, which means that we can use our shortcuts to find the other two side lengths. 

body_diagram_problem_8.1

We know that an isosceles right triangle has side lengths of $x, x$, and $x√2$. Since we already know that the hypotenuse is $8√2$, we can say that the other two sides both measure 8. 

body_diagram_problem_8.2

Now, we can add together the legs to find the perimeter. 

$8 + 8 + 8√2$

Our final answer is E, $16 + 8√2$

4)  Before we do anything else, let us fill in our given information. 

body_diagram_problem_9.1

Now, we can know the triangles and the exterior angle are all collinear, which means that the angles that create the line will sum to 180°. This means we can find angle CBD by subtracting our exterior angle from 180. 

$180 - 140$

body_diagram_problem_9.2

Now that we have two interior angle measures in triangle DCB, we can find the measure of the third (because the interior angles in a triangle will always add up to 180).

$180 - 40 - 47$

body_diagram_problem_9.3

[Note: you may notice that the sum of the two angles not touching the exterior angle sum up to equal the exterior angle--$47 + 93 = 140$. This is not a coincidence. It will always be the case that the two non-connected angles will sum to equal the exterior angle of any type of triangle.)

Now we again have two angles that create a straight line, which means that we can find the measure of angle CDA by subtracting our known angle from 180°. 

body_diagram_problem_9.4

And finally, CAD forms a triangle, which means that its interior angles will sum to equal 180. We can find angle ACD by subtracting our two known values from 180°. 

$180 - 76 - 87$

Our final answer is H, 17°. 

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Aw, yea. You've earned that nap.

The Take-Aways

Whether it be a trigonometry problem or a geometry problem, you’ll see triangles several times on any given ACT. Though most triangle problems are fairly straight forward, you’ll need to know the basic building blocks of triangles and geometry in order to understand how to solve them.

Know your definitions, memorize your formulas, and do your best to keep a clear head as you go through your test. And, as always, practice, practice, practice! The more experience you get in solving the variety of triangle questions the ACT can think to put in front of you, the better off you’ll be.

What’s Next?

Whoo! You took on triangles and won (give yourself a round of applause)! In the mood for more geometry? Hop on over to our guides on ACT circles , polygons , and solid geometry and round off all your geometry studies in one go.

Not sure what topic to tackle next? Make sure you've got a clear idea of all the math topics you'll be tested on and check out all of our ACT math guides for reference and practice. Each guide has definitions, formulas, and real ACT practice questions and will break down the solving process step-by-step. 

Been procrastinating? Check out our guide on how to take back your study time and beat back those procrastination demons.

Looking to get a perfect score? Our guide to getting a 36 on the ACT math (written by a perfect-scorer!) will help get you where you need to go.

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Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. She is passionate about bringing education and the tools to succeed to students from all backgrounds and walks of life, as she believes open education is one of the great societal equalizers. She has years of tutoring experience and writes creative works in her free time.

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IMAGES

  1. Angles in Triangles (Worksheets with Answers)

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  2. Constructing Triangles Textbook Exercise

    angles in triangles problem solving tes

  3. 4 2 Practice Angles Of Triangles Worksheet Answers

    angles in triangles problem solving tes

  4. Angles in Triangles (Worksheets with Answers)

    angles in triangles problem solving tes

  5. Angles in Triangles (Worksheets with Answers)

    angles in triangles problem solving tes

  6. Angles on a Triangle Solving problem examination Question Assessment

    angles in triangles problem solving tes

VIDEO

  1. FINI TES TRIANGLES MAINTENANT

  2. Angles And Triangles Part -1

  3. Explaining the “How Many Triangles?” Problem @normalotherheartedtherian

  4. Trigonometry 1.2 Similar Triangles Problem 7

  5. Trigonometry 1.2 Similar Triangles Problem 13

  6. Trigonometry 1.2 Similar Triangles Problem 12

COMMENTS

  1. Angles in Triangles (Worksheets with Answers)

    Age range: 11-14 Resource type: Lesson (complete) File previews pdf, 984.51 KB pdf, 828.97 KB pdf, 814.77 KB ppsx, 5.27 MB pdf, 1.05 MB Three differentiated worksheets ( with solutions) that allow students to take the first steps, then strengthen and extend their skills in working with angles in triangles.

  2. Angles Complete Lessons for KS2 to GCSE Maths

    Complete set of comprehensive angle lessons covering angle basics, angle rules, angles in triangles and angles in quadrilaterals. They cover everything you need to know about these topics, from KS2 level through to GCSE foundation level. These are ideal to use within the classroom, for home educating families, or for tutors to use in lessons!

  3. Angles in a Triangle Textbook Exercise

    Next: Parts of a Circle Textbook Exercise GCSE Revision Cards. 5-a-day Workbooks

  4. PDF Angles in a Triangle

    Question 2: James says that a triangle is right angled. Olivia says that the same triangle is isosceles. They are both correct. Explain how. Question 3: The ratio of three angles in a triangle are 1:2:3. Work out the size of each angle. Question 4: An isosceles triangle has one angle of 52°. Write down the possible sizes of the other two ...

  5. Angles In A Triangle

    All triangles have interior angles that add up to 180º180º. Angles in a triangle are the sum (total) of the angles at each vertex in a triangle. We can use this fact to calculate missing angles by finding the total of the given angles and subtracting it from 180º180º. This is true for all types of triangles. Right Angle Triangle: One 90°

  6. PDF Year 6 Angles in a Triangle 2 Reasoning and Problem Solving

    Reasoning and Problem Solving Angles in a Triangle 2 Developing 1a. No, because two of the angles would have to be equal for this to be an isosceles triangle. 2a. 1B, 2A, 3C 3a. Isosceles, 70 degrees Expected 4a. No, because two angles have to be the same and the total for all three angles must be 180 degrees. 5a. 1B, 2C, 3A

  7. Angles of a triangle (review)

    Example: Find the value of x in the triangle shown below. 106 ∘ x ∘ 42 ∘. We can use the following equation to represent the triangle: x ∘ + 42 ∘ + 106 ∘ = 180 ∘. The missing angle is 180 ∘ minus the measures of the other two angles: x ∘ = 180 ∘ − 106 ∘ − 42 ∘. x = 32. The missing angle is 32 ∘ .

  8. Find angles in triangles (practice)

    Lesson 2: Triangle angles. Angles in a triangle sum to 180° proof. Find angles in triangles. Isosceles & equilateral triangles problems. Find angles in isosceles triangles. Triangle exterior angle example. Worked example: Triangle angles (intersecting lines) Worked example: Triangle angles (diagram) Finding angle measures using triangles.

  9. Missing Angles Practice Questions

    . Click here for Answers . angle, right, straight line, point, full turn, vertically, opposite, basic, facts, triangle, quadrilateral Tessellations Practice Questions The Corbettmaths Practice Questions and Answers on missing angles

  10. Worked example: Triangle angles (diagram) (video)

    To answer your question: Yes you could treat the 90 as +90 and the 5 as -5. Try it. -5+90 and you still get the same answer as you would get with 90-5. Try another: -80 + 10 which would equal -70. If you flipped the problem but kept the signs it would be (+)10 - 80 which would also give you -70. hope this helped!!

  11. Working Out Angles in a Triangle

    The angles in any triangle will always add up to 180°. Equilateral triangles have three equal sides and angles, so each angle is always 60°. Isosceles triangles have two equal angles. You'll need to know at least one of the angles in order to solve the other two. Scalene triangles all have different angles, so you'll need to know two of ...

  12. Resourceaholic: Shape

    Angles on a straight line worksheets - Maths4Everyone on TES; Angles around a point worksheets - Maths4Everyone on TES; Angles at a point, angles at a point on a straight line, vertically opposite angles - Boss Maths; Angles in triangles (worksheet bundle) - Maths4Everyone on TES; Angles in a triangle extra practice - Maths4Everyone on TES

  13. Angles

    Angles in a triangle; The sum of angles in a triangle is \bf{180^{o}} . x + y + z = 180^o . ... Solve the problem and give reasons where applicable. As the sum of the two angles is 90 degrees, forming an equation, we have. x+10+3x=90, or. 4x+10=90. Solving this for x, we have

  14. Angles, Polygons and Geometrical Proof Short Problems

    This is part of our collection of Short Problems. You may also be interested in our longer problems on Angles, Polygons and Geometrical Proof Age 11-14 and Age 14-16. Printable worksheets containing selections of these problems are available here: Stage 3 ★. Sheet 1.

  15. Find angles in congruent triangles (practice)

    Course: High school geometry > Unit 3. Lesson 5: Working with triangles. Corresponding angles in congruent triangles. Find angles in congruent triangles. Isosceles & equilateral triangles problems. Find angles in isosceles triangles. Finding angles in isosceles triangles. Finding angles in isosceles triangles (example 2) Math >.

  16. Angles In Polygons

    4 Multiply the number of triangles by 180 to get the sum of the interior angles. 180∘ ×2 = 360∘ 180 ∘ × 2 = 360 ∘. 5 State your findings e.g. sides, regular/irregular, the sum of interior angles. The polygon is a irregular quadrilateral (specifically called a parallelogram as both opposite sides are parallel) with a sum of interior ...

  17. Triangle Solving Practice

    Triangle Solving Practice. Practice solving triangles. You only need to know: Angles Add to 180°. The Law of Sines. The Law of Cosines. Try to solve each triangle yourself first, using pen and paper. Then use the buttons to solve it step-by-step (more Instructions below).

  18. Triangles on ACT Math: Geometry Guide and Practice Problems

    19 29 We know, based on our rules for the side lengths of triangles, that the sum of two sides must be greater than the third. Because we are trying to find the smallest perimeter, we must find our missing side by taking the difference of our two leg lengths: 9 − 6 = 3