GCSE Interior Exterior Angles Dr J Frost jfrosttiffin

GCSE : : Interior & Exterior Angles Dr J Frost (jfrost@tiffin. kingston. sch. uk) Objectives: Determine interior/exterior angles of a polygon, the number of sides of a polygon and consider tessellation of shapes. Last modified: 24 th December 2018

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Overview We will consider the angles inside and outside regular and irregular polygons. ? Possible lesson structure: Lesson 1: Interior angles. Lesson 2: Regular Polygons & Exterior angles. Lesson 3: Tessellation and Problem Solving.

STARTER : : Interior angles of quadrilateral The interior angles of a quadrilateral add up to 360. ? 1 An interior angle is just an angle between two sides within the shape. Parallelogram 2 ? 3 ? 4 Trapezium ? ? Kite ?

Sum of interior angles Can you guess what the angles add up to in a pentagon? How would you prove it?

Sum of interior angles Click to Fromanimate ?

Examples 1 What is the total interior, and therefore, each interior angle, of each of these regular polygons? Regular Polygon Total interior 2 Fro Tip: Memorise these for pentagon, hexagon and octagon. Each Interior Angle Equilateral triangle (3) Square (4) Pentagon (5) ? ? Hexagon (6) ? ? Octagon (8) ? ? Nonagon (9) ? ? Decagon (10) ? ? ?

Test Your Understanding A regular dodecagon (12 sides). 160° 80° 130° 120° ? ? 120° 40° 100° ? 40°

Exercise 1 1 a (On supplied sheet) b c b = 260 ? x = 75 ? x = 25 ? d e a = 100 ? f x = 222 ? x = 309 ?

Exercise 1 g h x = 54 ? 2 i x = 120 ? x = 252 ? ? N 1 ?

Exercise 1 N 2 ?

Exterior Angles exterior interior ? Which of these are exterior angles of the polygon? ? NO ? YES NO?

Total Exterior Angle ? ! The sum of the exterior angles of any polygon is 360°. Click to Start Damonimation

Each Exterior/Interior Angle of Regular Polygon If the pentagon is regular, then each exterior angles is the same. Therefore: ? ? This therefore gives us an alternative way to work out the interior angle of a regular polygon:

Practicing Angles in Regular Polygons Sides Exterior Angle Interior Angle 3 Equilateral Triangle ? ? 4 Square ? ? 5 Pentagon ? ? 6 Hexagon ? ? 8 Octagon ? ? 9 Nonagon ? ? 10 Decagon ? ? Bonus Question: What is the largest number of sides a shape can have such that its interior angle is an integer? 360 sides. The interior ? angle will be 179°.

Number of sides of a regular polygon The advantage of this new method is that it works backwards: we can easily work out the number of sides of the polygon given an exterior or interior angle. ? interior exterior ?

Quickfire Questions Your teacher will fire these questions at a variety of you – try to do in your head if you can! ? interior exterior ? ?

Test Your Understanding 1 4 ? 2 ? 5 ? 3 ? ?

Exercise 2 1 4 ? ? ? 2 Determine how many sides a regular polygon with the following exterior angle would have: 30 12 sides 45 8 sides 12 30 sides 9 40 sides 5 Determine how many sides a regular polygon with the following interior angle would have: 156 15 sides 162 20 sides 144 10 sides 175 72 sides ? ? ? 3 ? 6 ?

Exercise 2 7 9 ? ? 8 10 ? ?

Problem Solving with Interior/Exterior Angles There are variety of skills that harder questions involving interior/exterior angles might involve: #1: Tessellation B C B A B B A C #2: Using isosceles triangles Shapes ‘tessellate’ if they fit together, without overlap, to form a repeating pattern. C B A B “The above repeating pattern consists of three regular polygons, A (hexagon), B (square) and C. Determine how many sides C has. ”

Problem Solving with Interior/Exterior Angles There are variety of skills that harder questions involving interior/exterior angles might involve: #3: Dealing with a mixture of both irregular and regular polygons.

#1 : : Tessellation [Edexcel GCSE Nov 2012 -1 H Q 18] The pattern is made from two types of tiles, tile A and tile B. Both tile A and tile B are regular polygons. Work out the number of sides tile A has. ? ? We’ve recently seen a method for finding the number of sides of a regular polygon given its interior angle! ? ?

Further Example B C C A B B A C B B A B ? C The above repeating pattern consists of three regular polygons, A (hexagon), B (square) and C. Determine how many sides C has.

Test Your Understanding ? ?

#2 : : Irregular Polygons within Regular Polygons ? ? ? ?

Test Your Understanding ? ? ? ? ? ?

#3 : : Isosceles Triangles When you have regular polygons in contact with each other, then their lengths are all the same. This regularly results in isosceles triangles. ?

Test Your Understanding ?

Exercise 3 1 3 ? 4 ? 2 ? ?

Exercise 3 5 ? 6 ?

Exercise 3 9 7 ? [IMC 2005 Q 14] Ten stones, of identical shape and size, are used to make an arch, as shown in the diagram. Each stone has a crosssection in the shape of a trapezium with three equal sides. What is the size of the smallest angles of the trapezium? 8 [JMO 2014 B 1] The figure shows an equilateral triangle ABC, a square BCDE, and a regular pentagon BEFGH. What is the difference between the sizes of ∠ADE and ∠AHE? 10 ? ? ?

Exercise 2 11 ? N ?

FINAL TOPIC REVIEW Vote with your diaries! A B C D

What is the total exterior angle of a polygon in terms of the number of sides n? 360 n 360 n 180(n-2)

What is the total interior angle of a 20 sided polygon? 3600 3240 6480

The interior angle of a polygon is 178. How many sides does it have? 20 40 90 180

What is the interior angle of a 90 sided regular polygon? 172 176 178 179

Determine the angle . 61 29 105 120 215 223 225 235