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GCSE :: Interior & Exterior Angles

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1 GCSE :: Interior & Exterior Angles
Dr J Frost Objectives: Determine interior/exterior angles of a polygon, the number of sides of a polygon and consider tessellation of shapes. Last modified: 24th December 2018

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3 Overview We will consider the angles inside and outside regular and irregular polygons. Types of questions we will explore: What is each angle inside a regular decagon (10 sides)? If the each angle inside a regular polygon is 175ยฐ, how many sides does it have? If I put a bunch of regular pentagons in a circle, what shape would be formed in the middle? ? Possible lesson structure: Lesson 1: Interior angles. Lesson 2: Regular Polygons & Exterior angles. Lesson 3: Tessellation and Problem Solving.

4 STARTER :: Interior angles of quadrilateral
An interior angle is just an angle between two sides within the shape. The interior angles of a quadrilateral add up to 360๏‚ฐ. ? Parallelogram 1 2 ๐‘ฆ 100ยฐ ๐‘ฅ 50ยฐ ๐‘ฅ ๐‘ฅ=130ยฐ ? ๐‘ฅ=100ยฐ ๐‘ฆ=80ยฐ ? ? 3 4 Trapezium Kite ๐‘ฅ ๐‘ฅ=120ยฐ ? ๐‘ฅ 95ยฐ 55ยฐ 60ยฐ Cointerior angles (โ€œCโ€ angles) add to 180ยฐ. ๐‘ฅ=105ยฐ ?

5 Sum of interior angles ๐‘›=๐Ÿ‘ ๐‘›=๐Ÿ’ Total of interior angles = ๐Ÿ‘๐Ÿ”๐ŸŽยฐ
Can you guess what the angles add up to in a pentagon? How would you prove it?

6 ! For an ๐‘›-sided shape, the sum of the interior angles is:
Sum of interior angles Click to Fromanimate We can cut a pentagon into three triangles. Notice that the angles combined in the three triangles are the total interior angle in the pentagon. The sum of the interior angles of the triangles is: 3ร—180ยฐ=540ยฐ ! For an ๐‘›-sided shape, the sum of the interior angles is: 180(๐‘›โˆ’2) ? (Notice the number of triangles we can form is 2 less than the number of sides. For a pentagon it was 5โˆ’2=3 triangles)

7 Examples What is the total interior, and therefore, each interior angle, of each of these regular polygons? 1 2 Fro Tip: Memorise these for pentagon, hexagon and octagon. 70ยฐ Determine ๐‘ฅ. ๐‘ฅ Regular Polygon Total interior Each Interior Angle Equilateral triangle (3) ๐Ÿ๐Ÿ–๐ŸŽร—๐Ÿ=๐Ÿ๐Ÿ–๐ŸŽยฐ ๐Ÿ๐Ÿ–๐ŸŽยฐรท๐Ÿ‘=๐Ÿ”๐ŸŽยฐ Square (4) ๐Ÿ๐Ÿ–๐ŸŽร—๐Ÿ=๐Ÿ‘๐Ÿ”๐ŸŽยฐ ๐Ÿ‘๐Ÿ”๐ŸŽยฐรท๐Ÿ’=๐Ÿ—๐ŸŽยฐ Pentagon (5) ๐Ÿ๐Ÿ–๐ŸŽร—๐Ÿ‘=๐Ÿ“๐Ÿ’๐ŸŽยฐ ๐Ÿ“๐Ÿ’๐ŸŽยฐรท๐Ÿ“=๐Ÿ๐ŸŽ๐Ÿ–ยฐ Hexagon (6) ๐Ÿ๐Ÿ–๐ŸŽร—๐Ÿ’=๐Ÿ•๐Ÿ๐ŸŽยฐ ๐Ÿ•๐Ÿ๐ŸŽยฐรท๐Ÿ”=๐Ÿ๐Ÿ๐ŸŽยฐ Octagon (8) ๐Ÿ๐Ÿ–๐ŸŽร—๐Ÿ”=๐Ÿ๐ŸŽ๐Ÿ–๐ŸŽยฐ ๐Ÿ๐ŸŽ๐Ÿ–๐ŸŽยฐรท๐Ÿ–=๐Ÿ๐Ÿ‘๐Ÿ“ยฐ Nonagon (9) ๐Ÿ๐Ÿ–๐ŸŽร—๐Ÿ•=๐Ÿ๐Ÿ๐Ÿ”๐ŸŽยฐ ๐Ÿ๐Ÿ๐Ÿ”๐ŸŽยฐรท๐Ÿ—=๐Ÿ๐Ÿ’๐ŸŽยฐ Decagon (10) ๐Ÿ๐Ÿ–๐ŸŽร—๐Ÿ–=๐Ÿ๐Ÿ’๐Ÿ’๐ŸŽยฐ ๐Ÿ๐Ÿ’๐Ÿ’๐ŸŽยฐรท๐Ÿ๐ŸŽ=๐Ÿ๐Ÿ’๐Ÿ’ยฐ 60ยฐ 110ยฐ 130ยฐ ? 6-sided so total interior angle: ๐Ÿ๐Ÿ–๐ŸŽร— ๐Ÿ”โˆ’๐Ÿ =๐Ÿ•๐Ÿ๐ŸŽยฐ ๐’™=๐Ÿ•๐Ÿ๐ŸŽโˆ’ ๐Ÿ—๐ŸŽ+๐Ÿ๐Ÿ๐ŸŽ+๐Ÿ๐Ÿ‘๐ŸŽ+๐Ÿ”๐ŸŽ+๐Ÿ•๐ŸŽ =๐Ÿ•๐Ÿ๐ŸŽยฐโˆ’๐Ÿ’๐Ÿ”๐ŸŽยฐ =๐Ÿ๐Ÿ”๐ŸŽยฐ ? ? ? ? ? ? ? ? ? ?

8 Test Your Understanding
A regular dodecagon (12 sides). 160ยฐ ๐‘ฅ ๐‘ฅ 130ยฐ 80ยฐ 120ยฐ ๐‘ฅ=140ยฐ ? ๐‘ฅ=144ยฐ ? 120ยฐ ๐‘ฅ 40ยฐ 100ยฐ ๐‘ฅ=240ยฐ ? 40ยฐ

9 Exercise 1 ? ? ? ? ? ? (On supplied sheet) 1a b c b = 260๏‚ฐ x = 75๏‚ฐ
f a = 100๏‚ฐ ? x = 222๏‚ฐ ? x = 309๏‚ฐ ?

10 Exercise 1 ? ? ? ? ? g h i x = 120๏‚ฐ x = 252๏‚ฐ x = 54๏‚ฐ
The total of the interior angles of a polygon is 1260ยฐ. How many sides does it have? The interior angle of a regular polygon is 179ยฐ. How many sides does it have? 2 ? N1 ?

11 Exercise 1 [UKMT] If a ๐‘›-sided polygon has exactly 3 obtuse angles (i.e. 90ยฐ<๐œƒ<180ยฐ), the rest acute or right-angled, then determine the possible values of ๐‘› (Hint: determine the possible range for the sum of the interior angles, and use these inequalities to solve). N2 ?

12 Exterior Angles NO YES NO ? ? ? ?
An exterior angle of a polygon is an angle between the line extended from one side, and an adjacent side. ! At any vertex (i.e. corner): ๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘–๐‘œ๐‘Ÿ+๐‘’๐‘ฅ๐‘ก๐‘’๐‘Ÿ๐‘–๐‘œ๐‘Ÿ=180ยฐ exterior interior ? Which of these are exterior angles of the polygon? ? ? NO YES ? NO

13 Click to Start Damonimation
Total Exterior Angle To defeat Kim Jon Il, Matt Damon must encircle his pentagonal palace. What angle does Matt Damon turn in total? He does one full rotation, soโ€ฆ ๐Ÿ‘๐Ÿ”๐ŸŽยฐ ? ! The sum of the exterior angles of any polygon is 360ยฐ. Click to Start Damonimation

14 Each Exterior/Interior Angle of Regular Polygon
If the pentagon is regular, then each exterior angles is the same. Therefore: Each exterior angle of pentagon ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ“ =๐Ÿ•๐Ÿยฐ Each interior angle of pentagon =๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ•๐Ÿ=๐Ÿ๐ŸŽ๐Ÿ–ยฐ ? ๐Ÿ๐ŸŽ๐Ÿ–ยฐ ๐Ÿ•๐Ÿยฐ ? This therefore gives us an alternative way to work out the interior angle of a regular polygon: Previous method: Total interior =180 5โˆ’2 =540ยฐ Each interior: 540รท5=108ยฐ Alternative method: Each exterior: =360รท5=72ยฐ Each interior: =180โˆ’72=108ยฐ

15 Practicing Angles in Regular Polygons
Sides Exterior Angle Interior Angle 3 Equilateral Triangle 360 3 =120ยฐ 180โˆ’120=60ยฐ 4 Square 360 4 =90ยฐ 180โˆ’90=90ยฐ 5 Pentagon 360 5 =72ยฐ 180โˆ’72=108ยฐ 6 Hexagon 360 6 =60ยฐ 180โˆ’60=120ยฐ 8 Octagon 360 8 =45ยฐ 180โˆ’45=135ยฐ 9 Nonagon 360 9 =40ยฐ 180โˆ’40=140ยฐ 10 Decagon =36ยฐ 180โˆ’36=144ยฐ ? ? ? ? ? ? ? ? ? ? ? ? ? ? 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 179ยฐ. ?

16 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. Key formulae: Total exterior angle =360ยฐ Interior + Exterior =180ยฐ The exterior angle of a regular polygon is 15ยฐ. How many sides does it have? ? ๐Ÿ‘๐Ÿ”๐ŸŽรท๐’=๐Ÿ๐Ÿ“ยฐ Therefore ๐’=๐Ÿ‘๐Ÿ”๐ŸŽรท๐Ÿ๐Ÿ“=๐Ÿ๐Ÿ’ sides interior exterior The interior angle of a regular polygon is 144ยฐ. How many sides does it have? ? Exterior angle =๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ’๐Ÿ’=๐Ÿ‘๐Ÿ”ยฐ ๐’=๐Ÿ‘๐Ÿ”๐ŸŽยฐรท๐Ÿ‘๐Ÿ”ยฐ=๐Ÿ๐ŸŽ sides

17 Quickfire Questions ? ? ? ? ? Exterior angle =18ยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ๐Ÿ– =๐Ÿ๐ŸŽ
Your teacher will fire these questions at a variety of you โ€“ try to do in your head if you can! Key formulae: Total exterior angle =360ยฐ Interior + Exterior =180ยฐ Exterior angle =18ยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ๐Ÿ– =๐Ÿ๐ŸŽ ? interior exterior Interior angle =160ยฐ Ext =๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ”๐ŸŽ=๐Ÿ๐ŸŽยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ๐ŸŽ =๐Ÿ๐Ÿ– ? Exterior angle =45ยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ’๐Ÿ“ =๐Ÿ– ? Ext =๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ•๐Ÿ“=๐Ÿ“ยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ“ =๐Ÿ•๐Ÿ Interior angle =175ยฐ ? Ext =๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ“๐ŸŽ=๐Ÿ‘๐ŸŽยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ‘๐ŸŽ =๐Ÿ๐Ÿ Interior angle =150ยฐ ?

18 Test Your Understanding
Determine the exterior angle of a 40-sided regular polygon. ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ’๐ŸŽ =๐Ÿ—ยฐ 1 4 50ยฐ 85ยฐ 75ยฐ ? ๐‘=70ยฐ ? 80ยฐ ๐‘ Determine the interior angle of a 15-sided regular polygon. ๐‘ฌ๐’™๐’•= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ๐Ÿ“ =๐Ÿ๐Ÿ’ยฐ ๐‘ฐ๐’๐’•=๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ’=๐Ÿ๐Ÿ“๐Ÿ”ยฐ 2 5 ? ๐‘ฅ 3 The interior angle of a regular polygon is 178ยฐ. Determine how many sides it has. ๐‘ฌ๐’™๐’•=๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ•๐Ÿ–=๐Ÿยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ =๐Ÿ๐Ÿ–๐ŸŽ sides [Edexcel] The diagram shows a regular hexagon and a regular octagon. Calculate the size of the angle marked ๐‘ฅ. You must show all your working. Angles around a point sum to ๐Ÿ‘๐Ÿ”๐ŸŽยฐ. ๐’™=๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ๐Ÿ๐ŸŽโˆ’๐Ÿ๐Ÿ‘๐Ÿ“=๐Ÿ๐ŸŽ๐Ÿ“ยฐ ? ?

19 Exercise 2 4 Determine how many sides a regular polygon with the following exterior angle would have: 30๏‚ฐ sides 45๏‚ฐ 8 sides 12๏‚ฐ 30 sides 9๏‚ฐ 40 sides [Edexcel IGCSE Jan2014(R)-3H Q3b] The diagram shows a regular 6-sided polygon. Work out the value ofย ๐‘ฆ. ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ” =๐Ÿ”๐ŸŽยฐ 1 ? ? ? ? ? Determine how many sides a regular polygon with the following interior angle would have: 156๏‚ฐ 15 sides 162๏‚ฐ 20 sides 144๏‚ฐ 10 sides 175๏‚ฐ 72 sides [Edexcel IGCSE Jan2014(R)-3H Q3a] The diagram shows a regular 5-sided polygon. Work out the value of ๐‘ฅ. ๐Ÿ๐Ÿ–๐ŸŽโˆ’ ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ“ =๐Ÿ๐ŸŽ๐Ÿ–ยฐ 5 2 ? ? ? ? ? 3 6 ๐’‚=๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ—๐ŸŽโˆ’๐Ÿ–๐ŸŽโˆ’๐Ÿ–๐ŸŽ=๐Ÿ๐ŸŽ๐ŸŽยฐ ? [Edexcel GCSE Mar13-1H Q13] The diagram shows a square and 4 regular pentagons. Work out the size of the angle marked ๐‘ฅ. ๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ—๐ŸŽโˆ’๐Ÿ๐ŸŽ๐Ÿ–โˆ’๐Ÿ๐ŸŽ๐Ÿ– =๐Ÿ“๐Ÿ’ยฐ ๐‘ฅ ๐‘Ž 80ยฐ 80ยฐ ?

20 Exercise 2 [JMO 2007 A6] The sizes in degrees of the interior angles of a pentagon are consecutive whole numbers. What is the size of the largest of these angles? Total interior angle of pentagon: ๐Ÿ๐Ÿ–๐ŸŽ ๐Ÿ“โˆ’๐Ÿ =๐Ÿ“๐Ÿ’๐ŸŽยฐ ๐Ÿ“๐Ÿ’๐ŸŽ ๐Ÿ“ =๐Ÿ๐ŸŽ๐Ÿ–ยฐ If we space out angles evenly around ๐Ÿ๐ŸŽ๐Ÿ–ยฐ, they will still add to ๐Ÿ“๐Ÿ’๐ŸŽยฐ: ๐Ÿ๐ŸŽ๐Ÿ”,๐Ÿ๐ŸŽ๐Ÿ•,๐Ÿ๐ŸŽ๐Ÿ–,๐Ÿ๐ŸŽ๐Ÿ—,๐Ÿ๐Ÿ๐ŸŽ So largest angle is ๐Ÿ๐Ÿ๐ŸŽยฐ. 9 7 [Edexcel IGCSE May2015(R)-3H Q2b] The diagram shows 3 identical regular pentagons. Work out the value of ๐‘ฆ. ๐Ÿ‘๐Ÿ”๐ŸŽโˆ’ ๐Ÿ‘ร—๐Ÿ๐ŸŽ๐Ÿ– =๐Ÿ‘๐Ÿ”ยฐ ? ? 8 [Edexcel June2007-5Hย Q15] ๐ด๐ต๐ถ๐ท๐ธ๐น is a regular hexagon and ๐ด๐ต๐‘„๐‘ƒ is a square. Angle ๐ถ๐ต๐‘„=๐‘ฅยฐ. Work out the value ofย ๐‘ฅยฐ. ๐’™=๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ๐Ÿ๐ŸŽโˆ’๐Ÿ—๐ŸŽ=๐Ÿ๐Ÿ“๐ŸŽยฐ [IMC 2015 Q9]ย What is the value ofย ๐‘+๐‘ž+๐‘Ÿ+๐‘ +๐‘ก+๐‘ข+๐‘ฃ+๐‘ค+๐‘ฅ+๐‘ฆ in the diagram? We have two lots of exterior angles: ๐Ÿ‘๐Ÿ”๐ŸŽร—๐Ÿ=๐Ÿ•๐Ÿ๐ŸŽยฐ 10 ? ?

21 Problem Solving with Interior/Exterior Angles
There are variety of skills that harder questions involving interior/exterior angles might involve: #1: Tessellation #2: Using isosceles triangles Shapes โ€˜tessellateโ€™ if they fit together, without overlap, to form a repeating pattern. B ๐ด ๐ต C C A A ๐ท ๐ถ B B B B A A C ๐ธ ๐ป B B ๐น ๐บ โ€œThe above repeating pattern consists of three regular polygons, A (hexagon), B (square) and C. Determine how many sides C has.โ€ โ€œ๐ด๐ต๐ถ๐ท is a square and ๐ถ๐ท๐ธ๐น๐บ๐ป is a regular hexagon. Determine the angle ๐ถ๐ต๐ป.โ€

22 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. [IMC 2006 Q19] The diagram shows a regular pentagon and a regular hexagon which overlap. What is the value ofย ๐‘ฅ?

23 #1 :: Tessellation Key point: If shapes fit together, then the interior angles around any point where the shapes meet add to 360ยฐ. [Edexcel GCSE Nov2012-1H Q18] 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. This angle: ๐Ÿ”๐ŸŽยฐ (interior angle of equilateral triangle) Therefore interior angle of tile A: ๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ”๐ŸŽ ๐Ÿ =๐Ÿ๐Ÿ“๐ŸŽยฐ Exterior angle =๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ“๐ŸŽ=๐Ÿ‘๐ŸŽยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ‘๐ŸŽ =๐Ÿ๐Ÿ sides ? ? Weโ€™ve recently seen a method for finding the number of sides of a regular polygon given its interior angle! ? ?

24 Further Example ? B A C Interior angle of ๐‘จ: ๐Ÿ๐Ÿ–๐ŸŽโˆ’ ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ” =๐Ÿ๐Ÿ๐ŸŽยฐ
The above repeating pattern consists of three regular polygons, A (hexagon), B (square) and C. Determine how many sides C has. ๐Ÿ๐Ÿ๐ŸŽยฐ ๐Ÿ—๐ŸŽยฐ ๐Ÿ๐Ÿ“๐ŸŽยฐ ? Interior angle of ๐‘จ: ๐Ÿ๐Ÿ–๐ŸŽโˆ’ ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ” =๐Ÿ๐Ÿ๐ŸŽยฐ Therefore interior angle of ๐‘ช=๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ—๐ŸŽโˆ’๐Ÿ๐Ÿ๐ŸŽ=๐Ÿ๐Ÿ“๐ŸŽยฐ Exterior angle of ๐‘ช=๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ“๐ŸŽ=๐Ÿ‘๐ŸŽยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ‘๐ŸŽ =๐Ÿ๐Ÿ

25 Test Your Understanding
? The diagram shows 4 congruent regular pentagons that form the sides of an ๐‘›-sided regular polygon. Determine the value of ๐‘›. ๐Ÿ๐ŸŽ๐Ÿ–ยฐ ๐Ÿ๐ŸŽ๐Ÿ–ยฐ ๐Ÿ๐Ÿ’๐Ÿ’ยฐ ? Interior angle of pentagon: ๐Ÿ๐Ÿ–๐ŸŽโˆ’ ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ“ =๐Ÿ๐ŸŽ๐Ÿ–ยฐ Interior angle of large polygon: ๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ๐ŸŽ๐Ÿ–โˆ’๐Ÿ๐ŸŽ๐Ÿ–=๐Ÿ๐Ÿ’๐Ÿ’ยฐ Exterior angle of large polygon: ๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ’๐Ÿ’=๐Ÿ‘๐Ÿ”ยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ‘๐Ÿ”

26 #2 :: Irregular Polygons within Regular Polygons
[Edexcel GCSE June2016-2H Q12] The diagram shows a regular pentagon. ๐ด๐ต and ๐ถ๐ท are two of the lines of symmetry of the pentagon. Work out the size of the angle marked ๐‘ฅ. 90ยฐ ? ? 90ยฐ 108ยฐ ? ? 108ยฐ Total interior angle of this irregular polygon: ๐Ÿ๐Ÿ–๐ŸŽ ๐Ÿ“โˆ’๐Ÿ =๐Ÿ“๐Ÿ’๐ŸŽยฐ ๐’™=๐Ÿ“๐Ÿ’๐ŸŽโˆ’ ๐Ÿ—๐ŸŽ+๐Ÿ—๐ŸŽ+๐Ÿ๐ŸŽ๐Ÿ–+๐Ÿ๐ŸŽ๐Ÿ– =๐Ÿ๐Ÿ’๐Ÿ’ยฐ ? One method: Within the regular polygon is an irregular polygon for which we can determine each of the angles. Recall that total interior of ๐‘›-sided polygon: 180 ๐‘›โˆ’2 ?

27 Test Your Understanding
[IMC 2006 Q19] The diagram shows a regular pentagon and a regular hexagon which overlap. What is the value ofย ๐‘ฅ? The central shape is a pentagon. Its total interior angle =๐Ÿ“๐Ÿ’๐ŸŽยฐ ๐’™=๐Ÿ“๐Ÿ’๐ŸŽโˆ’๐Ÿ๐ŸŽ๐Ÿ–โˆ’๐Ÿ๐ŸŽ๐Ÿ–โˆ’๐Ÿ๐Ÿ๐ŸŽโˆ’๐Ÿ๐Ÿ๐ŸŽ =๐Ÿ–๐Ÿ’ยฐ ? 108ยฐ ? 120ยฐ ? ? 108ยฐ ? 120ยฐ ?

28 #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. ๐ด ๐ต ๐ถ ๐ท ๐ธ ๐น ๐บ ๐ป ? Since ๐‘ช๐‘ซ๐‘ฌ๐‘ญ๐‘ฎ๐‘ฏ is regular, ๐‘ช๐‘ซ=๐‘ช๐‘ฏ and since ๐‘จ๐‘ฉ๐‘ช๐‘ซ is regular, ๐‘ช๐‘ซ=๐‘ฉ๐‘ช. Therefore ๐‘ฉ๐‘ช=๐‘ช๐‘ฏ and so triangle ๐‘ฉ๐‘ช๐‘ฏ is isosceles. โˆ ๐‘ฉ๐‘ช๐‘ฏ=๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ—๐ŸŽโˆ’๐Ÿ๐Ÿ๐ŸŽ=๐Ÿ๐Ÿ“๐ŸŽยฐ (angles around point ๐‘ช sum to ๐Ÿ‘๐Ÿ”๐ŸŽยฐ) Therefore โˆ ๐‘ช๐‘ฉ๐‘ฏ= ๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ“๐ŸŽ ๐Ÿ =๐Ÿ๐Ÿ“ยฐ โ€œ๐ด๐ต๐ถ๐ท is a square and ๐ถ๐ท๐ธ๐น๐บ๐ป is a regular hexagon. Determine the angle ๐ถ๐ต๐ป.โ€

29 Test Your Understanding
[IMC 2009 Q12] The diagram shows a square inside a regular hexagon. What is the size of the marked angle atย ๐‘‹? ? ๐‘จ๐‘ฉ๐‘ฟ is isosceles (as with previous question) โˆ ๐‘ช๐‘ฉ๐‘ฟ=๐Ÿ๐ŸŽ๐Ÿ–ยฐ โˆดโˆ ๐‘จ๐‘ฉ๐‘ฟ=๐Ÿ๐ŸŽ๐Ÿ–โˆ’๐Ÿ—๐ŸŽ=๐Ÿ๐Ÿ–ยฐ As ๐‘จ๐‘ฉ๐‘ฟ is isosceles: โˆ ๐‘จ๐‘ฟ๐‘ฉ= ๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ– ๐Ÿ =๐Ÿ–๐Ÿยฐ ๐ด 81ยฐ ๐‘‹ 81ยฐ 108ยฐ 18ยฐ 90ยฐ ๐ถ ๐ต

30 Exercise 3 ? ? ? ? 3 1 4 2 [Edexcel IGCSE Nov-2010-4H Q13]
The size of each interior angle of a regular polygon is 11 times the size of each exterior angle. Work out the number of sides the polygon has. Interior : Exterior = 11 : 1. Exterior = ๐Ÿ๐Ÿ–๐ŸŽยฐรท๐Ÿ๐Ÿยฐ=๐Ÿ๐Ÿ“ โ†’ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ๐Ÿ“ =๐Ÿ๐Ÿ’ [KS3 SATs 2004 L6-L8 Paper 2 Q19 Edited] A pupil has three tiles. One is a regular octagon, one is a regular hexagon, and one is a square. The side length of each tile is the same. The pupil says the hexagon will fit exactly like this. Is the pupil correct? No, as ๐Ÿ๐Ÿ‘๐Ÿ“+๐Ÿ—๐ŸŽ+๐Ÿ๐Ÿ๐ŸŽ=๐Ÿ‘๐Ÿ’๐Ÿ“ยฐ which is not ๐Ÿ‘๐Ÿ”๐ŸŽยฐ. 1 ? 4 ? 2 [Edexcel GCSE Nov2014-1H Q17] ABCDEFGHย is a regular octagon. BCKFGJย is a hexagon. JK is a line of symmetry of the hexagon. Angleย ๐ต๐ฝ๐บ= angle ๐ถ๐พ๐น=140ยฐ. Work out the size of angleย KFE. Total interior angle of ๐‘ฉ๐‘ช๐‘ฒ๐‘ญ๐‘ฎ๐‘ฑ=๐Ÿ๐Ÿ–๐ŸŽ ๐Ÿ”โˆ’๐Ÿ =๐Ÿ•๐Ÿ๐ŸŽยฐ. โˆ ๐‘ฒ๐‘ญ๐‘ฎ= ๐Ÿ•๐Ÿ๐ŸŽโˆ’๐Ÿ๐Ÿ’๐ŸŽโˆ’๐Ÿ๐Ÿ’๐ŸŽ ๐Ÿ’ =๐Ÿ๐Ÿ๐ŸŽยฐ โ†’ โˆ ๐‘ฒ๐‘ญ๐‘ฌ=๐Ÿ๐Ÿ‘๐Ÿ“โˆ’๐Ÿ๐Ÿ๐ŸŽ=๐Ÿ๐Ÿ“ยฐ [Edexcel IGCSE Nov2009-3H Q3a] The diagram shows a regular octagon, with centre O. Work out the value of ๐‘ฅ. Half an interior angle of octagon: ๐Ÿ๐Ÿ‘๐Ÿ“ ๐Ÿ =๐Ÿ”๐Ÿ•.๐Ÿ“ยฐ ? ?

31 Exercise 3 5 A regular polygon ๐ด is surrounded by squares and equilateral triangles in an alternating pattern, as shown. Show that ๐ด is a hexagon. Interior angle of ๐‘จ=๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ—๐ŸŽโˆ’๐Ÿ—๐ŸŽโˆ’๐Ÿ”๐ŸŽ=๐Ÿ๐Ÿ๐ŸŽยฐ Exterior angle =๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ๐ŸŽ=๐Ÿ”๐ŸŽยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ”๐ŸŽ =๐Ÿ” so a hexagon. ? ๐ด A regular polygon ๐ต with ๐‘› sides is surrounded by squares and regular pentagons in an alternating pattern, as shown. Determine the value of ๐‘›. Interior angle of ๐‘ฉ=๐Ÿ‘๐Ÿ”๐ŸŽโˆ’๐Ÿ—๐ŸŽโˆ’๐Ÿ๐ŸŽ๐Ÿ–=๐Ÿ๐Ÿ”๐Ÿยฐ Exterior angle =๐Ÿ๐Ÿ–๐ŸŽโˆ’๐Ÿ๐Ÿ”๐Ÿ=๐Ÿ๐Ÿ–ยฐ ๐’= ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ๐Ÿ– =๐Ÿ๐ŸŽ sides 6 ? ๐ต

32 Exercise 3 9 [IMC 2005 Q14] Ten stones, of identical shape and size, are used to make an arch, as shown in the diagram. Each stone has a cross-section in the shape of a trapezium with three equal sides. What is the size of the smallest angles of the trapezium? 7 [IMC 2003 Q22] The diagram shows a regular dodecagon (a polygon with twelve equal sides and equal angles). What is the size of the marked angle? ๐Ÿ•๐Ÿ“ยฐ ? [JMO 2014 B1] The figure shows an equilateral triangleย ABC, a squareย BCDE, and a regular pentagonย BEFGH. What is the difference between the sizes ofย โˆ ADEย andย โˆ AHE? 8 ? ๐Ÿ—๐Ÿ—ยฐ 10 [IMC 2018 Q18]ย The diagram shows a regular pentagon and an equilateral triangle placed inside a square. What is the value ofย ๐‘ฅ? ๐Ÿ๐Ÿ’ยฐ (e.g. by first considering the angles in the bottom irregular pentagon) ? ๐ŸŽยฐ ?

33 Exercise 2 11 Find all regular polygons which tessellate (when restricted only to one type of polygon). Equilateral triangle, square, hexagon. By thinking about interior angles, prove that the regular polygons you identified above are the only regular polygons which tessellate. Method 1: The possible exterior angles of a regular polygons are the factors of 360 less than 180: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 This gives interior angles of 179, 178, ..., 140, 135, 120, 108, 90, 60. To tessellate, the interior angle has to divide 360. Only 120, 90 and 60 does. This corresponds to a hexagon, square and equilateral triangle. Method 2: 360 divided by the interior angle must give a whole number, in order for the regular polygon to tessellate. Interior angle is ๐Ÿ๐Ÿ–๐ŸŽโˆ’ ๐Ÿ‘๐Ÿ”๐ŸŽ ๐’ , so ๐Ÿ‘๐Ÿ”๐ŸŽ ๐Ÿ๐Ÿ–๐ŸŽโˆ’ ๐Ÿ‘๐Ÿ”๐ŸŽ ๐’ =๐’Œ for some constant ๐’Œ. Simplifying this gives ๐’Œ๐’โˆ’๐Ÿ๐’Œโˆ’๐Ÿ๐’=๐ŸŽ This factorises to ๐’Œโˆ’๐Ÿ ๐’โˆ’๐Ÿ =๐Ÿ’ This only numbers which multiply to give 4 are ๐Ÿร—๐Ÿ’ or ๐Ÿร—๐Ÿ or ๐Ÿ’ร—๐Ÿ. This ๐’=๐Ÿ”,๐Ÿ’ or ๐Ÿ‘ in each case. ? N ?

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

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

36 What is the total interior angle of a 20 sided polygon?
360 3600 3240 6480

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

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

39 Determine the angle . 61๏‚ฐ 29๏‚ฐ 105๏‚ฐ 120๏‚ฐ 215๏‚ฐ 223๏‚ฐ 225๏‚ฐ 235๏‚ฐ


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