Year 8: Geometric Reasoning Dr J Frost ([email protected]) Objectives: Be able to reason about sides and angles, and find interior/exterior angles of polygons. Last modified: 12th April 2014 STARTER: Identifying 2D polygons ! A polygon is a 2D shape with?straight sides. Sides: 3 Triangle ? Equilateral 4 Quadrilateral ? ? Square

? Parallelogram 5 6 7 Pentagon ? Hexagon ? Heptagon ? 8 9 10 ? ? Isosceles Scalene ?

Rectangle ? Trapezium Octagon ? Nonagon ? Decagon ? ? Rhombus ? Arrowhead 12 20 Dodecagon ? Kite

? Icosagon ? Properties of quadrilaterals Shape Name Lines of Num pairs of symmetry parallel sides Diagonals always equal? Diagonals perpendicular? Square 4 ? 2

? Yes? Yes? Rectangle 2 ? 2 ? Yes? No ? Kite 1 ? 0 ?

No ? Yes? Rhombus 2 ? 2 ? No ? Yes? Parallelogram 0 ? 2 ?

No ? No ? Arrowhead 1 ? 0 ? No ? Yes? RECAP: Interior angles of quadrilateral The interior angles of a quadrilateral add up to 360. ? 1 Parallelogram 2 y

100 x x 50 x = 130 ? 3 ? x = 100 Trapezium x x = 120 ? ? y = 80 4 Kite x 95

55 60 x = 105 ? Sum of interior angles n=3 Total of interior angles = 180 n=4 Total of interior angles = 360 Can you guess what the angles add up to in a pentagon? How would you prove it? Sum of interior angles Click to Bromanimate We can cut a pentagon into three triangles. The sum of the interior angles of the triangles is:

3 x 180 = 540 ! For an n-sided shape, the sum of the interior angles is: 180(n-2) ? Test Your Understanding A regular decagon (10 sides). 160 x x 80 130 120 x = 140 ? x = 144 ? 120 40 x = 240 ?

x 100 40 Exercise 1 1a b x = 75? c x = 25? d e f x = 222 ? x = 309 ? Exercise 1 g

h x = 54? 2 i x = 120 ? x = 252 ? The total of the interior angles of a polygon is . How many sides does it have? ? N1 The interior angle of a regular polygon is . How many sides does it have? ? Exercise 1 N2 If a n-sided polygon has exactly 3 obtuse angles (i.e. 90 < < 180), then
determine the possible values of (Hint: determine the possible range for the sum of the interior angles, and use these inequalities to solve). ? Interior Angles An exterior angle of a polygon is an angle between the line extended from one side, and an adjacent side. Which of these are exterior angles of the polygon? ? NO ? YES NO? Interior Angles To defeat Kim Jon Il, Matt Damon must encircle his pentagonal palace. What angle does Matt Damon turn in total? 360 ? ! The sum of the exterior angles of

any polygon is 360. Click to Start Damonimation Interior Angles If the pentagon is regular, then all the exterior angles are clearly the same. Therefore: Exterior angle of pentagon = 360 / 5 = 72 ? Interior angle of pentagon = 180 72 = 108 ? Angles in Regular Polygons Num Sides Name of Exterior Regular Polygon Angle Interior Angle 3

Triangle 60 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10

Decagon 120 ? 90 ? 72 ? 60 ? 51.4 ? 45 ? 40 ? 36 ? ? 90 ? 108 ? 120 ? 128.6? 135 ? 140 ? 144 ?

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. Test Your Understanding GCSE question The diagram shows a regular hexagon and a regular octagon. Calculate the size of the angle marked x. You must show all your working. x = 105 ? GCSE Question Hint: Fill in what angles you do know. You can work out what the interior angle of Tile A will be. Question: 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. Sides = 12 ?

Test Your Understanding Q1 Q2 50 85 a 75 80 80 80 c a = 110 ? Q3 What is the exterior angle of a 180-sided regular polygon? 360 180 ? = 2

c = 70? Q4 The interior angle of a regular polygon is 165. How many sides does it have? Interior angle = 180 165 = 15 n = 360 15 = 24 ? Alternative method: Total interior angle = 165n Then solve 180(n 2) = 165n Exercise 2 Q1 Determine how many sides a regular Q3 polygon with the following exterior angle would have: 30 12 sides 45 8 sides 12 30 sides

9 40 sides ? ? ? ? The diagram shows a regular hexagon and a regular octagon. Calculate the size of the angle marked. You must show all your working. Interior angle of hexagon: 180 (360/6) = 120 Interior angle of octagon: 180 (360/8) = 135 x = 360 120 135 = 105 Q2 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 ? ?

? ? ? Q4 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. Interior angle of A = (360 60)/2 = 150 Exterior angle = 30 Sides = 360/30 = 12 ? Exercise 2 Q5 A regular polygon is surrounded by squares and regular hexagons, alternating between the two. How many sides does this shape have?

Q6 ? Equilateral triangle, square, hexagon. N Interior angle = 360 90 120 = 150 n = 360 / 30 = 12 sides ? Find all regular polygons which tessellate (when restricted only to one type of polygon). 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 180 (360/n), so 360 / (180 (360/n)) = k for some constant k. Simplifying this gives kn 2k 2n = 0 This factorises to (k 2)(n 2) = 4 This only numbers which multiply to give 4 are 1 x 4 or 2 x 2 or 4 x 1. This n = 6, 4 or 3 in each case. TEST YOUR UNDERSTANDING 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

360 n 360n 180(n-2) What is the total interior angle of a 20 sided polygon? 360 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