Year 8: Geometric Reasoning Dr J Frost (jfrost@tiffin.kingston.sch.uk) Objectives: Be able to classify shapes, reason about sides and angles, and find interior/exterior angles of polygons. Last modified: 13th April 2016

STARTER: Identifying 2D polygons ! A polygon is a 2D shape with straight sides. ? Sides: 3 Triangle Equilateral ? Isosceles ? Scalene ? 4 ? Square Rectangle ? Rhombus ? Quadrilateral ? Parallelogram ? Trapezium ? Kite ? Arrowhead ? 5 Pentagon ? 8 12 Octagon ? Dodecagon ? 6 Hexagon ? 9 20 Nonagon ? Icosagon ? 7 Heptagon ? 10 Decagon ?

Click to Start Bromanimation A big debate we had in the maths office… Is a square a trapezium? It depends on the definition: “A trapezium is a quadrilateral with exactly one pair of parallel sides.” or “A trapezium is a quadrilateral with at least one pair of parallel sides.” A square would be a trapezium under the second definition. It seems odd that we might consider shapes with two pairs of parallel as trapeziums (as we’d usually call them parallelograms), but otherwise we’d have the following situation… Click to Start Bromanimation Trapezium? NO YES This lack of continuity (i.e. not being a trapezium for one specific case) is considered a bad thing in maths, so the weight of evidence is that a square IS a trapezium, and thus the second definition is the correct one.

Categorising Activity Shape Description 2D Shape Polygon A 2D shape with straight edges. Ellipse (Also known as an oval) Circle Quadrilateral Polygon with four edges. Square Regular polygon with four edges. Rectangle Two pairs of parallel sides, all angles equal. Oblong All rectangles which are not squares. Trapezium Quadrilateral with at least one pair of parallel sides. Parallelogram Quadrilateral with two pairs of parallel sides. Rhombus Quadrilateral with all edges the same length. Kite Quadrilateral where sides in adjacent pairs are equal in length. Arrowhead Kite with a reflex angle. Trapezium Square We’ve seen that all squares are trapeziums. We might use the arrow to mean “is a type of”. Using a full page of your book, form a ‘tree’ of all the shape classifications on the right. You may work in pairs. ‘2D Shape’ should be at the top of your tree/page, i.e.: Bro Tip: To see if for example all squares are rectangles, see if a square satisfies the definition of a rectangle. 2D Shape Polygon

Solution 2D Shape Polygon Ellipse Quadrilateral Circle Trapezium Kite Parallelogram Arrowhead Rhombus Rectangle Square Oblong

Properties of quadrilaterals Diagonals always equal in length? Shape Name Lines of symmetry Num pairs of parallel sides Diagonals perpen- dicular? Square Rectangle Kite Rhombus Parallelogram Arrowhead ? 4 2 1 ? 2 Yes No ? Yes No ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

RECAP: Interior angles of quadrilateral The interior angles of a quadrilateral add up to 360. ? Parallelogram 1 2 y 100° x x 50° x = 130° ? x = 100° y = 80° ? ? 3 4 Trapezium Kite x x = 120° ? x 55° 95° 60° x = 105° ?

Sum of interior angles n = 3 n = 4 Total of interior angles = 360° Can you guess what the angles add up to in a pentagon? How would you prove it?

! For an n-sided shape, the sum of the interior angles is: 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° x 40° 100° x = 240° ? 40°

Exercise 1 ? ? ? ? ? ? 1a b c b = 260 x = 75 x = 25 d e f a = 100

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

Exercise 1 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). N2 ?

Interior Angles NO YES NO ? ? ? 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

Click to Start Damonimation 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 ? ? Exterior angle of pentagon = 360 / 5 = 72° 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 Regular Polygon Exterior Angle Interior Angle 3 Triangle 120° 60° 4 Quadrilateral 90° 5 Pentagon 72° 108° 6 Hexagon 7 Heptagon 51.4° 128.6° 8 Octagon 45° 135° 9 Nonagon 40° 140° 10 Decagon 36° 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 75 a 80 80 80 c a = 110° ? c = 70° ? Q3 What is the exterior angle of a 180-sided regular polygon? 360 ÷ 180 = 2 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 Q3 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 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 Q1 ? ? 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 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 A regular polygon is surrounded by squares and regular hexagons, alternating between the two. How many sides does this shape have? Q5 Q6 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 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. ? N ? Interior angle = 360 – 90 – 120 = 150 n = 360 / 30 = 12 sides ?

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