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Year 8: Geometric Reasoning

Year 8: Geometric Reasoning. Dr J Frost (jfrost@tiffin.kingston.sch.uk). Objectives: Be able to reason about sides and angles, and find interior/exterior angles of polygons. Last modified: 12 th April 2014. STARTER : Identifying 2D polygons. !.

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Year 8: Geometric Reasoning

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  1. Year 8: Geometric Reasoning Dr J Frost (jfrost@tiffin.kingston.sch.uk) Objectives: Be able to reason about sides and angles, and find interior/exterior angles of polygons. Last modified: 12th April 2014

  2. 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 ?

  3. Properties of quadrilaterals Shape Name Lines of symmetry Num pairs of parallel sides Diagonals always equal? Diagonals perpen- dicular? Square Rectangle Kite Rhombus Parallelogram Arrowhead ? 4 2 1 2 0 1 ? 2 2 0 2 2 0 Yes Yes No No No No ? ? Yes No Yes Yes No Yes ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

  4. 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° ?

  5. Sum of interior angles n = 3 n = 4 Total of interior angles = 360° Total of interior angles = 180° Can you guess what the angles add up to in a pentagon? How would you prove it?

  6. 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) ?

  7. 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°

  8. Exercise 1 1a b c x = 75 ? x = 25 ? d e f x = 222 ? x = 309 ?

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

  10. 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 ?

  11. 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

  12. 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

  13. 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° ? ?

  14. Angles in Regular Polygons ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 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°. ?

  15. 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° ?

  16. 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 ?

  17. 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 ? ?

  18. 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 ? ? ? ?

  19. 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 ?

  20. TEST YOUR UNDERSTANDING Vote with your diaries! A B C D

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

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

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

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

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

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