Year 8: Geometric Reasoning Dr J Frost
Description: Year 8: Geometric Reasoning Dr J Frost (jfrosttiffin.kingston.sch.uk) Last modified: 12th April 2014 Objectives: Be able to reason about sides and angles, and find interiorexterior angles of polygons. ! Sides: 3 Triangle Scalene Isosceles
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slide1. Year 8: Geometric Reasoning Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 12th April 2014 Objectives: Be able to reason about sides and angles, and find interior/exterior angles of polygons.<br>
slide2. ! Sides: 3 Triangle Scalene Isosceles Equilateral 4 Quadrilateral Square Rectangle Rhombus Parallelogram Trapezium Kite 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon 20 Icosagon Arrowhead ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? A polygon is a 2D shape with straight sides. ?<br>
slide3. 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 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?<br>
slide4. x = 100° y = 80° ? 50° x x = 130° 100° x y ? ? x 60° x = 120° ? 95° 55° x x = 105° ? Kite Parallelogram 1 2 3 4 Trapezium The interior angles of a quadrilateral add up to 360. ?<br>
slide5. 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?<br>
slide6. 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) ? Click to Bromanimate<br>
slide7. 130° 120° 80° 160° x x = 140° ? A regular decagon (10 sides). x x = 144° ? 120° 100° 40° 40° x x = 240° ?<br>
slide8. x = 75 x = 25 x = 222 x = 309 ? ? ? ? 1a b c d e f<br>
slide9. x = 54 x = 120 x = 252 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? ? ? ? g h i 2 N1 ? ?<br>
slide10. 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 ?<br>
slide11. 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 ? ? ?<br>
slide12. Click to Start Damonimation 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°. ?<br>
slide13. 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° ? ?<br>
slide14. 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°. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?<br>
slide15. 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<br>
slide16. 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 ? GCSE Question<br>
slide17. 80 80 a 50 75 85 80 c 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 Q1 Q2 Q4 ? What is the exterior angle of a 180-sided regular polygon?
360 ÷ 180 = 2 Q3 ? a = 110° ? c = 70° ?<br>
slide18. 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 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 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 Q1 Q2 Q3 Q4 ? ? ? ? ? ? ? ? ? ?<br>
slide19. A regular polygon is surrounded by squares and regular hexagons, alternating between the two. How many sides does this shape have? 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).
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. Q5 Q6 N ? ? ?<br>
slide20. A B C D TEST YOUR UNDERSTANDING
Vote with your diaries!<br>
slide21. 360 360
n 360n 180(n-2) What is the total exterior angle of a polygon in terms of the number of sides n?<br>
slide22. 360 3600 3240 6480 What is the total interior angle of a 20 sided polygon?<br>
slide23. 20 40 90 180 The interior angle of a polygon is 178. How many sides does it have?<br>
slide24. 172 176 178 179 What is the interior angle of a 90 sided regular polygon?<br>
slide25. 215 223 225 235 Determine the angle . 61 105 29 120<br>
slide2. ! Sides: 3 Triangle Scalene Isosceles Equilateral 4 Quadrilateral Square Rectangle Rhombus Parallelogram Trapezium Kite 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon 20 Icosagon Arrowhead ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? A polygon is a 2D shape with straight sides. ?<br>
slide3. 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 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?<br>
slide4. x = 100° y = 80° ? 50° x x = 130° 100° x y ? ? x 60° x = 120° ? 95° 55° x x = 105° ? Kite Parallelogram 1 2 3 4 Trapezium The interior angles of a quadrilateral add up to 360. ?<br>
slide5. 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?<br>
slide6. 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) ? Click to Bromanimate<br>
slide7. 130° 120° 80° 160° x x = 140° ? A regular decagon (10 sides). x x = 144° ? 120° 100° 40° 40° x x = 240° ?<br>
slide8. x = 75 x = 25 x = 222 x = 309 ? ? ? ? 1a b c d e f<br>
slide9. x = 54 x = 120 x = 252 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? ? ? ? g h i 2 N1 ? ?<br>
slide10. 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 ?<br>
slide11. 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 ? ? ?<br>
slide12. Click to Start Damonimation 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°. ?<br>
slide13. 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° ? ?<br>
slide14. 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°. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?<br>
slide15. 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<br>
slide16. 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 ? GCSE Question<br>
slide17. 80 80 a 50 75 85 80 c 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 Q1 Q2 Q4 ? What is the exterior angle of a 180-sided regular polygon?
360 ÷ 180 = 2 Q3 ? a = 110° ? c = 70° ?<br>
slide18. 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 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 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 Q1 Q2 Q3 Q4 ? ? ? ? ? ? ? ? ? ?<br>
slide19. A regular polygon is surrounded by squares and regular hexagons, alternating between the two. How many sides does this shape have? 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).
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. Q5 Q6 N ? ? ?<br>
slide20. A B C D TEST YOUR UNDERSTANDING
Vote with your diaries!<br>
slide21. 360 360
n 360n 180(n-2) What is the total exterior angle of a polygon in terms of the number of sides n?<br>
slide22. 360 3600 3240 6480 What is the total interior angle of a 20 sided polygon?<br>
slide23. 20 40 90 180 The interior angle of a polygon is 178. How many sides does it have?<br>
slide24. 172 176 178 179 What is the interior angle of a 90 sided regular polygon?<br>
slide25. 215 223 225 235 Determine the angle . 61 105 29 120<br>