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.

STARTER: Identifying 2D polygons

!Sides:

3 Triangle

ScaleneIsoscelesEquilateral

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

? ? ?

? ? ?

? ? ? ????

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A polygon is a 2D shape with straight sides.?

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

?

?

?

?

?

?

?

?

?

?

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Properties of quadrilaterals

x = 100° y = 80°?

50°x

x = 130°100°

x

y

??

x

60°x = 120°?

95° 55°x

x = 105°?

Kite

Parallelogram

RECAP: Interior angles of quadrilateral

1 2

3 4Trapezium

The interior angles of a quadrilateral add up to 360.?

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

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)?

Sum of interior angles

Click to Bromanimate

130°120° 80°

160°x

x = 140°?

A regular decagon (10 sides).

x

x = 144°?

Test Your Understanding

120°100°40°

40° x

x = 240°?

Exercise 1

x = 75 x = 25

x = 222

x = 309

? ?

??

1a b c

de f

Exercise 1

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

?

?

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

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

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°.

?

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°

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?

Angles in Regular Polygons

Num Sides Name of Regular Polygon

Exterior Angle

Interior Angle

3 Triangle 120° 60°

4 Quadrilateral 90° 90°

5 Pentagon 72° 108°

6 Hexagon 60° 120°

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°.

? ?? ?? ?? ?? ?? ?? ?? ?

?

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° ?

Test Your UnderstandingGCSE 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?

GCSE Question

Test Your Understanding

80

80

a

50

7585

80c

The interior angle of a regular polygon is 165. How many sides does it have?Interior angle = 180 – 165 = 15n = 360 ÷ 15 = 24

Alternative method:Total interior angle = 165nThen 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°?

Exercise 2Determine how many sides a regular polygon with the following exterior angle would have:

30 12 sides45 8 sides12 30 sides9 40 sides

Determine how many sides a regular polygon with the following interior angle would have:

156 15 sides162 20 sides144 10 sides175 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) = 120Interior angle of octagon: 180 – (360/8) = 135x = 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 = 150Exterior angle = 30Sides = 360/30 = 12

Q1

Q2

Q3

Q4

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Exercise 2A 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 = 150n = 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, 120This 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 = 0This factorises to (k – 2)(n – 2) = 4This 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

?

?

?

A B C D

TEST YOUR UNDERSTANDINGVote with your diaries!

360 360n 360n 180(n-2)

What is the total exterior angle of a polygon in terms of the number of

sides n?

360 3600 3240 6480

What is the total interior angle of a 20 sided polygon?

20 40 90 180

The interior angle of a polygon is 178. How many sides does it have?

172 176 178 179

What is the interior angle of a 90 sided regular polygon?

215 223 225 235

Determine the angle .

61

10529

120