Triangles and polygons -...

1 Triangles and polygons 11

A Special triangles and quadrilaterals level 6

1 Triangles and polygons You will revise the names and properties of special triangles and quadrilaterals.

This work will help you

• use the sum of the interior angles of a triangle and of a quadrilateral

• use the interior and exterior angles of a polygon, including a regular polygon

A1 This regular hexagon has been split into two trapeziums.

Draw sketches to show how a regular hexagoncan be split into each of the following.Use triangular dotty paper if you like.

(a) Three rhombuses

(b) Six equilateral triangles

(c) Four trapeziums

(d) A kite and two isosceles triangles

(e) A rectangle and two isosceles triangles

(f) An equilateral triangle and three isosceles triangles

(g) An isosceles triangle and two trapeziums

A2 This square has been divided into two right-angled triangles and a parallelogram.

Draw sketches to show how a squarecan be split into each of the following.Use square dotty paper if you like.

(a) Two right-angled triangles

(b) Three right-angled triangles

(c) Two isosceles triangles and two trapeziums

(d) A kite and two right-angled triangles

(e) An isosceles triangle and two right-angled triangles

(f) Two trapeziums

(g) A square and four right-angled triangles

A3 A quadrilateral has rotation symmetry of order 4.

(a) What is the name of the quadrilateral?

(b) Describe some other special properties that it has.

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12 1 Triangles and polygons

105° 40°

a

43°

92°d

76°

44°

c

65°b

a b

c

38°

72° 67°

e

d

47°

g

f

B1 Work out the missing angles.

B3 Work out the angles marked with small letters.Explain how you worked out each angle.

A

C

DB

60°

100°a

H

G

E

F

b

140°

I

L K J

c

40° 70°

P

O

NM

gd

ef

20°

55°

The sum of the angles of a triangle is 180°.

B Angles of a triangle level 6

B2 Work out the missing angles in these isosceles triangles.

B4 Find the values of x and y.(y – 4)°87°

2x°x°

(4y – 1)°

(2y + 3)°







C1 You can split a quadrilateral into two triangles like this.

(a) What is the sum of the ‘black’ angles?

(b) What is the sum of the ‘white’ angles?

(c) What is the sum of the interior angles of a quadrilateral?

C2 Find the angles marked with letters.Explain how you worked out each angle.

C3 (a) Draw any polygon and record how many sides it has.

Choose a vertex. Draw straight lines from it to all the other vertices.

(b) How many triangles have you made?

(c) What is the total of all the angles in the triangles?

(d) From your answer to (c) write the total ofall the interior angles of your polygon.

(e) Do (a) to (d) again, for a polygon with a different number of sides.

(f) Suppose the polygon you start with has n sides.Go through (a) to (d) again to find a formula for the total of the interior angles.

75°

100°

160°A B

D

Cp

90°

165°

155°

E

F

I H

G

q

(a) (b)

N M L K

J

O

s 80°

150°

140°

(c)

C Angles of a polygon level 6

C4 What is the sum of the interior angles of an 11-sided polygon?

The method of reasoning in C3 (f) is an example of mathematical deduction.We assumed it’s true that the angles of a triangle add up to 180°.We then used this assumption in a sequence of steps leading to a statementabout any polygon that can be split into triangles as in the diagram:

The sum of the interior angles of a polygon with n sides is 180(n – 2)°.







C5 For each of these polygons,

(i) record the number of sides

(ii) work out the sum of the interior angles

(iii) work out the missing angle

C6 Each of these polygons has one line of reflection symmetry.Work out the missing angles.

C7 The sum of the interior angles of a certain polygon is 2160°.How many sides does it have?

C8 Find the size of each interior angle of

(a) a regular hexagon (b) a regular nonagon (nine sides)

If you extend a side of a polygon,the angle made is called an exterior angle.

C9 If a pencil is moved around the sides of a polygon,at each vertex it turns through the exterior angle.

(a) When the pencil gets back to where it started from,it will be pointing in the same direction as before.What angle has it turned through?

(b) What is the sum of the exterior angles of a polygon?

140° 110°

75° 105°

?(a) (b) 95°

90°

225°

60° 100°

?

(c) 150°

120° 150°

140°

110°

115°

?

75°

124°

a

bc

132°

e

f

d

94°

142°

A regular polygon has all its sides equal and all its angles equal.

Start andfinish







C10 For each of these polygons,

(i) find the missing exterior angle or angles

(ii) work out the interior angle at each vertex

(iii) work out the total of the interior angles.Use the formula from C3 to check whether this total agrees with the number of sides of the polygon.

C11 This is a regular 12-sided polygon (a dodecagon).Each exterior angle is marked e.

Calculate the size of e.

68° 82°

100°

CD

A

B

w

(a) (b)

I

E

H

F

G

74°

96°

50°

77°

x

J

LO

P K

M

N

102°

y

y

y

y

y

y(c) (d)

Q R

SV

U

T

2z

z

65°

75°

34°

3z

The sum of the exterior angles of a polygon is 360°.This is true however many sides it has.

e e

e

e

e

ee

e

e

e

e

e

Angles with the same letter are the same size







C12 Calculate the size of each exterior angle of a regular polygon with

(a) 5 sides (b) 24 sides (c) 30 sides (d) 45 sides

C13 This is a close-up a vertex of the dodecagon in C11.What is the size of the interior angle, i ?

C14 Calculate the interior angle for each regular polygon in C12.

C15 This is part of a regular decagon (a 10-sided shape).Calculate the size of the interior angle ofthe regular decagon.

OCR

C16 Each exterior angle of a certain regular polygon is 40°.How many sides must this polygon have?

C17 How many sides does a regular polygon have if each exterior angle is

(a) 9° (b) 24° (c) 10° (d) 18°

C18 How many sides does a regular polygon have if each interior angle is

(a) 135° (b) 160° (c) 175° (d) 174°

C19 A regular polygon has n sides. Write an expression in n for

(a) the size of an exterior angle (b) the size of an interior angle

i

Not toscale

D Mixed questions

D1 PQRST is a regular pentagon.Giving reasons for your answers, calculate

(a) angle a (b) angle b

D2 Two interior angles of a kite are 70° and 100°.What could be the other two angles?(Draw sketches and give reasons.)

a

b

P

Q

RS

T







Test yourself

T1 Calculate the lettered angles.

T2 The diagram shows part of a regular polygon.Each interior angle is 162°.Calculate the number of sides of the polygon.

AQA

T3 ABCDEFGHI is a regular nonagon.

Giving reasons, calculate angles x, y and z.

A B

C

D

E

F

G

H

Iy

xz

50°

110°

60°

a

90°

90°

54°

d

c

b

51°

136°

62°

g

e

h

f

D3 ABCD is a rhombus and ABCE is a kite.Work out the value of x.

AQA

*D4 A polygon like this cannot be divided into triangles by drawing lines from one vertex as was done in question C3.

Does the formula 180(n – 2)° for the sum of the interior angles ofa polygon still apply? Give your reasons.

Not drawn accurately162°

Not drawn accurately

110°

50°

A

D

B

C

E

x






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