Angles and Symmetry
Angle Geometry

There are a number of important results concerning angles in different shapes, at a point and on a line. In this section the following results will be used.

1.

Angles at a Point

The angles at a point will always add up to 360°. It does not matter how many angles are formed at the point – their total will always be 360°.

2.

Angles on a Line

Any angles that form a straight line add up to 180°.

3.

Angles in a Triangle

The angles in any triangle add up to 180°.

4.

Angles in an Equilateral Triangle

In an equilateral triangle all the angles are 60° and all the sides are the same length.

5.

Angles in an Isosceles Triangle

In an isosceles triangle two sides are the same length and two angles are the same size.

6.

Angles in a Quadrilateral

The angles in any quadrilateral add up to 360°.

Worked Examples

1

Find the sizes of angles a and b in the diagram below.

Show me

First consider the quadrilateral. All the angles of this shape must add up to 360°, so

60° + 120° + 80° + a = 360°
260° + a = 360°
a = 360° − 260°
= 100°

Then consider the straight line formed by the angles a and b. These two angles must add up to 180° so,

a + b = 180°
but a = 100°, so
100°+ b = 180°
b = 180° − 100°
= 80°
2

Find the angles a, b, c and d in the diagram.

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First consider the triangle shown.

The angles of this triangle must add up to 180°, so

40° + 30° + a = 180°

Next consider the angles round the point shown.

The three angles must add up to 360°, so

120° + b + a = 360°
but a = 110°, so
120° + 110° + b = 360°
230° + b = 360°
b = 360° − 230°
= 130°

Finally, consider the second triangle.

The angles must add up to 180° , so

c + b + d = 180°

As this is an isosceles triangle the two angles, c and d, must be equal, so using c = d and the fact that b = 130°, gives

c + 130° + c = 180°
2c = 180° − 130°
= 50°
c = 25°

As c = 25°, d = 25°.
3

In the figure below, not drawn to scale, ABC is an isosceles triangle with ∠CAB = p° and ∠ABC = (p + 3)°.

(a)

Write an expression in terms of p for the value of the angle at C.

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As ABC is an isosceles triangle,

∠ACB = p + 3°
(b)

Determine the size of EACH angle in the triangle.

Show me

For triangle ABC,

p + (p + 3) + (p + 3) = 180°
3p + 6° = 180° (take 6 from each side)
3p = 180° − 6°
3p = 174° (divide both sides by 3)
p = 58°

Exercises

Find the size of the angles marked with a letter in each diagram.

(a)

a = °

(b)

x = °

(c)

b = °

(d)

a = °

(e)

a = °

(f)

x = °

(g)

x = °

(h)

a = °

(i)

x = °

(j)

a = °

b = °

(k)

a = °

b = °

c = °

(l)

a = °

b = °

(m)

a = °

(n)

x = °

(o)

x = °

(a)

For each triangle, find the angles marked a and b .

(i)

a = °

b = °

(ii)

a = °

b = °

(iii)

a = °

b = °

(b)

Find the size of the angle b in each problem below without working out the size of any other angles.

(i)
b = °
(ii)
b = °
(iii)
b = °

Find the angles marked with letters in each of the following diagrams.

In each diagram the lines all lie inside a rectangle.

(a)

a = °

b = °

c = °

d = °

(b)

a = °

b = °

c = °

d = °

e = °

f = °

g = °

(c)

a = °

b = °

c = °

d = °

e = °

(d)

a = °

b = °

c = °

d = °

e = °

f = °

A swing is built from two metal frames. A side view of the swing is shown below.

The lengths of AB and AE of the swing are the same and the lengths of AC and AD of the swing are the same. Find the sizes of the angles a, b, c, d, e and f.

a = °

b = °

c = °

d = °

e = °

f = °

The diagram shows a wooden frame that forms part of the roof of a house.

Find the sizes of the angles a, b, c, d, e and f.

a = °

b = °

c = °

d = °

e = °

f = °

Find the value of x in each diagram.

(a)
x = °
(b)
x = °
(c)
x = °
(d)
x = °
(e)
x = °
(f)
x = °
(g)
x = °
(h)
x = °
(i)
x = °
(j)
x = °
(k)
x = °
(l)
x = °

The diagram shows a regular hexagon.

O is the point at the centre of the hexagon.

A and B are two vertices.

(a)

Write down the order of rotational symmetry of the regular hexagon.

(b)

Draw the lines from O to A and from O to B.

(i)

Write down the size of angle AOB.

°
(ii)

What is the mathematical name for triangle AOB.