Lesson 9, Topic 2
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# Exterior and Interior Opposite Angles

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In triangle XYZ below on the left, the exterior angle is “a” and the two opposite interior angles are x and y.

To prove that a = p + q, draw a line at Z (i.e. line ZW) to divide the exterior angle “a” into angles p and q and let this line be parallel to line XY (see the diagram above).

From the diagram on the right,

x = p (corresponding angles ZW//XY)

y = q (alternate angles ZW//XY)

Hence: a = x + y = p + q

Therefore, the exterior angle of a triangle is equal to the sum of the two opposite interior angles.

### Example 1

Find the size of angle x in this triangle

Solution

x + 53º + 87º = 180º (sum of angles of a Triangle)

x + 140º = 180º

x = 180º – 140º

x = 40º

### Example 2

a. In the diagram below, find the value of x

b. Hence, use the value of x to find the actual values of the interior angles

Solution

a. <ABC = 2x (vertically opposite angles)

2x + 3x + 4x = 180º (sum of angles of a triangle)

9x = 180º

Divide both sides by 9

$$\frac{9x}{9} = \frac{180}{9}$$

x = $$\frac{180}{9}$$

x = 20º

b. If a =  20º,  then 2x = 2 x 20º = 40º

3x = 3 x 20º = 60º

4x = 4 x 20º = 80º

The angles are 40º, 60º, 80º

### Example 3

State the sizes of the lettered angles and give reasons

Solution

140º + a = 180º (angles on a straight line is 180º)

a  =  180º – 140º

a = 40º

a + b + c = 180 (sum of angles in a triangle)

40 + b + c = 180

b + c = 180 – 40

b + c = 140

b = 70, c = 70  (base angles in an isosceles triangle are equal)

b = c {sum of angles of an isosceles triangle base}

OR

140 = b + c (the exterior angle of a triangle is equal to the sum of the two opposite interior angles)

b = 70, c = 70  (base angles in an isosceles triangle are equal)

### Example 4

Find angles x, y, z

Solution

Consider Δ ABC

x + 32º + 90º = 180º (sum of angles in a triangle)

x = 180º – 90º – 32º

x = 180º – 122º

x = 58º

x + y = 180 (angles on a straight line)

58 + y = 180

y = 180 – 58

y = 122º

Consider Δ ACD

z + 30 + 122 = 180 (sum of angles in a triangle)

z + 152 = 180

z = 180 – 152

z = 28º

Therefore: x = 58º, y = 122º, z = 28º

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