IGCSE Maths / Working with shape and space

Exterior angle of a triangle revision notes

Edexcel International GCSE Maths (4MA1), chapter 11: Working with shape and space, specification point 4.1c. Free lesson.

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An exterior angle is formed when you extend a side of a triangle beyond a corner. It equals the sum of the two opposite interior angles, which gives you a quick way to find missing angles.

What an exterior angle is

An exterior angle is the angle outside a triangle made when you extend one of its sides past a corner. The two interior angles at the far end of the triangle are called the opposite interior angles. The interior angle beside the exterior angle is not one of the opposite angles.

How the long method leads to the rule

You can always find an exterior angle in two steps. First use the fact that the angles inside a triangle add to 180°. Then use the fact that angles on a straight line add to 180°. For example, if ∠BAC = 54° and ∠ABC = 71°, the interior angle at C is 180° − 54° − 71° = 55°. The exterior angle is 180° − 55° = 125°. This is the same as adding 54° and 71°. This method works even if you cannot remember the exterior angle property.

The exterior angle property

The two-step method shows the shortcut: the exterior angle of a triangle equals the sum of the two opposite interior angles. In the diagram, ∠ACD = ∠BAC + ∠ABC. The nearby interior angle, ∠ACB, is not included. Add the two angles at the far end from the exterior angle.

Using the rule in either direction

When the exterior angle is unknown, add the two opposite interior angles. When one opposite interior angle is unknown, subtract the known angle from the exterior angle. Name the property in your reason in both cases.

Check which angle the question asks for

The interior angle beside the exterior angle and the exterior angle together make 180° because they lie on a straight line. If you find the interior angle first, you still need to subtract it from 180° to get the exterior angle. For angles of 48° and 67°, the interior angle at the extended corner is 180° − 48° − 67° = 65°. The exterior angle is 48° + 67° = 115°, not 65°.