Exterior Angle of a Triangle

Statement

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. In symbols:

\[\text{Exterior angle} = \text{Interior angle A} + \text{Interior angle B}\]

This is a fundamental property of triangles that follows from the fact that the interior angles of a triangle always sum to \(180^\circ\).

Why it works

• Suppose a triangle has interior angles \(A, B, C\).
• The sum of these angles is \(A + B + C = 180^\circ\).
• If we extend one side to form an exterior angle, say at \(C\), the exterior angle and interior angle \(C\) form a straight line, so together they sum to \(180^\circ\).
• Therefore, exterior angle at \(C = 180^\circ - C\).
• But from the triangle sum, \(A + B = 180^\circ - C\).
• Hence, exterior angle at \(C = A + B\).

Recipe (Steps to Apply)

1. Identify which exterior angle is required (the angle formed outside the triangle when you extend one side).
2. Identify the two opposite interior angles (the other two angles inside the triangle).
3. Add these two opposite interior angles together.
4. That sum is the exterior angle.

Spotting it

Use this fact whenever a triangle problem involves an exterior angle or asks for missing angles in a diagram where one side has been extended.

Common pairings

• Angle sum of a triangle (\(180^\circ\)).
• Angles on a straight line (\(180^\circ\)).
• Parallel line angle facts (alternate and corresponding angles).

Mini examples

1. Given: Interior opposite angles are \(50^\circ\) and \(60^\circ\). Find: Exterior angle. Answer: \(110^\circ\).
2. Given: Interior opposite angles are \(40^\circ\) and \(80^\circ\). Find: Exterior angle. Answer: \(120^\circ\).

Pitfalls

• Mixing up which angles are “opposite.” The opposite interior angles are the two inside the triangle not adjacent to the exterior angle.
• Forgetting that an exterior angle is outside the triangle.
• Adding the wrong pair of angles (students sometimes add adjacent angles incorrectly).

Exam Strategy

• Mark all known angles on the diagram first.
• Look for the straight-line rule (interior + exterior at a vertex = 180°).
• Check that your answer is bigger than each of the opposite angles individually (since it is their sum).

Worked Micro Example

A triangle has interior angles \(A=55^\circ\), \(B=65^\circ\), and \(C=60^\circ\). Find the exterior angle at \(C\).

Opposite interior angles are \(A\) and \(B\): \(55^\circ + 65^\circ = 120^\circ\).

So, exterior angle at \(C = 120^\circ\).

Summary

The exterior angle of a triangle equals the sum of the two opposite interior angles. This rule is a direct consequence of the interior angle sum (\(180^\circ\)) and the straight-line angle fact. It is one of the most useful facts in GCSE geometry problems involving triangles and angle chasing.