Sheet № 222 · Foundation + Higher · AQA · Edexcel · OCR
Exterior Angle Theorem –
The exterior angle theorem is a key angle fact tested at Foundation and Higher tiers across all GCSE Maths exam boards. It states that an exterior angle of a triangle equals the sum of the two opposite interior angles. Understanding this theorem helps you solve multi-step angle problems efficiently and provides a basis for angle proofs. T
§Key definitions
Question:
In triangle ABC, angle A = 55° and angle B = 70°. Side BC is extended to point D. Find the exterior angle ACD.
Answer:
The exterior angle ACD = 125°.
Q1 (Foundation):
A triangle has angles 62° and 83°. Find the exterior angle at the third vertex.
Q2 (Foundation):
The exterior angle of a triangle is 118°. One of the opposite interior angles is 53°. Find the other.
Q3 (Higher):
In triangle XYZ, the exterior angle at Z is (5x + 10)°. Angle X = (2x + 5)° and angle Y = (2x + 25)°. Find x and all three interior angles.
§Formulas to memorise
Exterior angle of a triangle = sum of the two opposite interior angles
D = A + B
Use the exterior angle theorem: exterior angle = sum of the two opposite interior angles.
Worked example
In triangle ABC, angle A = 55° and angle B = 70°. Side BC is extended to point D. Find the exterior angle ACD.
Working:
⚠ Common mistakes
- ✗Using the adjacent interior angle instead of the opposite ones. The exterior angle equals the sum of the two non-adjacent (remote) interior angles, not the angle next to it.
- ✗Confusing exterior angles with reflex angles. An exterior angle is formed by extending one side — it is supplementary to the adjacent interior angle (they add to 180°).
- ✗Not stating the theorem name. In "give a reason" questions, write "exterior angle of a triangle equals the sum of the two opposite interior angles" for the full mark.
✦ Exam tips
- →The exterior angle theorem is a shortcut — you could always use "angles in a triangle = 180°" and "angles on a straight line = 180°" instead, but the theorem is faster.
- →In multi-step angle problems, look for extended sides that create exterior angles — this often simplifies the solution.
- →Higher-tier questions may ask you to prove the theorem (as in Worked Example 3) using parallel lines or the angle sum of a triangle.
- →This theorem applies only to triangles, not to polygons in general.