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. This guide explains the theorem, proves it using basic angle facts, and gives you worked examples and practice questions.

What Is the Exterior Angle Theorem?

When one side of a triangle is extended beyond a vertex, the angle formed between the extended side and the adjacent side is called an exterior angle.

The Theorem

If a triangle has interior angles A, B, and C, and the exterior angle at C is called D, then:

Proof Using Angle Facts

Step 1 — The interior angles of a triangle sum to 180°: A + B + C = 180°.

Step 2 — The exterior angle D and interior angle C form a straight line: C + D = 180°.

Step 3 — From Step 1: A + B = 180° − C.

Step 4 — From Step 2: D = 180° − C.

Step 5 — Therefore D = A + B.

Step-by-Step Method

Finding an Exterior Angle

1. Identify the exterior angle and the two interior angles that are "opposite" to it (i.e. not adjacent to it).
2. Add the two opposite interior angles together.
3. The result is the exterior angle.

Finding a Missing Interior Angle Using an Exterior Angle

1. Use the exterior angle theorem: exterior angle = sum of the two opposite interior angles.
2. If one interior angle is known, subtract it from the exterior angle to find the other.

Worked Example 1 — Foundation Level

Question: In triangle ABC, angle A = 55° and angle B = 70°. Side BC is extended to point D. Find the exterior angle ACD.

Working:

Step 1 — The exterior angle at C (angle ACD) equals the sum of the two opposite interior angles: A + B.

Step 2 — Angle ACD = 55° + 70° = 125°.

Answer: The exterior angle ACD = 125°.

Worked Example 2 — Higher Level

Question: In triangle PQR, the exterior angle at R is 120°. Angle P is twice angle Q. Find angles P and Q.

Working:

Step 1 — By the exterior angle theorem: P + Q = 120°.

Step 2 — P = 2Q, so 2Q + Q = 120°. Therefore 3Q = 120°.

Step 3 — Q = 40°. P = 2 × 40° = 80°.

Step 4 — Check: angle R = 180° − 120° = 60°. Angle sum: 80 + 40 + 60 = 180°. Correct.

Answer: Angle Q = 40° and angle P = 80°.

Worked Example 3 — Exam Style

Question: In the diagram, triangle ABC has angle BAC = 48°. Side BC is extended to D. Line CE is drawn so that CE is parallel to BA. Prove that angle ACD = angle BAC + angle ABC.

Working:

Step 1 — Since CE is parallel to BA, angle ACE = angle BAC = 48° (alternate angles).

Step 2 — Also, angle ECD = angle ABC (corresponding angles, since CE ∥ BA and BD is a transversal).

Step 3 — Angle ACD = angle ACE + angle ECD = angle BAC + angle ABC.

Answer: This proves the exterior angle theorem: angle ACD = angle BAC + angle ABC.

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.

Practice Questions

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.

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Related Topics

• Angles in a Triangle — the 180° angle sum rule.
• Angles: Basic Rules and Parallel Lines — alternate, corresponding, and co-interior angles.
• Angles in Polygons — exterior angles of polygons sum to 360°.

Summary

The exterior angle theorem states that an exterior angle of a triangle equals the sum of the two opposite interior angles. It follows directly from the angle sum of a triangle (180°) and angles on a straight line (180°). This shortcut saves time in multi-step angle problems. Always state the theorem clearly in "give a reason" questions, and remember it applies specifically to triangles. At Higher tier, you may be asked to prove the theorem using parallel lines.