Triangles
26. The Exterior Angle Secret · always equals the sum of the two opposite interior angles
The exterior angle ∠ABD always equals the sum of interior angles ∠A and ∠C.
The Exterior Angle Theorem: an exterior angle of a triangle equals the sum of the two remote (non-adjacent) interior angles. Drag A, B or C — ∠ABD always equals ∠A + ∠C, no matter the triangle's shape.
What this lesson covers
Try to break it
What happens if you stretch the triangle until it's very flat? Does the relationship still hold? Try dragging B close to the line AC.
How you build it
Make a triangle and extend side AB.
- Place vertex A.
- Place vertex B.
- Place vertex C.
- Draw side AB.
- Draw side BC.
- Draw side CA.
- Place point D on line AB extended past B.
- Extend AB to D. Exterior ∠CBD = ∠A + ∠C.
The proof, step by step
Prove that exterior angle ∠ABD always equals the sum of interior angles ∠A and ∠C.
- Through B, draw a line parallel to AC.
- ∠A = the angle between this parallel and AB (alternate interior angles).
- ∠C = the angle between this parallel and BC (corresponding angles).
- The exterior angle ∠ABD is the sum of these two angles.
- Therefore, ∠ABD = ∠A + ∠C.
Worked example
In ΔXYZ, side YZ is extended to W. If ∠X = 55° and ∠Y = 65°, what is the measure of the exterior angle ∠XZW?
By the Exterior Angle Theorem, the exterior angle ∠XZW equals the sum of the two opposite interior angles ∠X and ∠Y. So, ∠XZW = 55° + 65° = 120°.
- 110°
- 120° — correct
- 130°
- 140°