Exterior Angle Theorem
An exterior angle of a triangle is formed by extending one side. Its measure equals the sum of the two non-adjacent (remote) interior angles.
Introduction
When you extend a side of a triangle past a vertex, the angle formed on the outside is called an exterior angle. The Exterior Angle Theorem gives us a powerful shortcut: instead of subtracting from 180°, we can relate the exterior angle directly to the two remote interior angles.
Past Knowledge
Triangle Angle Sum (5.1.2). Linear pairs. Supplementary angles.
Today's Goal
Prove and apply the Exterior Angle Theorem to find missing angle measures.
Future Success
Isosceles Triangle Theorem (5.1.4), inequality theorems, and proofs.
Key Concepts
Vocabulary
- Exterior angle — the angle formed between one side of a triangle and the extension of an adjacent side.
- Remote interior angles (non-adjacent interior angles) — the two interior angles that are not adjacent to the exterior angle.
- Adjacent interior angle — the interior angle that forms a linear pair with the exterior angle.
Exterior Angle Theorem
The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.
where and are the remote interior angles.
Corollary
An exterior angle of a triangle is always greater than either remote interior angle individually.
Theorem & Proof
Two-Column Proof: Exterior Angle Theorem
Given: with exterior angle formed by extending through to point .
Prove:
Click each step to follow the proof. ∠A + ∠B = ∠ACD (exterior)
| # | Statement | Reason |
|---|---|---|
| 1 | Triangle Angle Sum Theorem | |
| 2 | and form a linear pair. | Definition of linear pair (, , are collinear) |
| 3 | Linear Pair Postulate (supplementary angles) | |
| 4 | Subtraction Property of Equality (from step 3) | |
| 5 | Substitution (step 4 into step 1) | |
| 6 | Subtraction Property of Equality (subtract 180° from both sides, rearrange) |
∎ The exterior angle equals the sum of the two remote interior angles.
Worked Examples
Finding an Exterior Angle
In , and . Find the exterior angle at .
By the Exterior Angle Theorem:
The exterior angle at is .
Solving for a Variable
An exterior angle of a triangle measures . The two remote interior angles are and . Find and all three measures.
Remote interiors: ,
Exterior angle:
Check: ✓
. Remote interiors: 40° and 50°. Exterior angle: 90°.
Back-Solving for an Interior Angle
In , the exterior angle at is 142°. If , find and .
Find : By the Exterior Angle Theorem:
Find : The interior at is supplementary to the exterior:
Verify: ✓
Common Pitfalls
Using the Adjacent Interior Angle Instead of Remote
The theorem says exterior = sum of the two remote interior angles. The adjacent angle is supplementary to the exterior — that's a different relationship.
Confusing Interior and Exterior
The exterior angle is always larger than either remote interior angle. If your answer for an exterior angle is less than one of the interior angles, recheck.
Real-Life Applications
Billiards & Angle of Incidence
When a billiard ball bounces off a rail, the triangle formed by its path has an exterior angle equal to the sum of the incidence and reflection angles. Players intuitively use the exterior angle theorem to predict where balls will travel.
Aviation — Descent Angles
Pilots use the exterior angle relationship when calculating approach paths. The angle at which the plane appears to descend from the control tower is the exterior angle formed by the glide path and the ground — determined by the remote angles of elevation and climb.
Practice Quiz
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