Lesson 4 · The Triangle

Section 02

The Exterior Angle of a Triangle

Extend one side of a triangle and a new angle appears outside it. That angle carries information about the two angles farthest from it.

What counts as an exterior angle?

If ABC is a triangle and D lies on ray CB but outside segment CB (that is, D ∈ CB→ and D ∉ CB), then ∠ABD is an exterior angle of triangle ABC.

Careful: an angle drawn outside the triangle is not automatically an exterior angle. It must be formed by one side and the extension of an adjacent side. So if D is not on the ray BC, then ∠ACD is not an exterior angle of the triangle.

4123DBCAm∠4 = m∠2 + m∠3
Side CB extended to D creates the exterior angle ∠ABD (angle 4)

Rule

The measure of an exterior angle of any triangle equals the sum of the measures of the two non-adjacent (remote) interior angles.

The Proof

  1. ∵ m(∠1) + m(∠2) + m(∠3) = 180° (interior angles of the triangle) … (1)
  2. ∵ ∠1 and ∠4 form a straight angle
  3. ∴ m(∠1) + m(∠4) = 180° … (2)
  4. From (1) and (2): m(∠4) = m(∠2) + m(∠3) ∎

In words: the exterior angle at B equals the sum of the interior angles at A and C.

Example 3 — Two quick applications

1) Exterior angle unknown

In triangle ABC the interior angles at A and B are 37° and 62°, and ∠ACD is an exterior angle of measure x°.

  1. ∵ ∠ACD is an exterior angle of △ABC
  2. ∴ x = 37 + 62 = 99

2) Interior angles unknown

∠ABD is an exterior angle of measure 74°, while the two remote interior angles are (4x − 1)° and (2x + 15)°.

  1. ∴ (4x − 1) + (2x + 15) = 74
  2. ∴ 6x + 14 = 74
  3. ∴ 6x = 60
  4. ∴ x = 10

Try it yourself

In triangle LMN, P lies on LN extended. The remote interior angles measure 38° and 78°, and the exterior angle measures (5x − 4)°. Find x. (5x − 4 = 38 + 78 = 116, so 5x = 120 and x = 24)

Useful consequence

Because an exterior angle equals the sum of two positive interior angles, every exterior angle of a triangle is greater than either of its remote interior angles. This gives you a fast sanity check on any answer.