The exterior angle of a triangle is the angle formed between one side and the extension of its adjacent side.

Shown below is the ΔABC where ∠ACD is the exterior angle formed by the side AC and the extension of the side BC to D.

How Many Exterior Angles Does a Triangle Have

Every triangle has three exterior angles, two at each vertex.

Properties

An exterior angle of a triangle is equal to the sum of the two opposite interior angles, thus an exterior angle is greater than any of its two opposite interior angles; for example, in ΔABC, ∠5 = ∠a + ∠b

The sum of an exterior angle and its adjacent interior angle is equal to 180 degrees; for example, ∠5 + ∠c = 180°

All exterior angles of a triangle add up to 360° (taken one angle at each vertex); ∠1 + ∠3 +∠5 = 360°

The two exterior angles at each vertex are equal (vertically opposite angles); ∠1 = ∠2, ∠3 =∠4, & ∠5 = ∠6

Triangle Exterior Angle Theorem

Triangle Exterior Angle Theorem Proof

Prove Triangle Exterior Angle Theorem

To prove:

∠ACD = ∠ABC + ∠CAB

Proof:

Given: ΔABC, ABD is a line segment ∠ACD + ∠BCA = 180° … (1) [∠ACD & ∠BCA form a linear pair, they are supplementary] ∠ABC + ∠BCA + ∠CAB = 180° …. (2) [Angle Sum Property of a Triangle] Substituting (1) in (2), we get, ∠ACD + ∠BCA = ∠ABC + ∠BCA + ∠CAB ∠ACD = ∠ABC + ∠CAB Hence Proved

Given below are some solved examples to understand the concept.

How to Find the Exterior Angles of a Triangle

Here, we will apply the exterior angle theorem to find the missing interior or exterior angles in a triangle.

In the given triangle, if m∠1 = 40° and m∠2 = 80°, find m∠4.

Solution:

As we know, from the exterior angle theorem, m∠4 = m∠1 + m∠2, here m∠1 = 40°, m∠2 = 80° m∠4 = 40° + 80° = 120°

Solve for x.

Solution:

As we know, from the exterior angle theorem, (x – 2) + (2x + 10) = 140° => 3x + 8 = 140° => 3x = 132 => x = 44

If the measure of the exterior angle is (5x – 10) degrees, and the measure of the two remote interior angles are 30 degrees and (x + 20) degrees, find x.

Solution:

As we know, from the exterior angle theorem, (5x – 10) = 30° + (x + 20) 5x – x = 30° + 20° + 10° 4x = 60° x = 15°

Calculate values of x and y in the given triangle.

Solution:

In the given triangle, x is the exterior angle and y is the interior angle As we know, from the exterior angle theorem, x = 80° + 50° = 130° Now, as x and y are linear pairs that add up to 180° Thus, x + y = 180°, here x = 130° 130° + y = 180° y = 180° – 130° = 50°