Exterior Angle of a Triangle

Exterior Angle of a Triangle

How to solve the exterior angle of a triangle problems: formula, proof, examples, and their solutions.

Formula

The measure of an exterior angle of a triangle is equal to the sum of the measures of nonadjacent interior angles of a triangle.

The measure of an exterior angle (red) of a triangle
is equal to
the sum of the measures of
nonadjacent interior angles of a triangle.
(blue, green)

Proof

Proving the formula using the interior angles of a triangle formula and the linear pair formula (Two Column Proof)

Draw the brown angle.

m∠(blue) + m∠(green) + m∠(brown) = 180

Interior angles of a triangle

m∠(brown) + m∠(red) = 180

Linear pair

The right side of line 1 and 2 are 180.
So m∠(blue) + m∠(green) + m∠(brown)
= m∠(brown) + m∠(red).

To cancel m∠(brown),
write m∠(brown) = m∠(brown),
and do the subtraction.

Then m∠(blue) + m∠(green) = m∠(red)

Switch both sides.
(Symmetric property)

Then m∠(red) = m∠(blue) + m∠(green).

Example 1

Find the value of x. The measures of the interior angles of a triangle: 60, 47. The measures of the exterior angle of the triangle: 10x + 7.

The red angle is the exterior angle of this triangle.

So 10x + 7 = 60 + 47.

Example 2

Find the value of x. The measures of the interior angles of a triangle: 30, 7x + 3. The measures of the exterior angle of the triangle: 68.

The red angle is the exterior angle of this triangle.

So 68 = 30 + (7x + 3).