The two interior opposite angles of an exterior angle of a triangle are 75° and 85°. Find the measure of the exterior angle

Exterior angle = m∠160°

Step-by-step explanation:

Given the measures of two interior opposite angles, m75° and m∠85°, and that the prompt requires us to find the measure of the exterior angle of a triangle:

In order to find the measure of the exterior angle, we must apply the Exterior Angle Theorem which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of two non-adjacent remote interior angles.  

Let x = exterior angle of a triangle

In order to solve for the measure of x, we can set up the following equation:

x = m∠75° +  m∠85°

x = 160°  ⇒ Measure of the exterior angle of a triangle.

We must verify whether we have the correct value for the exterior angle of a triangle. In reference to the Triangle Sum Theorem, which states that the sum of all interior angles of a triangle is equal to 180°.

Since we already have the measures of the two remote interior angles and the exterior angle of a triangle, we can find the measure of the third interior angle of a triangle by setting up the following equation:

m∠75° +  m∠85° +  m∠y° = 180°

Where ∠y is the unknown third interior angle of a triangle.

m∠75° +  m∠85° +  m∠y° = 180°

160° +  m∠y° = 180°

Subtract 160° from both sides of the equation:

160° - 160° +  m∠y° = 180° - 160°

m∠y° = 20°  ⇒  Third interior angle of a triangle.

Hence, the measures of the three interior angles of a triangle are: 75°, 85°, and 20°, for which the third angle (∠y) and the exterior angle of a triangle (∠x) are supplements:

m∠x°  + m∠y°  = 180°

160° + 20° = 180°

180° = 180°  (True statement).

Therefore, the measure of the exterior angle of a triangle is 160°.