In triangle PQR, the exterior angle at Q measures 115° and the exterior angle at R measures 135°. Find the measure of ∠QPR.

The correct answer is 70.

Step 1: Understand the interior and exterior angle relationship.
An interior angle and its adjacent exterior angle at any vertex of a triangle form a linear pair on a straight line. Therefore, their sum is equal to 180°.

Step 2: Calculate the interior angles at vertices Q and R.
We are given that the exterior angle at vertex Q is 115° and the exterior angle at vertex R is 135°.

To find the interior angle ∠PQR:

$\angle \mathrm{PQR}=\mathrm{180°}-\mathrm{115°}=\mathrm{65°}$

To find the interior angle ∠PRQ:

$\angle \mathrm{PRQ}=\mathrm{180°}-\mathrm{135°}=\mathrm{45°}$

Step 3: Calculate the measure of angle ∠QPR using the Angle Sum Property.
The sum of all interior angles inside any triangle is always 180°.

$\angle \mathrm{QPR}+\angle \mathrm{PQR}+\angle \mathrm{PRQ}=\mathrm{180°}$

Substitute the calculated values of ∠PQR and ∠PRQ into the equation:

$\angle \mathrm{QPR}+\mathrm{65°}+\mathrm{45°}=\mathrm{180°}$

$\angle \mathrm{QPR}+\mathrm{110°}=\mathrm{180°}$

Subtract 110° from both sides to find ∠QPR:

$\angle \mathrm{QPR}=\mathrm{180°}-\mathrm{110°}=\mathrm{70°}$

Thus, the measure of ∠QPR is 70°.