In a triangle PQR, ∠ QOR = 130°, where O is the in-centre of triangle PQR. The measure of ∠ QPR is:

The correct answer is 80°.

Step-by-Step Explanation:

Step 1: Understand the geometric property of the in-center of a triangle.

The in-center (O) of a triangle is the point where the angle bisectors of the interior angles intersect.

There is a well-known relation between the angle formed by the in-center with two vertices (∠QOR) and the opposite vertex angle (∠QPR):

$\angle \text{QOR}={90}^{\circ }+\frac{1}{2}\angle \text{QPR}$

Step 2: Substitute the given values into the formula.

We are given that ∠QOR = 130°.

${130}^{\circ }={90}^{\circ }+\frac{1}{2}\angle \text{QPR}$

Step 3: Solve for ∠QPR.

Subtract 90° from both sides of the equation:

${130}^{\circ }-{90}^{\circ }=\frac{1}{2}\angle \text{QPR}$

${40}^{\circ }=\frac{1}{2}\angle \text{QPR}$

Multiply both sides by 2:

$\angle \text{QPR}=2\times {40}^{\circ }={80}^{\circ }$

Thus, the measure of ∠QPR is 80°.