In triangle `PQR`, the sides `PQ` and PR are produced to A and B respectively. The bisectors of `/_AQR `and `/_BRQ` intersect at point O. If `/_QOR = 50^@`, what is the value (in degree) of `/_QPR`?

Options

A. `50`
B. `60`
C. `80`
D. `100 `

Answer

To solve the problem, we need to find the value of angle \( \angle QPR \) in triangle \( PQR \) given that the angle \( \angle QOR = 50^\circ \). We know that \( O \) is the intersection of the angle bisectors of \( \angle AQR \) and \( \angle BRQ \).

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have triangle \( PQR \).
- The sides \( PQ \) and \( PR \) are extended to points \( A \) and \( B \) respectively.
- The angle bisectors of \( \angle AQR \) and \( \angle BRQ \) intersect at point \( O \).
- We are given that \( \angle QOR = 50^\circ \).

2. **Use the Property of the Excenter**:
- The point \( O \) is the excenter of triangle \( PQR \) opposite to vertex \( P \).
- The property of the excenter states that:
\[ \angle QOR = 90^\circ - \frac{1}{2} \angle QPR \]

3. **Set Up the Equation**:
- From the property mentioned, we can set up the equation:
\[ 50^\circ = 90^\circ - \frac{1}{2} \angle QPR \]

4. **Rearrange the Equation**:
- Rearranging the equation gives:
\[ \frac{1}{2} \angle QPR = 90^\circ - 50^\circ \]
\[ \frac{1}{2} \angle QPR = 40^\circ \]

5. **Solve for \( \angle QPR \)**:
- Multiplying both sides by 2:
\[ \angle QPR = 2 \times 40^\circ = 80^\circ \]

### Final Answer:
Thus, the value of \( \angle QPR \) is \( 80^\circ \).

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