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

Options

A. 1) 50
B. 2) 60
C. 3) 80
D. 4)100

Answer

To solve the problem, we need to find the value of angle QPR in triangle PQR given that angle QOR is 50 degrees. We will use the properties of angle bisectors and the relationships between angles in a triangle.

### Step 1: Understand the Geometry
We have triangle PQR, and the sides PQ and PR are extended to points A and B respectively. The angle bisectors of angles AQR and BRQ intersect at point O. We are given that angle QOR = 50 degrees.

### Step 2: Apply the Angle Bisector Theorem
From the properties of angle bisectors, we know that:
- The angle formed at point O by the bisectors of angles AQR and BRQ can be expressed in terms of angle QPR.
- Specifically, if we denote angle QPR as angle P, then the relationship is given by:
\[ \angle QOR = 90^\circ - \frac{\angle P}{2} \]

### Step 3: Substitute the Given Value
We know that angle QOR = 50 degrees. Thus, we can set up the equation:
\[ 50^\circ = 90^\circ - \frac{\angle P}{2} \]

### Step 4: Solve for Angle P
Rearranging the equation to isolate angle P:
\[ \frac{\angle P}{2} = 90^\circ - 50^\circ \]
\[ \frac{\angle P}{2} = 40^\circ \]
Now, multiply both sides by 2 to find angle P:
\[ \angle P = 80^\circ \]

### Conclusion
Thus, the value of angle QPR is:
\[ \angle QPR = 80^\circ \]