Internal bisectors of `angle Q` and `angle R` of traingle PQR intersect at O.If angle ROQ= 96 then the value of angle RPQ is

Options

A. `36^(@)`
B. `24^(@)`
C. `12^(@)`
D. `6^(@)`

Answer

To solve the problem, we need to find the value of angle RPQ in triangle PQR, given that the internal bisectors of angles Q and R intersect at point O and angle ROQ is 96 degrees.

### Step-by-Step Solution:

1. **Understanding the Angles**:
We know that angle ROQ is given as 96 degrees. Since O is the intersection of the internal bisectors of angles Q and R, we can denote angle Q as \( \angle PQO \) and angle R as \( \angle PRO \).

2. **Using the Angle Bisector Theorem**:
The angle bisector theorem states that the angle formed at the intersection of the angle bisectors can be expressed in terms of the angles of the triangle. Specifically, we have:
\[ \angle ROQ = 90^\circ + \frac{\angle RPQ}{2} \]

3. **Setting Up the Equation**:
Since we know that \( \angle ROQ = 96^\circ \), we can substitute this value into our equation:
\[ 96^\circ = 90^\circ + \frac{\angle RPQ}{2} \]

4. **Solving for Angle RPQ**:
To isolate \( \angle RPQ \), we first subtract 90 degrees from both sides:
\[ 96^\circ - 90^\circ = \frac{\angle RPQ}{2} \]
\[ 6^\circ = \frac{\angle RPQ}{2} \]
Now, multiply both sides by 2 to solve for \( \angle RPQ \):
\[ \angle RPQ = 6^\circ \times 2 = 12^\circ \]

5. **Conclusion**:
Therefore, the value of angle RPQ is \( 12^\circ \).

### Final Answer:
The value of angle RPQ is \( 12^\circ \).