A circle passing through points Q and R of triangle PQR, cut the sides PQ and PR at point X and Y respectively. If PQ = PR, then what is the value (in degrees) of `angle PRQ + angle QXY` ?

Options

A. 120
B. 150
C. 240
D. 180

Answer

To solve the problem, we need to analyze the properties of the triangle and the cyclic quadrilateral formed by the points on the circle.

### Step-by-Step Solution:

1. **Understand the Triangle and Circle**: We have triangle PQR where PQ = PR. A circle passes through points Q and R and intersects sides PQ and PR at points X and Y respectively.

2. **Draw the Diagram**: Visualize the triangle PQR with points Q and R on the circle. Mark points X and Y where the circle intersects sides PQ and PR.

3. **Identify the Angles**: We need to find the sum of angles \( \angle PRQ \) and \( \angle QXY \).

4. **Use the Property of Cyclic Quadrilaterals**: Since points Q, R, X, and Y lie on the circumference of the circle, we can use the property of cyclic quadrilaterals. The property states that the sum of the opposite angles in a cyclic quadrilateral is 180 degrees.

5. **Apply the Property**: In our case, the angles of interest are:
- \( \angle PRQ \) (which is one angle of triangle PQR)
- \( \angle QXY \) (which is the angle formed by the intersection of the circle at points X and Y)

According to the cyclic quadrilateral property:
\[ \angle PRQ + \angle QXY = 180^\circ \]

6. **Conclusion**: Therefore, the value of \( \angle PRQ + \angle QXY \) is \( 180^\circ \).

### Final Answer:
\[ \angle PRQ + \angle QXY = 180^\circ \]