A triangle ABC is inscribed in a circle and the bisectors of the angles A, B and C meet the cir cumference at P, Q and R respec tively. The angles of the triangle PQR respectively are

Options

A. `90^@ - A/2, 90^@ + A/2 90^@ + C/2`
B. `90^@ + A/2 , 90^@ - B/2 , 90^@ - C`
C. `90^@ - A/2 , 90^@ - B/2 , 90^@ - C/2`
D. None of these

Answer

To find the angles of triangle PQR formed by the angle bisectors of triangle ABC inscribed in a circle, we can follow these steps:

### Step 1: Understand the Setup
We have triangle ABC inscribed in a circle. The angle bisectors of angles A, B, and C intersect the circumference of the circle at points P, Q, and R respectively.

### Step 2: Identify the Angles at Points P, Q, and R
- The angle at point P (∠PQR) corresponds to the angle at vertex A (∠BAC) of triangle ABC.
- The angle at point Q (∠QRP) corresponds to the angle at vertex B (∠ABC) of triangle ABC.
- The angle at point R (∠RPQ) corresponds to the angle at vertex C (∠ACB) of triangle ABC.

### Step 3: Use the Angle Bisector Theorem
According to the property of angle bisectors:
- ∠BQP = ∠BAP = 1/2 ∠A
- ∠BQR = ∠BCR = 1/2 ∠C
- ∠PQR = ∠PAB + ∠BQR = (1/2 ∠A) + (1/2 ∠C)

### Step 4: Calculate ∠PQR
Using the angles from triangle ABC:
- ∠PQR = (1/2 ∠A) + (1/2 ∠C) = 1/2 (∠A + ∠C)

Since the sum of angles in triangle ABC is 180 degrees:
- ∠A + ∠B + ∠C = 180 degrees
- Thus, ∠A + ∠C = 180 degrees - ∠B

Substituting this into our equation for ∠PQR:
- ∠PQR = 1/2 (180 degrees - ∠B) = 90 degrees - (1/2 ∠B)

### Step 5: Calculate Other Angles
Using similar reasoning:
- ∠QRP = 90 degrees - (1/2 ∠C)
- ∠RPQ = 90 degrees - (1/2 ∠A)

### Conclusion
Thus, the angles of triangle PQR are:
- ∠PQR = 90 degrees - (1/2 ∠B)
- ∠QRP = 90 degrees - (1/2 ∠C)
- ∠RPQ = 90 degrees - (1/2 ∠A)

### Final Answer
The angles of triangle PQR are:
- P = 90 - (1/2 ∠A)
- Q = 90 - (1/2 ∠B)
- R = 90 - (1/2 ∠C)