In a circle with centre `P. AB` and `CD` are congruent chords. If `/_PAB = 40^(@)`, then `/_CPD= `

Options

A. `40^(@)`
B. `80^(@)`
C. `100^(@)`
D. `50^(@)`

Answer

To solve the problem, we need to find the measure of angle \( \angle CPD \) given that \( \angle PAB = 40^\circ \) and that chords \( AB \) and \( CD \) are congruent.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have a circle with center \( P \).
- Chords \( AB \) and \( CD \) are congruent.
- \( \angle PAB = 40^\circ \).

2. **Recognize Properties of the Circle**:
- Since \( AB \) and \( CD \) are congruent chords, the distances from the center \( P \) to the chords are equal.
- The triangle \( APB \) is isosceles because \( PA = PB \) (both are radii of the circle).

3. **Determine Angles in Triangle \( APB \)**:
- In triangle \( APB \), since \( PA = PB \), we have \( \angle PAB = \angle PBA = 40^\circ \).
- The sum of angles in a triangle is \( 180^\circ \). Therefore:
\[ \angle APB + \angle PAB + \angle PBA = 180^\circ \]
\[ \angle APB + 40^\circ + 40^\circ = 180^\circ \]
\[ \angle APB + 80^\circ = 180^\circ \]
\[ \angle APB = 180^\circ - 80^\circ = 100^\circ \]

4. **Use the Property of Vertically Opposite Angles**:
- The angle \( \angle CPD \) is vertically opposite to \( \angle APB \).
- Therefore, we have:
\[ \angle CPD = \angle APB = 100^\circ \]

5. **Conclusion**:
- The measure of angle \( \angle CPD \) is \( 100^\circ \).

### Final Answer:
\[ \angle CPD = 100^\circ \]