Prove that equal chords of congruent circles subtend equal angles at their centre.

Answer

To prove that equal chords of congruent circles subtend equal angles at their centers, we can follow these steps:

### Step-by-Step Solution:

1. **Draw Two Congruent Circles**:
- Let’s denote the first circle as Circle O with center O and the second circle as Circle O' with center O'.

2. **Draw Equal Chords**:
- Draw a chord AB in Circle O and a chord CD in Circle O'.
- Given that the lengths of the chords are equal, we have AB = CD.

3. **Join the Centers to the Chords**:
- Join the center O to points A and B, creating line segments OA and OB.
- Similarly, join the center O' to points C and D, creating line segments O'C and O'D.

4. **Identify Radii**:
- Since the circles are congruent, the radii of both circles are equal. Therefore, we have:
- OA = O'C (both equal to the radius R)
- OB = O'D (both equal to the radius R)

5. **Establish Congruent Triangles**:
- Now we have two triangles: Triangle AOB and Triangle O'CD.
- We can establish the following equalities:
- AB = CD (given)
- OA = O'C (equal radii)
- OB = O'D (equal radii)
- Thus, by the Side-Side-Side (SSS) congruence criterion, we can conclude that:
- Triangle AOB ≅ Triangle O'CD.

6. **Conclude Equal Angles**:
- Since the triangles AOB and O'CD are congruent, the corresponding angles are equal:
- Therefore, ∠AOB = ∠O'CD.

### Conclusion:
We have proved that equal chords of congruent circles subtend equal angles at their centers.