Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

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

Let's assume we have two congruent circles with equal radii, and two chords AB and CD in these circles that are equal in length.

Draw a diagram: Draw two congruent circles with their centers O₁ and O₂. Mark the radii OA and OB in the first circle and OC and OD in the second circle. Connect these radii to the chords AB and CD, respectively.

Prove that the triangles are congruent: Since the circles are congruent, their radii are equal, i.e., OA = OC and OB = OD.

Prove that the angles are equal: Now, we need to show that the angles ∠AOB and ∠COD are equal.

Using the congruence of triangles, we can conclude that:

∠O₁AO₂ = ∠O₁CO₂ (since OA = OC, by congruence of radii)
∠O₁BO₂ = ∠O₁DO₂ (since OB = OD, by congruence of radii)
Prove that the angles at the centers are equal: Since ∠O₁AO₂ = ∠O₁CO₂ and ∠O₁BO₂ = ∠O₁DO₂, we can add these angles together to get:

∠O₁AO₂ + ∠O₁BO₂ = ∠O₁CO₂ + ∠O₁DO₂

∠AOB = ∠COD

So, we have shown that equal chords AB and CD of congruent circles subtend equal angles ∠AOB and ∠COD at their respective centers O₁ and O₂.