Question

Given: ⊙O with central angles ∠AOC ≅ ∠BOD

Prove: AC ≅ BD

Circle O is shown. Line segments O A, O C, O B, and O D are radii. Line segments connect points A and C and points B and D to form 2 triangles inside of the circle. Angles A O C and B O D are congruent.

Complete the missing parts of the paragraph proof.

Proof:

We know that central angles
are congruent, because it is given. We can say that segments AO, CO, BO, and DO are congruent because
. Then by the
congruency theorem, we know that triangle AOC is congruent to triangle BOD. Finally, we can conclude that chord AC is congruent to chord BD because
.

Answer 1

We know that central angles ∠AOC and ∠BOD are congruent, because it is given. We can say that segments AO, CO, BO, and DO are congruent because they are radii of circle O. Then by the SAS congruency theorem, we know that triangle AOC is congruent to triangle BOD. Finally, we can conclude that chord AC is congruent to chord BD because corresponding parts of congruent triangles are congruent (CPCTC).