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Proving That Congruent Central Angles Have Congruent Chords

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

**Prove:** AC ≅ BD

*Circle O is shown. Line segments OA, OC, OB, and OD are radii. Line segments connect points A and C and points B and D to form two triangles inside the circle. Angles AOC and BOD are congruent.*

Complete the missing parts of the paragraph proof.

**Proof:**

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 all radii of the same circle. Then, by the Side-Angle-Side (SAS) congruence 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).

Answer :

By recognizing that the radii of a circle are congruent and using the SAS congruency theorem, we prove that triangles AOC and BOD are congruent. Consequently, the chords AC and BD are proven to be congruent as well.

Central angles in a circle are formed by two radii and have their vertices at the center of the circle. When two congruent central angles are given, it implies that the corresponding chords are also congruent. This is because congruent central angles subtend congruent arcs on the circle.

We know that central angles ∠AOC ≅ ∠BOD are congruent, because it is given. We can say that segments AO, CO, BO, and DO are congruent because they are all radii of the same circle. Then by the Side-Angle-Side (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).

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Rewritten by : Jeany

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