Question

given: ⊙o with central angles ∠aoc ≅ ∠bod
prove: \overline{ac} ≅ \overline{bd}
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 all radii of a circle are congruent. then by the dropdown congruency theorem, we know that triangle dropdown congruent to triangle bod. finally, we can dropdown that chord ac is congruent to chord bd dropdown.
dropdown options for congruency theorem: aas, sas, sss

Explanation:

Step1: Identify triangle sides/angle

We have \( AO \cong BO \), \( CO \cong DO \) (radii), and \( \angle AOC \cong \angle BOD \) (given).

Step2: Determine congruence theorem

The SAS (Side - Angle - Side) theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. Here, for \( \triangle AOC \) and \( \triangle BOD \), \( AO \cong BO \), \( \angle AOC \cong \angle BOD \), \( CO \cong DO \), so SAS applies.

Step3: Corresponding parts of congruent triangles

If \( \triangle AOC \cong \triangle BOD \) (by SAS), then their corresponding sides \( \overline{AC} \) and \( \overline{BD} \) are congruent (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Answer:

First dropdown (congruency theorem): SAS
Second dropdown (reason for \( \overline{AC} \cong \overline{BD} \)): Corresponding Parts of Congruent Triangles are Congruent (CPCTC)