Question

proving that congruent central angles have congruent chords
given: ⊙o with central angles ∠aoc ≅ ∠bod
prove: \\(\overline{ac} \cong \overline{bd}\\)
complete the missing parts of the paragraph proof.
proof:
we know that central angles dropdown are congruent,
because it is given. we can text, co, bo, and do
are congruent because dropdown. then
by the dropdown congruency text that triangle aoc is
congruent to triangle bod. finally, we can conclude that chord ac is
congruent to chord bd because
dropdown.

Explanation:

Step1: Identify Given Angles

The problem states $\angle AOC \cong \angle BOD$ (given). So the first dropdown for "central angles" should be "AOC and BOD".

Step2: Identify Radii Congruence

AO, CO, BO, DO are radii of $\odot O$. All radii of a circle are congruent, so $AO \cong BO$ and $CO \cong DO$ (radii of same circle are congruent).

Step3: Triangle Congruence Criterion

We have two sides (radii) and included angle (given central angles) congruent. So by SAS (Side - Angle - Side) congruence criterion, $\triangle AOC \cong \triangle BOD$.

Step4: Corresponding Parts of Congruent Triangles

Since $\triangle AOC \cong \triangle BOD$, their corresponding sides $\overline{AC}$ and $\overline{BD}$ are congruent (CPCTC: Corresponding Parts of Congruent Triangles are Congruent).

Answer:

1. First dropdown: AOC and BOD
2. Radii congruence: radii of the same circle are congruent
3. Congruence criterion: SAS
4. Final conclusion reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)