Question

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 say that segments ao, co, bo, and do
are congruent because dropdown. then
by the dropdown 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 dropdown.

Explanation:

Step1: Identify given central angles

The given central angles are $\angle AOC \cong \angle BOD$, so the first blank is filled with $\angle AOC$ and $\angle BOD$.

Step2: Reason for congruent radii

Segments $AO$, $CO$, $BO$, and $DO$ are radii of the same circle $\odot O$, and all radii of a circle are congruent. So the second blank is "all radii of a circle are congruent".

Step3: Congruency theorem for triangles

We have two sides (radii) and the included angle (given central angles) congruent, so the congruency theorem is SAS (Side - Angle - Side). So the third blank is "SAS".

Step4: Reason for congruent chords

Since $\triangle AOC \cong \triangle BOD$ (by SAS), their corresponding sides (chords $AC$ and $BD$) are congruent. So the fourth blank is "corresponding parts of congruent triangles are congruent (CPCTC)".

Answer:

1. $\boldsymbol{\angle AOC \cong \angle BOD}$
2. all radii of a circle are congruent
3. SAS
4. corresponding parts of congruent triangles are congruent (CPCTC)