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 (\boxed{quad}) are congruent,
because it is given. we can say that segments ao, co, bo, and do
are congruent because (\boxed{quad}). then
by the (\boxed{quad}) 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 (\boxed{quad}).

Explanation:

Step1: Identify given central angles

The given central angles are $\angle AOC$ and $\angle BOD$, so we know $\angle AOC \cong \angle BOD$ (given).

Step2: Identify radii congruence

Segments $AO$, $CO$, $BO$, and $DO$ are radii of $\odot O$. By the definition of a circle, all radii of a circle are congruent. So $AO \cong CO \cong BO \cong DO$.

Step3: Identify congruence theorem

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

Step4: Conclude chord congruence

Since $\triangle AOC \cong \triangle BOD$, their corresponding sides (the chords $AC$ and $BD$) are congruent. This is because corresponding parts of congruent triangles are congruent (CPCTC).

Answer:

1. $\boldsymbol{\angle AOC \text{ and } \angle BOD}$ (because it is given that $\angle AOC \cong \angle BOD$)
2. $\boldsymbol{\text{all radii of a circle are congruent}}$ (since $AO, CO, BO, DO$ are radii of $\odot O$)
3. $\boldsymbol{SAS}$ (because we have two sides (radii) and the included angle (central angle) congruent)
4. $\boldsymbol{\text{corresponding parts of congruent triangles are congruent (CPCTC)}}$ (since $\triangle AOC \cong \triangle BOD$, their corresponding chords $AC$ and $BD$ are congruent)