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 dropdown are congruent, because it is given. segments ao, co, bo, and do reason because reason. by the dropdown congruency theorem, triangle aoc is congruent to triangle dropdown. we can conclude that chord ac is congruent to chord bd because dropdown.

Explanation:

Step1: Identify Central Angles

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

Step2: Identify Congruency Theorem

Segments $AO$, $CO$, $BO$, $DO$ are radii of the circle, so $AO = BO$ and $CO = DO$. With $\angle AOC \cong \angle BOD$, by the SAS (Side - Angle - Side) congruency theorem, $\triangle AOC \cong \triangle BOD$. So the second blank is "SAS".

Step3: Chord Congruence Reason

If two chords are corresponding parts of congruent triangles (here, $\triangle AOC \cong \triangle BOD$), then the chords are congruent. Also, in a circle, congruent central angles intercept congruent chords. So the last blank is "congruent central angles intercept congruent chords (or corresponding parts of congruent triangles are congruent)".

Answer:

First blank: AOC and BOD; Second blank: SAS; Third blank: congruent central angles intercept congruent chords (or corresponding parts of congruent triangles are congruent)