Question

given: ⊙o with central angles ∠aoc≅∠bod. prove: (overline{ac}congoverline{bd}). complete the missing parts of the paragraph proof. proof. we know that central angles are congruent, because it is given. we can say that segments ao, co, bo, and do are congruent because. then by the 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 all chords of a circle are congruent; cpctc; of the definition of chords

Explanation:

Step1: Identify given information

We are given that $\angle AOC\cong\angle BOD$.

Step2: Recall property of circle - radii

Segments $AO$, $CO$, $BO$, and $DO$ are radii of the circle $\odot O$. Radii of the same circle are congruent.

Step3: Determine congruency theorem

Since we have two pairs of congruent sides (radii) and the included - angles are congruent ($\angle AOC\cong\angle BOD$), by the Side - Angle - Side (SAS) congruency theorem, $\triangle AOC\cong\triangle BOD$.

Step4: Use corresponding parts of congruent triangles

If two triangles are congruent, then their corresponding parts are congruent (CPCTC). Since $\triangle AOC\cong\triangle BOD$, and $AC$ and $BD$ are corresponding parts of these congruent triangles, $AC\cong BD$.

Answer:

We know that central angles $\angle AOC$ and $\angle BOD$ are congruent, because it is given. We can say that segments $AO$, $CO$, $BO$, and $DO$ are congruent because they are radii of the same circle. Then by the SAS 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 CPCTC.