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Given: ⊙O with central angles ∠AOC ≅ ∠BOD
Prove: AC ≅ BD

Circle O is shown. Line segments OA, OC, OB, and OD are radii. Line segments connect points A and C and points B and D to form two triangles inside of the circle. Angles AOC and BOD are congruent.

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 OA, OC, OB, and OD are congruent because they are all radii of the circle. Then by the Side-Angle-Side (SAS) congruence theorem, we know that triangle AOC is congruent to triangle BOD. Finally, we can conclude that chord AC is congruent to chord BD because corresponding parts of congruent triangles are congruent (CPCTC).

Answer :

Answer:

We know that central angles

AOC and BOD

are congruent, because it is given. We can say that segments

AO

,

CO

,

BO

, and

DO

are congruent because

all radii of a circle are congruent

. 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

Step-by-step explanation:

Congruence of Central Angles:

Central angles AOC and BOD are congruent.

Congruence of Segments:

Segments AO, CO, BO, and DO are congruent because all radii of a circle are congruent.

Triangle Congruence:

By the SAS (Side-Angle-Side) congruency theorem, triangle AOC is congruent to triangle BOD.

Chord Congruence:

Finally, it is concluded that chord AC is congruent to chord BD using CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

.

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Rewritten by : Jeany

The missing statements and reasons in the two column proof are;

1. AOC and BOD.

2. All radii of a circle are congruent

3. SAS congruency theorem

4. CPCTC

How to complete two column proofs?

A two-column proof is defined as a geometric proof consists of a list of statements, and the reasons that we know those statements are true.

The two column proof of the angles is;

Statement 1: Central angles ∠AOC and ∠BOD are congruent

Reason 1: Given

Statement 2: The segments AO, CO, BO, and DO are congruent

Reason 2: All radii of a circle are congruent

Statement 3: Triangle AOC is congruent to triangle BOD

Reason 3: SAS congruency theorem

Statement 4: Chord AC is congruent to chord BD

Reason 4: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Read more about Two Column Proof at; https://brainly.com/question/1788884

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