Question

1. select the correct proof from the options listed.

given: $overline{bc}$ is a diameter of circle $o$; $overline{ab} \parallel \overline{fo}$; $\overline{db} \parallel \overline{go}$; $\overline{cf} \cong \overline{cg}$
prove: $overline{ca} \cong \overline{cd}$
a.

statements	reasons
2. $\angle cfo \cong \angle cgo$; $\angle cof \cong \angle cog$	2. evident from diagram
3. $overline{co} \cong \overline{co}$	3. reflexive property
4. $\triangle cfo \cong \triangle cgo$	4. angle - angle - side theorem
5. $overline{fo} \cong \overline{go}$	5. c.p.c.t.c
6. $overline{ca} \cong \overline{cd}$	6. in the same circle, chords equidistant from the center of the circle are congruent.

b.

statements	reasons
2. $overline{ca} \cong \overline{cd}$; $overline{ab} \cong \overline{db}$	2. evident from diagram
3. $overline{cb} \cong \overline{cb}$	3. reflexive property
4. $\triangle cab \cong \triangle cdb$	4. side - side - side postulate
5. $overline{ca} \cong \overline{cd}$	5. c.p.c.t.c

Explanation:

Step1: Analyze Option A logic

1. $\overline{BC}$ is diameter, $\overline{AB}\parallel\overline{FO}$, $\overline{DB}\parallel\overline{GO}$, $\overline{CF}\cong\overline{CG}$ (Given)
2. $\angle CFO$ and $\angle CGO$ are right angles (since $\overline{AB}\parallel\overline{FO}$, $\overline{DB}\parallel\overline{GO}$, and $\overline{BC}$ is a diameter, so $\angle CAB=\angle CDB=90^\circ$, making $\angle CFO=\angle CGO=90^\circ$; $\overline{CO}\cong\overline{CO}$ (Reflexive), $\overline{CF}\cong\overline{CG}$ (Given), so $\triangle CFO\cong\triangle CGO$ by HL, but the reason says AAS which is also valid for right triangles)
3. $\overline{FO}\cong\overline{GO}$ (CPCTC)
4. $\overline{FO}$ is distance from $O$ to $\overline{CA}$, $\overline{GO}$ is distance from $O$ to $\overline{CD}$. Since $\overline{FO}\cong\overline{GO}$, $\overline{CA}\cong\overline{CD}$ (chords equidistant from center are congruent)

Step2: Analyze Option B logic

1. Step 2 states $\overline{CA}\cong\overline{CD}$ as "evident from diagram", which is the statement we need to prove. This is circular reasoning and invalid.

Answer:

A.
Statements

1. $\overline{BC}$ is a diameter of circle $O$; $\overline{AB} \parallel \overline{FO}$; $\overline{DB} \parallel \overline{GO}$; $\overline{CF} \cong \overline{CG}$
2. $\angle CFO \cong \angle CGO$; $\angle COF \cong \angle COG$
3. $\overline{CO} \cong \overline{CO}$
4. $\triangle CFO \cong \triangle CGO$
5. $\overline{FO} \cong \overline{GO}$
6. $\overline{CA} \cong \overline{CD}$

Reasons

1. Given
2. Evident from diagram
3. Reflexive Property
4. Angle-Angle-Side Theorem
5. C.P.C.T.C
6. In the same circle, chords equidistant from the center of the circle are congruent.