Question

given: \\(\overline{am} \cong \overline{cp}\\), \\(\overline{cm} \cong \overline{gp}\\), \\(c\\) is the midpoint of \\(\overline{ag}\\)\
prove: \\(\triangle acm \cong \triangle\\)\
\

statements	reasons	\
---	---	\
1.	1. given	\
2.	2. given	\
3. \\(\underline{\quad} \cong \underline{\quad}\\)	3. def. of a \underline{\quad}	\
4.	4.	\
5.	5.

Explanation:

Step1: Analyze Given Information

We know \( \overline{AM} \cong \overline{CP} \) (given), \( \overline{CM} \cong \overline{GP} \) (given), and \( C \) is the midpoint of \( \overline{AG} \), so \( \overline{AC} \cong \overline{CG} \) (by definition of midpoint).

Step2: Identify Congruence Criterion

We have three pairs of congruent sides: \( \overline{AM} \cong \overline{CP} \), \( \overline{CM} \cong \overline{GP} \), \( \overline{AC} \cong \overline{CG} \). So by SSS (Side - Side - Side) congruence criterion, \( \triangle ACM \cong \triangle CGP \) (wait, maybe a typo, but following the problem's prove \( \triangle ACM \cong \triangle... \)). Let's fill the table:

1. Statement 1: \( \overline{AM} \cong \overline{CP} \) (Reason: Given)
2. Statement 2: \( \overline{CM} \cong \overline{GP} \) (Reason: Given)
3. Statement 3: \( \overline{AC} \cong \overline{CG} \) (Reason: Def. of a midpoint, since \( C \) is midpoint of \( \overline{AG} \))
4. Statement 4: \( \triangle ACM \cong \triangle CGP \) (Wait, maybe the other triangle, but assuming the correct triangles, using SSS)
5. Reason for Step 4: SSS (Side - Side - Side) Congruence Postulate

Answer:

1. Statement: \( \overline{AM} \cong \overline{CP} \); 2. Statement: \( \overline{CM} \cong \overline{GP} \); 3. Statement: \( \overline{AC} \cong \overline{CG} \), Reason: midpoint; 4. Statement: \( \triangle ACM \cong \triangle [correct triangle] \), Reason: SSS. (Filling the table as per geometric proof steps)