The problem provided in the image gives the following:

Given:

• $\overline{AB} \cong \overline{CD}$

To Prove:

• $\overline{AC} \cong \overline{BD}$

From the diagram, points $A$, $B$, $C$, and $D$ are collinear with $\overline{AD}$ being a straight segment.

Step-by-step proof:

1. Given: $\overline{AB} \cong \overline{CD}$
   Reason: This is the information provided in the problem.

2. AC and BD are segments of the straight line AD:

   • The segments $\overline{AC}$ and $\overline{BD}$ can be considered as composed of smaller parts. Specifically, $\overline{AC} = \overline{AB} + \overline{BC}$ and $\overline{BD} = \overline{BC} + \overline{CD}$.
     Reason: By segment addition postulate.

3. Substitute the congruent parts:

   • Since $\overline{AB} \cong \overline{CD}$, substitute into the equations for $\overline{AC}$ and $\overline{BD}$.
     This gives:
     $\overline{AC} = \overline{AB} + \overline{BC}$ and
     $\overline{BD} = \overline{AB} + \overline{BC}$.
     Reason: Substitution property of congruence.

4. Conclude that $\overline{AC} \cong \overline{BD}$:

   • Since $\overline{AC}$ and $\overline{BD}$ are composed of the same segments, they must be congruent.
     Reason: Definition of congruent segments (equal parts sum to equal wholes).

Thus, you have proven that $\overline{AC} \cong \overline{BD}$.

Would you like more details on any step? Let me know if you have questions!

Here are 5 related questions:

1. What is the segment addition postulate?
2. How do you prove two segments are congruent?
3. Can this proof be adapted using coordinate geometry?
4. What are other ways to prove congruence between line segments?
5. How does congruence differ from equality in geometry?

Tip: In geometry, when proving congruence between segments or shapes, always rely on clear postulates and properties, such as segment addition and substitution.