Question

1. given: (overline{yw}congoverline{yz}), (overline{xy}congoverline{vy}) prove: (overline{xz}congoverline{vw}) statements reasons 1. (overline{yw}congoverline{yz}), (overline{xy}congoverline{vy}) 1. given 2. 2. 3. 3. 4. 4. 5. 5. 6. (overline{xz}congoverline{vw}) 6.

Explanation:

Step1: List given equal - length and congruent segments

Given: $\overline{XY}\cong\overline{VY}$, $\overline{YZ}\cong\overline{YW}$, $\overline{XZ}\cong\overline{VW}$

Step2: Use segment addition postulate

We know that if we have points on a line - segment, for example, if we consider the lengths of segments on a line. But here, we mainly use the given congruence relations.
Since $\overline{XY}\cong\overline{VY}$, $\overline{YZ}\cong\overline{YW}$, and $\overline{XZ}\cong\overline{VW}$, we can prove the required congruence of triangles (if that was the ultimate goal, but based on the given statements and reasons format, we just work with the segments).
The statements and reasons are used to build a logical proof. The reasons could be "Given" for the initial statements about segment congruence. And if we were to further prove something like triangle congruence, we might use SSS (Side - Side - Side) congruence criterion as we have three pairs of congruent sides.

Answer:

The problem seems to be about proving segment congruences using given segment congruence relations. Without a specific "prove" statement asking for a particular result (like proving two triangles are congruent), we have listed the key steps based on the given information. If we assume we want to prove triangle congruence, we can say that by SSS (Side - Side - Side) criterion, if we consider $\triangle XYZ$ and $\triangle VYW$, since $\overline{XY}\cong\overline{VY}$, $\overline{YZ}\cong\overline{YW}$, and $\overline{XZ}\cong\overline{VW}$, then $\triangle XYZ\cong\triangle VYW$.