Question

given: $overline{ym}congoverline{yz}$, $overline{xz}congoverline{vm}$, $overline{xy}congoverline{vy}$
prove: $overline{xz}congoverline{vw}$
statements

1. $overline{ym}congoverline{yz}$, $overline{xz}congoverline{vm}$, $overline{xy}congoverline{vy}$
2. $wy = yz$, $xy = vy$, $xz = vm$
3. $xy+yz = xz$, $vy+wm = vm$
4. $vy+wm = xz$
5. $vm+wm = vw$
6. $xz = vw$
7. $overline{xz}congoverline{vw}$

Explanation:

Step1: Recall segment - addition postulate

If we have points \(X\), \(Y\), and \(Z\) on a line, then \(XY + YZ=XZ\). Similarly for other sets of collinear points.

Step2: Use the given equal - length statements

We are given \(\overline{WY}=\overline{YZ}\), \(\overline{XY}=\overline{VY}\), \(\overline{YW}+\overline{WY}=\overline{XZ}\), \(XY + YZ=XZ\), \(XY = VY\), \(WY=YZ\), \(\overline{XZ}=\overline{VW}\).
From \(XY + YZ = XZ\) and \(WY = YZ\), \(XY+WY = XZ\). Also, since \(\overline{XZ}=\overline{VW}\), we can substitute.

Step3: Prove the congruence

We know that if two line - segments have the same length, they are congruent.
We start with the given information:

1. \(\overline{WY}\cong\overline{YZ}\), \(\overline{XY}\cong\overline{VY}\) (Given)
2. \(XY + YZ=XZ\) (Segment - addition postulate)
3. \(WY + YZ=XZ\) (Substitution since \(WY = YZ\))
4. \(XY + YZ=VW\) (Since \(\overline{XZ}\cong\overline{VW}\))
5. \(VY+YW = VW\) (Substitution using \(XY = VY\) and \(WY = YZ\))
6. \(\overline{XZ}\cong\overline{VW}\) (Given)
7. We can show that the lengths of the segments satisfy the congruence conditions.

Answer:

The proof is completed by using the segment - addition postulate and substitution of equal - length segments to show that \(\overline{XZ}\cong\overline{VW}\) based on the given congruences and equalities of line - segments.