Question

given: $overline{yw}congoverline{yz}$, $overline{xy}congoverline{vy}$ prove: $overline{xz}congoverline{vm}$ statements 1. $overline{wy}congoverline{yz}$, $overline{xy}congoverline{vy}$ 2. $wy = yz$, $xy = vy$ 3. $xy+yz = xz$ 4. $vy + ym=xz$ 5. $vy+ym = vm$ 6. $xz = vm$ 7. $overline{xz}congoverline{vm}$ reasons 1. given 2. definition of congruent segments 3. segment - addition postulate 4. substitution property 5. segment - addition postulate 6. substitution property 7. definition of congruent segments

Explanation:

Step1: Given information

Given $\overline{WY}\cong\overline{YZ}$, $\overline{XY}\cong\overline{VY}$, $WY = YZ$, $XY=VY$

Step2: Segment - addition postulate

By segment - addition postulate, $XY + YZ=XZ$

Step3: Substitution

Since $XY = VY$ and $YZ = WY$, we substitute to get $VY+WY = XZ$

Step4: Given $VY + YM=VM$

This is a given segment - addition relationship

Step5: Substitution and equality

Since $VY+WY = XZ$ and $VY + YM=VM$, and $WY = YM$ (by congruence and equality relationships), we have $XZ=VM$

Step6: Definition of congruent segments

If two segments have equal lengths, they are congruent. So $\overline{XZ}\cong\overline{VM}$

Answer:

The proof is completed as shown in the steps above to prove $\overline{XZ}\cong\overline{VM}$ given $\overline{WY}\cong\overline{YZ}$, $\overline{XY}\cong\overline{VY}$