Question

6 given: $ywcong yz$, $xy = vy$ prove: $overline{xz}congoverline{vw}$ statements reasons 1. $overline{wy}congoverline{yz}$, $overline{xy}congoverline{vy}$ 1. given 2. $wy = yz$, $xy = vy$ 2. 3. $xy + yz=xz$ 3. 4. $vy + yw = xz$ 4. 5. $vy + yw = vw$ 5. 6. $xz = vw$ 6. 7. $overline{xz}congoverline{vw}$ 7.

Explanation:

Step1: Definition of congruent segments

If two segments are congruent, their lengths are equal. So if $\overline{WY}\cong\overline{YZ}$ and $\overline{XY}\cong\overline{VY}$, then $WY = YZ$ and $XY=VY$.

Step2: Segment - addition postulate

The sum of the lengths of two adjacent segments on a line is equal to the length of the combined segment. So $XY + YZ=XZ$.

Step3: Substitution property

Since $WY = YZ$ and $XY = VY$, we can substitute $WY$ for $YZ$ and get $VY+YW = XZ$.

Step4: Segment - addition postulate

The sum of the lengths of two adjacent segments on a line is equal to the length of the combined segment. So $VY + YW=VW$.

Step5: Transitive property of equality

Since $VY + YW=XZ$ and $VY + YW=VW$, then $XZ = VW$.

Step6: Definition of congruent segments

If the lengths of two segments are equal, then the segments are congruent. Since $XZ = VW$, then $\overline{XZ}\cong\overline{VW}$.

Answer:

1. Definition of congruent segments
2. Segment - addition postulate
3. Substitution property
4. Segment - addition postulate
5. Transitive property of equality
6. Definition of congruent segments