Question

1. developing proof write a complete proof by rearranging the reasons listed on the pieces of paper. given > uv ≅ xy, vw ≅ wx, wx ≅ yz prove > uw ≅ xz statements reasons 1. uv ≅ xy, vw ≅ wx, wx ≅ yz transitive property of segment congruence 2. vw ≅ yz addition property of equality 3. uv = xy, vw = yz definition of congruent segments 4. uv + vw = xy + yz given 5. uv + vw = uw, xy + yz = xz segment addition postulate 6. uw = xz definition of congruent segments 7. uw ≅ xz substitution property of equality

Explanation:

Step1: Start with given

Given $\overline{UV}\cong\overline{XY},\overline{VW}\cong\overline{WX},\overline{WX}\cong\overline{YZ}$.

Step2: Use definition of congruent segments

By the definition of congruent segments, if $\overline{UV}\cong\overline{XY}$ then $UV = XY$, if $\overline{VW}\cong\overline{WX}$ then $VW = WX$ and if $\overline{WX}\cong\overline{YZ}$ then $WX=YZ$. So we have $UV = XY$ and $VW = YZ$.

Step3: Apply addition property of equality

Adding the two - equalities $UV = XY$ and $VW = YZ$, we get $UV + VW=XY + YZ$ by the addition property of equality.

Step4: Use segment addition postulate

By the segment addition postulate, $UV + VW = UW$ and $XY + YZ = XZ$.

Step5: Use substitution

Substituting $UV + VW$ with $UW$ and $XY + YZ$ with $XZ$ in the equation $UV + VW=XY + YZ$, we get $UW = XZ$.

Step6: Use definition of congruent segments again

Since $UW = XZ$, by the definition of congruent segments, $\overline{UW}\cong\overline{XZ}$.

Answer:

The proof is as follows:

Statements	Reasons
2. $UV = XY,VW = YZ$	Definition of congruent segments
3. $UV + VW=XY + YZ$	Addition property of equality
4. $UV + VW = UW,XY + YZ = XZ$	Segment Addition Postulate
5. $UW = XZ$	Substitution
6. $\overline{UW}\cong\overline{XZ}$	Definition of congruent segments