Question

given: (overline{xy}congoverline{uv}), (overline{yz}congoverline{tu}) prove: (overline{xz}congoverline{tv}) statements: 1. (overline{xy}congoverline{uv}), (overline{yz}congoverline{tu}) 2. (xy = uv), (yz = tu) 3. (xy+yz = xz), (tu + uv=tv) 4. (uv + yz=xz), (yz + uv = tv) 5. (xz = tv) 6. (overline{xz}congoverline{tv}) reasons: 1. given 2. definition of congruent segments 3. segment - addition postulate 4. substitution property 5. commutative property of addition 6. definition of congruent segments

Explanation:

Step1: Given information

Given $\overline{XY}\cong\overline{UV}$, $\overline{YZ}\cong\overline{TU}$, $XY = UV$, $YZ=TU$.

Step2: Segment - addition postulate

By the segment - addition postulate, $XY + YZ=XZ$ and $TU + UV = TV$.

Step3: Substitute equal segments

Since $XY = UV$ and $YZ = TU$, we can substitute into the above equations. So $UV+YZ = XZ$ and $YZ + UV=TV$.

Step4: Transitive property of equality

If $UV + YZ=XZ$ and $YZ + UV = TV$, then $XZ = TV$.

Step5: Definition of congruent segments

If $XZ = TV$, then $\overline{XZ}\cong\overline{TV}$ by the definition of congruent segments (two segments are congruent if they have the same length).

Answer:

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