Question

complete the proof that (overline{rx}congoverline{tu}).
1 (overline{uv}congoverline{wx})
2 (angle rcongangle t)
3 (angle twucongangle rvx)
4 (vx = wx+vw)
5 (uw = uv+vw)
6 (vx = uv+vw)
7 (uw = vx)
8 (\triangle rvxcong\triangle twu)
9 (overline{rx}congoverline{tu})

Explanation:

Step1: Given

Statement 1: $\overline{UV}\cong\overline{WX}$ is given.

Step2: Given

Statement 2: $\angle R\cong\angle T$ is given.

Step3: Vertical angles are congruent

$\angle TWU$ and $\angle RVX$ are vertical angles, so $\angle TWU\cong\angle RVX$ (Vertical Angles Theorem).

Step4: Segment addition postulate

For segment $VX$, by the segment addition postulate, $VX = WX+VW$.

Step5: Segment addition postulate

For segment $UW$, by the segment addition postulate, $UW = UV + VW$.

Step6: Substitution

Since $\overline{UV}\cong\overline{WX}$ (from step 1), substitute $WX$ with $UV$ in $VX = WX + VW$, so $VX=UV + VW$.

Step7: Transitive property of equality

From $UW = UV+VW$ (step 5) and $VX = UV + VW$ (step 6), we get $UW = VX$ (Transitive property: if $a=b$ and $b = c$, then $a=c$).

Step8: AAS (Angle - Angle - Side) congruence criterion

We have $\angle R\cong\angle T$ (step 2), $\angle RVX\cong\angle TWU$ (step 3), and $UW = VX$ (step 7). So, $\triangle RVX\cong\triangle TWU$ by AAS (Angle - Angle - Side) congruence.

Step9: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since $\triangle RVX\cong\triangle TWU$, then $\overline{RX}\cong\overline{TU}$ (CPCTC).

Answer:

1. Given
2. Given
3. Vertical Angles Theorem
4. Segment Addition Postulate
5. Segment Addition Postulate
6. Substitution (because $\overline{UV}\cong\overline{WX}$)
7. Transitive Property of Equality
8. AAS (Angle - Angle - Side)
9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)