Question

statement
1 ∠rvx ≅ ∠twu
2 ∠r ≅ ∠t
3 overline{wx} ≅ overline{uv}
4 vx = wx + vw
5 uw = uv + vw
6 vx = uv + vw
7 uw = vx
8 △rvx ≅ △twu
9 overline{rv} ≅ overline{tw}
reason

Explanation:

Step1: Vertical angles are congruent

$\angle RVX\cong\angle TWU$ because vertical angles are congruent.

Step2: Given

$\angle R\cong\angle T$ is given.

Step3: Given

$\overline{WX}\cong\overline{UV}$ is given.

Step4: Segment addition postulate

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

Step5: Segment addition postulate

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

Step6: Substitution

Since $WX\cong UV$, substitute $WX$ with $UV$ in $VX = WX + VW$, so $VX=UV + VW$.

Step7: Transitive property of equality

Since $UW = UV+VW$ and $VX = UV + VW$, then $UW = VX$ by the transitive property of equality.

Step8: AAS (Angle - Angle - Side)

In $\triangle RVX$ and $\triangle TWU$, $\angle R\cong\angle T$, $\angle RVX\cong\angle TWU$, and $VX = UW$. So, $\triangle RVX\cong\triangle TWU$ by AAS (Angle - Angle - Side) congruence criterion.

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

Since $\triangle RVX\cong\triangle TWU$, then $\overline{RV}\cong\overline{TW}$ by CPCTC.

Answer:

1. Vertical angles are congruent; 2. Given; 3. Given; 4. Segment addition postulate; 5. Segment addition postulate; 6. Substitution; 7. Transitive property of equality; 8. AAS (Angle - Angle - Side); 9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)