Question

∠srw ≅ ∠tuv, ∠rws ≅ ∠utv, and \\( \overline { vw } \cong \overline { st } \\). complete the proof that \\( \overline { uv } \cong \overline { rs } \\).

1 ∠srw ≅ ∠tuv
2 ∠rws ≅ ∠utv
3 \\( \overline { vw } \cong \overline { st } \\)
4 sw = vw + sv
5 tv = st + sv
6 sw = st + sv
7 tv = sw
8 △rsw ≅ △uvt
9 \\( \overline { uv } \cong \overline { rs } \\)

Explanation:

Step1: Given

\(\angle SRW\cong\angle TUV\) (Given)

Step2: Given

\(\angle RWS\cong\angle UTV\) (Given)

Step3: Given

\(\overline{VW}\cong\overline{ST}\) (Given)

Step4: Segment Addition Postulate

\(SW = VW+SV\) (Segment Addition Postulate)

Step5: Segment Addition Postulate

\(TV = ST + SV\) (Segment Addition Postulate)

Step6: Substitution

Since \(VW = ST\) (from \(\overline{VW}\cong\overline{ST}\)), substitute \(VW\) with \(ST\) in \(SW = VW+SV\) to get \(SW=ST + SV\)

Step7: Transitive Property

From \(TV = ST + SV\) and \(SW=ST + SV\), we have \(TV = SW\) (Transitive Property of Equality)

Step8: ASA (Angle - Side - Angle)

In \(\triangle RSW\) and \(\triangle UVT\), \(\angle SRW\cong\angle TUV\), \(SW = TV\), \(\angle RWS\cong\angle UTV\). So, \(\triangle RSW\cong\triangle UVT\) (ASA Congruence Criterion)

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

Since \(\triangle RSW\cong\triangle UVT\), then \(\overline{UV}\cong\overline{RS}\) (CPCTC)

Answer:

The reasons for each step are: 1. Given; 2. Given; 3. Given; 4. Segment Addition Postulate; 5. Segment Addition Postulate; 6. Substitution Property of Equality; 7. Transitive Property of Equality; 8. ASA (Angle - Side - Angle) Congruence Criterion; 9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)