Question

1 △stw ≅ △utv given
2 (overline{sw}congoverline{uv}) cpctc
3 (overline{st}congoverline{tu}) cpctc
4 (overline{tw}congoverline{tv}) cpctc
5 (sv = st + tv) additive property of length
6 (uw=tu + tw) additive property of length
7 (sv = tu + tw) substitution
8 (sv = uw) transitive property of equality
9 (overline{vw}congoverline{vw}) reflexive property of congruence
10 (△uvwcong△swv)

Explanation:

Step1: Recall the congruence criteria

We know that for two triangles to be congruent, we have several criteria like SSS (Side - Side - Side), SAS (Side - Angle - Side), ASA (Angle - Side - Angle), AAS (Angle - Angle - Side) and HL (Hypotenuse - Leg) for right - angled triangles.

Step2: Analyze the given information

We have \(\overline{SW}\cong\overline{UV}\) (from step 2), \(\overline{VW}\cong\overline{VW}\) (from step 9) and \(SV = UW\) (from step 8). Since \(SV=ST + TV\) and \(UW=TU + TW\), and \(\overline{ST}\cong\overline{TU}\), \(\overline{TW}\cong\overline{TV}\) (from steps 3 and 4), we can rewrite \(SV\) and \(UW\) in terms of congruent segments.

In \(\triangle UVW\) and \(\triangle SWV\), we have \(\overline{SW}\cong\overline{UV}\), \(\overline{VW}\cong\overline{VW}\) and \(\overline{SV}\cong\overline{UW}\)

Step3: Apply the SSS criterion

The SSS (Side - Side - Side) congruence criterion states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Answer:

\(\triangle UVW\cong\triangle SWV\) by the SSS (Side - Side - Side) congruence criterion.