Question

∠svt ≅ ∠quw, ∠t ≅ ∠w, and \overline{tu} ≅ \overline{vw}. complete the proof that △quw ≅ △svt.

1 ∠svt ≅ ∠quw
2 ∠t ≅ ∠w
3 \overline{tu} ≅ \overline{vw}
4 uw = vw + uv
5 tv = tu + uv
6 uw = tu + uv
7 tv = uw
8 △quw ≅ △svt

Explanation:

Step1: Given

\(\angle SVT\cong\angle QUW\) (Given)

Step2: Given

\(\angle T\cong\angle W\) (Given)

Step3: Given

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

Step4: Segment Addition Postulate

\(UW = VW+UV\) (Segment Addition Postulate)

Step5: Segment Addition Postulate

\(TV = TU + UV\) (Segment Addition Postulate)

Step6: Substitution

Since \(TU\cong VW\), substitute \(VW\) with \(TU\) in \(UW = VW+UV\) gives \(UW=TU + UV\)

Step7: Transitive Property of Equality

From \(TV = TU + UV\) and \(UW=TU + UV\), we have \(TV = UW\) (Transitive Property of Equality)

Step8: ASA (Angle - Side - Angle)

In \(\triangle QUW\) and \(\triangle SVT\), \(\angle QUW\cong\angle SVT\), \(UW = TV\), \(\angle W\cong\angle T\). So \(\triangle QUW\cong\triangle SVT\) (ASA Congruence Criterion)

Answer:

1. Given
2. Given
3. Given
4. Segment Addition Postulate
5. Segment Addition Postulate
6. Substitution Property
7. Transitive Property of Equality
8. ASA (Angle - Side - Angle) Congruence Criterion