Question

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

statement\treason
1 vw ≅ tu\tgiven
2 ∠quw ≅ ∠svt\tgiven
3 ∠t ≅ ∠w\tgiven
4 uw = vw + uv
5 tv = tu + uv
6 uw = tu + uv
7 tv = uw
8 △quw ≅ △svt

Explanation:

Step1: State segment - addition postulate

By the segment - addition postulate, if we have three collinear points \(V\), \(W\), \(U\), then \(UW = VW+UV\).

Step2: State segment - addition postulate

By the segment - addition postulate, if we have three collinear points \(T\), \(U\), \(V\), then \(TV = TU + UV\).

Step3: Substitute \(VW\) with \(TU\)

Since \(VW\cong TU\) (given), we can substitute \(VW\) with \(TU\) in the equation \(UW = VW+UV\), getting \(UW = TU + UV\).

Step4: Transitive property of equality

From \(TV = TU + UV\) and \(UW = TU + UV\), by the transitive property of equality, we have \(TV = UW\).

Step5: Angle - Side - Angle (ASA) congruence criterion

We know that \(\angle QUW\cong\angle SVT\) (given), \(TV = UW\) (proven above), and \(\angle T\cong\angle W\) (given). So, by the Angle - Side - Angle (ASA) congruence criterion, \(\triangle QUW\cong\triangle SVT\).

Answer:

1. Segment - addition postulate
2. Segment - addition postulate
3. Substitution property (since \(VW\cong TU\))
4. Transitive property of equality
5. ASA (Angle - Side - Angle) congruence criterion