Question

tv ⊥ tu, sv ⊥ su, and sw ≅ tw. complete the proof that uw ≅ vw.

1 tv ⊥ tu
2 sv ⊥ su
3 sw ≅ tw
4 ∠s ≅ ∠t
5 ∠swv ≅ ∠twu
6 △svw ≅ △tuw
7 uw ≅ vw

Explanation:

Step1: Identify given information

Statements 1 - 3 are given as per the problem.

Step2: Congruent right angles

Since \( \overline{TV}\perp\overline{TU}\) and \( \overline{SV}\perp\overline{SU}\), \( \angle S\) and \( \angle T\) are right angles. Right angles are congruent.

Step3: Vertical angles congruence

\( \angle SWV\) and \( \angle TWU\) are vertical angles. Vertical angles are congruent.

Step4: Triangle congruence

We have two angles (\( \angle S\cong\angle T\), \( \angle SWV\cong\angle TWU\)) and a non - included side (\( \overline{SW}\cong\overline{TW}\)). By AAS (Angle - Angle - Side) Congruence Theorem, \( \triangle SVW\cong\triangle TUW\).

Step5: CPCTC

Since \( \triangle SVW\cong\triangle TUW\), their corresponding parts \( \overline{UW}\) and \( \overline{VW}\) are congruent (CPCTC).

Answer:

1. Given
2. Given
3. Given
4. Right angles are congruent
5. Vertical angles are congruent
6. AAS (Angle - Angle - Side) Congruence Theorem
7. Corresponding parts of congruent triangles are congruent (CPCTC)