Question

st ⊥ tu and tu ⊥ qu. complete the proof that δstu ≅ δqut.

1 ∠q ≅ ∠s given
2 st ⊥ tu given
3 tu ⊥ qu given
4 ∠qut ≅ ∠stu
5 tu ≅ tu reflexive property of congruence
6 δstu ≅ δqut

Explanation:

Step1: Determine the type of angles

Since \(\overline{ST}\perp\overline{TU}\) and \(\overline{TU}\perp\overline{QU}\), \(\angle STU = 90^{\circ}\) and \(\angle QUT=90^{\circ}\). So \(\angle QUT\cong\angle STU\) (All right - angles are congruent).

Step2: Use the AAS (Angle - Angle - Side) congruence criterion

We know that \(\angle Q\cong\angle S\) (given), \(\angle QUT\cong\angle STU\) (proven above), and \(\overline{TU}\cong\overline{TU}\) (reflexive property of congruence). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle STU\cong\triangle QUT\).

Answer:

1. All right - angles are congruent; 6. AAS (Angle - Angle - Side) congruence criterion.