Question

uv ⊥ su, ∠utv ≅ ∠stw, sw ≅ uv, and su ⊥ sw. complete the proof that △stw ≅ △utv.

statement
1 uv ⊥ su
2 ∠utv ≅ ∠stw
3 sw ≅ uv
4 su ⊥ sw
5 ∠s ≅ ∠u
6 △stw ≅ △utv

reason
given
given
given
given

Explanation:

Step1: Determine the measure of angles \( \angle U \) and \( \angle S \)

Since \( \overline{UV}\perp\overline{SU} \), by the definition of perpendicular lines, \( \angle U = 90^{\circ} \). Since \( \overline{SU}\perp\overline{SW} \), by the definition of perpendicular lines, \( \angle S=90^{\circ} \). Then, by the transitive property of equality (\( 90^{\circ}=90^{\circ} \)), \( \angle S\cong\angle U \).

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

We know that \( \angle UTV\cong\angle STW \) (given), \( \overline{SW}\cong\overline{UV} \) (given), and \( \angle S\cong\angle U \) (from step 1). The AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Answer:

1. Reason: Definition of perpendicular (to get \( \angle U = 90^{\circ},\angle S = 90^{\circ}\)) and transitive property of congruence (\( \angle S\cong\angle U \))
2. Reason: AAS (Angle - Angle - Side) congruence criterion.