Question

$\overline{sw}\cong\overline{su}$. complete the proof that $\triangle stw\cong\triangle sxu$.

Explanation:

Step1: Analyze ∠U ≅ ∠W

From the diagram, we can see that ∠U and ∠W are marked with the same tick mark, which indicates that these two angles are given as congruent (or it can be inferred from the diagram's angle markings). So the reason for ∠U ≅ ∠W is "Given (or marked angles are congruent)".

Step2: Analyze ∠S ≅ ∠S

∠S is a common angle to both triangles \( \triangle STW \) and \( \triangle SXU \). The reason for a common angle being congruent to itself is the "Reflexive Property of Congruence" (which states that any geometric figure is congruent to itself, so \( \angle S \cong \angle S \) by reflexive property).

Step3: Analyze \( \triangle STW \cong \triangle SXU \)

We have \( \overline{SW} \cong \overline{SU} \) (given), \( \angle U \cong \angle W \) (from step 1), and \( \angle S \cong \angle S \) (from step 2). So by the "Angle - Angle - Side (AAS) Congruence Criterion", which states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent. Here, the two angles are \( \angle U \cong \angle W \), \( \angle S \cong \angle S \) and the non - included side is \( \overline{SW} \cong \overline{SU} \), so we can conclude \( \triangle STW \cong \triangle SXU \) by AAS.

Answer:

1. Reason for \( \overline{SW} \cong \overline{SU} \): Given
2. Reason for \( \angle U \cong \angle W \): Marked angles are congruent (or Given from diagram)
3. Reason for \( \angle S \cong \angle S \): Reflexive Property of Congruence
4. Reason for \( \triangle STW \cong \triangle SXU \): AAS (Angle - Angle - Side) Congruence Criterion