Question

△svw is equilateral. complete the proof that △uvw ≅ △uts.

(image of a geometric figure with points w, s, v, t, u and a table with statements and reasons:

1. △svw is equilateral - given
2. ∠uvw ≅ ∠tsu - given
3. ∠vwu ≅ ∠sut - vertical angle theorem
4. ( overline{uw} ≅ overline{su} ) - definition of equilateral triangle
5. △uvw ≅ △uts - blank for reason)

Explanation:

Step1: Identify congruent parts

We have \( \angle UNV \cong \angle TSU \) (given), \( \angle VUW \cong \angle SUT \) (vertical angles), and \( \overline{UW} \cong \overline{SU} \) (from equilateral triangle, so sides are equal).

Step2: Apply ASA congruence

The Angle - Side - Angle (ASA) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Here, for \( \triangle UVW \) and \( \triangle UTS \), we have two angles (\( \angle UNV \cong \angle TSU \), \( \angle VUW \cong \angle SUT \)) and the included side (\( \overline{UW} \cong \overline{SU} \)) congruent. So by ASA, \( \triangle UVW \cong \triangle UTS \).

Answer:

Angle - Side - Angle (ASA) Congruence Criterion