Question

st ≅ rv, and ∠urv ≅ ∠tsu. complete the proof that △stu ≅ △rvu.

1 △rsu is equilateral given
2 st ≅ rv given
3 ∠urv ≅ ∠tsu given
4 ru ≅ su
5 △stu ≅ △rvu

Explanation:

Step1: Recall the property of an equilateral triangle

In an equilateral triangle, all sides are equal. Since \(\triangle RSU\) is equilateral (given), then \(\overline{RU}\cong\overline{SU}\) (by the definition of an equilateral triangle: all sides of an equilateral triangle are congruent).

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

We have \(\overline{ST}\cong\overline{RV}\) (given), \(\angle URV\cong\angle TSU\) (given), and \(\overline{RU}\cong\overline{SU}\) (from step 1).
By the SAS (Side - Angle - Side) congruence criterion (\(SAS\): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent), \(\triangle STU\cong\triangle RVU\).

Answer:

For statement 4, the reason is "Definition of equilateral triangle". For statement 5, the reason is "\(SAS\) (Side - Angle - Side) Congruence Criterion".