Question

δrst and δpqt are equilateral. complete the proof that δqrt ≅ δpst.
1 δrst is equilateral
2 δpqt is equilateral
3 \overline{rt} ≅ \overline{st}
4 \overline{pt} ≅ \overline{qt}
5 ∠pts ≅ ∠qtr
6 δqrt ≅ δpst

Explanation:

Step1: Given

Statement 1: \(\triangle RST\) is equilateral. Given in the problem.
Statement 2: \(\triangle PQT\) is equilateral. Given in the problem.

Step2: Property of equilateral triangle

Statement 3: \(\overline{RT}\cong\overline{ST}\). In an equilateral triangle \(\triangle RST\), all sides are equal.
Statement 4: \(\overline{PT}\cong\overline{QT}\). In an equilateral triangle \(\triangle PQT\), all sides are equal.

Step3: Vertical angles

Statement 5: \(\angle PTS\cong\angle QTR\). Vertical angles are congruent.

Step4: SAS (Side - Angle - Side) congruence criterion

Statement 6: \(\triangle QRT\cong\triangle PST\). We have \(\overline{RT}\cong\overline{ST}\), \(\angle QTR\cong\angle PTS\), \(\overline{QT}\cong\overline{PT}\). By the SAS (Side - Angle - Side) congruence criterion for triangles.

Answer:

1. Given
2. Given
3. All sides of an equilateral triangle are equal
4. All sides of an equilateral triangle are equal
5. Vertical angles are congruent
6. SAS (Side - Angle - Side) congruence criterion