Question

△pqs is equilateral. complete the proof that △pst ≅ △qsr.

statement
1 △pqs is equilateral
2 ∠spt ≅ ∠rqs
3 ◽{qr} ≅ ◽{pt}
4 ◽{ps} ≅ ◽{qs}
5 △pst ≅ △qsr

reason

Explanation:

Step1: Given

Statement 1 is given in the problem.

Step2: Given

Statement 2 is given in the problem (marked angles in the diagram).

Step3: Given

Statement 3 is given in the problem (marked segments in the diagram).

Step4: Definition of equilateral triangle

Since \(\triangle PQS\) is equilateral, all its sides are equal. So, \(PS = QS\) (by the definition of an equilateral triangle).

Step5: SAS (Side - Angle - Side) Congruence Criterion

We have \(PT\cong QR\) (from statement 3), \(\angle SPT\cong\angle RQS\) (from statement 2), and \(PS\cong QS\) (from statement 4). By the SAS (Side - Angle - Side) congruence criterion, \(\triangle PST\cong\triangle QSR\).

Answer:

1. Given
2. Given
3. Given
4. Definition of equilateral triangle
5. SAS (Side - Angle - Side) Congruence Criterion