Question

δpqs is equilateral and s is the midpoint of \\(\overline{rt}\\). complete the proof that \\(\angle pst\cong\angle qsr\\).

Explanation:

Step1: Use the property of an equilateral triangle

Since \(\triangle PQS\) is equilateral, \(PS = QS\) (all sides of an equilateral triangle are equal).

Step2: Prove \(\triangle PST\cong\triangle QSR\)

We have \(PS = QS\) (from Step 1), \(ST=RS\) (given, \(S\) is the mid - point of \(RT\)), and \(PT = QR\) (given). By the Side - Side - Side (SSS) congruence criterion, \(\triangle PST\cong\triangle QSR\).

Step3: Use the property of congruent triangles

If \(\triangle PST\cong\triangle QSR\), then their corresponding angles are equal. So, \(\angle PST\cong\angle QSR\) (corresponding parts of congruent triangles are congruent).

Answer:

\(\angle PST\cong\angle QSR\) because \(\triangle PST\cong\triangle QSR\) (by SSS congruence, since \(PS = QS\) (equilateral triangle property), \(ST = RS\) (mid - point definition), \(PT=QR\) (given)) and corresponding parts of congruent triangles are congruent.