Question

△pqs is equilateral and s is the midpoint of rt. complete the proof that ∠spt ≅ ∠rqs.

statement
1 △pqs is equilateral
2 s is the midpoint of rt
3 pt ≅ qr
4 st ≅ rs
5 ps ≅ qs
6 △pst ≅ △qsr
7 ∠spt ≅ ∠rqs

reason
given
given
given

sss

Explanation:

Step1: Mid - point property

Since \(S\) is the mid - point of \(\overline{RT}\), by the definition of a mid - point, \(\overline{ST}\cong\overline{RS}\).

Step2: Equilateral triangle property

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

Step3: Corresponding parts of congruent triangles

Since \(\triangle PST\cong\triangle QSR\) (by SSS congruence criterion), then \(\angle SPT\cong\angle RQS\) (by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem).

Answer:

1. Definition of mid - point; 5. Definition of equilateral triangle; 7. CPCTC.