Question

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

	statement	reason
2	\\(\triangle pqs\\) is equilateral	given
3	\\(overline{pt} \cong overline{qr}\\)	given
4	\\(overline{st} \cong overline{rs}\\)
5	\\(overline{ps} \cong overline{qs}\\)	definition of equilateral triangle
6	\\(\triangle pst \cong \triangle qsr\\)

Explanation:

Step1: Justify $\overline{ST} \cong \overline{RS}$

Definition of midpoint (a midpoint divides a segment into two congruent parts, so since $S$ is the midpoint of $\overline{RT}$, $\overline{ST} \cong \overline{RS}$)

Step2: Justify $\triangle PST \cong \triangle QSR$

We have three pairs of congruent sides: $\overline{PT} \cong \overline{QR}$ (given), $\overline{ST} \cong \overline{RS}$ (from Step1), $\overline{PS} \cong \overline{QS}$ (from equilateral triangle definition). This satisfies the Side-Side-Side (SSS) Congruence Postulate.

Answer:

1. Reason for Statement 4: Definition of midpoint
2. Reason for Statement 6: SSS (Side-Side-Side) Congruence Postulate