Question

given p is the midpoint of RT and SC what congruence theorem justifies the congruence statement triangle RSP is congruent to triangle TCP

Answer 1

If P is the midpoint of RT and SC, then this point divides both lines into two equal segments:

RT is divided into RP and PT → RP=PT

SC is divided into SP and PC → SP=PC

Next, ∠RPS, marked in the sketch with red, and ∠CPT, marked in the sketch with yellow, are on opposite sides of point P, i.e. they share the vertex at point P, which makes them vertically opposite angles.

Vertically opposite angles are congruent so that ∠RPS=∠CPT

The corresponding sides are equal:

RP=PT

SP=PC

And the corresponding angles between the sides are also equal:

∠RPS = ∠CPT

You can say that ΔRPS and ΔTPC are congruent by the Side-Angle-Side postulate.