Geometry

Triangles RST and XYZ are congruent.  Point $P$ lies on $\overline{ST},$ and point $W$ lies on $\overline{YZ}.$  Select all the true statements.

If XY = ST, then \triangle RSP is similar to \triangle XYW.
If \overline{RP} \perp \overline{XW} and \overline{ST} \perp \overline{WZ}, then triangle RSP is isoceles
If $\overline{RP}$ bisects $\angle SRT,$ then \triangle RSP is isosceles
If P is the midpoint of \overline{ST}, then $\triangle RSP is scalene