Find the error in the following proof

Question

In ∆PQR, let ray RS bisect angle PRQ and be perpendicular to line PQ at S. Then ∠PRS ≅ ∠QSR since

all right angles are congruent. Therefore ∆PQR ≅ ∆QRS by ASA, making PR = QR. This “proves” that all triangles are isosceles.

Answer 1

The postulate says, point (R in this case) not

on the line (PQ), there is a unique perpendicular height,

but cannot guarantee THAT height bisects the angle

There is a Unique angle bisector and a unique perpendicular

height, but such rays are NOT always the same.