Quickly !

questions:

∠bar ≅ ∠car
∠bpr ≅ ∠cpr
∠apc ≅ ∠bpr
∠apb ≅ ∠apc

 ∠BAR ≅ ∠CAR

Explanation:
Recall what happens when a line bisects an angle. If a line bisects an angle, it forms two equal angles on either side of the bisector. Since you need to prove that line AR bisects ∠BAC.

So in order to prove that it bisects, you'd need to prove that ΔRBA is congruent to ΔRCA

Hence, ∠BAR ≅ ∠CAR