In the triangle GHJ, PQ || GH, PR || GJ, HP = PJ and P is a point on HJ. Prove that triangle GHJ is isosceles.

We will prove these two triangles are congruent.

• $\angle HQP = \angle JRP = 90^\circ$ (Given: PQ $\perp$ GH and PR $\perp$ GJ).
• $HP = PJ$ (Given).
• $\angle QPH = \angle RPJ$ (Given: $\angle P_1 = \angle P_2$).

Therefore, $\triangle HPQ \cong \triangle JPR$ (Angle-Angle-Side (AAS) congruence criterion).

Since the triangles are congruent, their corresponding angles are equal.
Thus, $\angle H = \angle J$ (Corresponding angles of congruent triangles).

Since two base angles of $\triangle GHJ$ are equal ($\angle H = \angle J$), the sides opposite these angles must also be equal ($GJ = GH$). Therefore, $\triangle GHJ$ is an isosceles triangle.