In the figure, it is given that BR = PC and ∠ACB= ∠QRP and AB ∥ PQ. Prove that
AC = QR.

Given: BR = PC and ∠ ACB = ∠ QRP , AB || PQ

To Prove: AC = QR

Proof:

In Δ ABC, we have

BC = BR + RC

In Δ PQR

PR = PC + RC

But , BR = PC [Given]

So, BC = PC + RC and PR = BR + RC

⇒ BC = PR

So, in Δ ABC and Δ PQR, we have

∠ ACB = ∠ QRP [Given]

BC = PR [Proved Above]

∠ ABC = ∠ QPR [AB || PQ, alternate interior angles]

Thus, Δ ABC ≅ Δ PQR [Angle – Side – Angle]

∴ AC = QR [C. P. C. T]