In the given figure, ABCD is a quadrilateral in which AB‖DC and P is the midpoint of BC. On producing, AP and DC meet at Q. prove that (i) AB=CQ, (ii) DQ=DC+AB.

Question

In the given figure, ABCD is a quadrilateral in which AB &#x2016; DC and P is the midpoint of BC. On producing, AP and DC meet at Q. prove that (i) AB=CQ, (ii) DQ=DC+AB.

Philoid · Accepted Answer

Given: ABCD is a quadrilateral in which AB‖DC and BP = PC

To prove: AB=CQ and DQ=DC+AB

Proof:

In ∆ABP and ∆PCQ we have,

∠PAB = ∠PQC …alternate angles

∠APB = ∠CPQ … vertically opposite angles

BP = PC … given

Thus by AAS property of congruence,

∆ABP ≅ ∆PCQ

Hence, we know that, corresponding parts of the congruent triangles are equal

∴ AB = CQ …(1)

But, DQ = DC + CQ

= DC + AB …from 1