Question

In the following diagram, AP and BQ are equal and parallel to each other. 

Prove that:  

1. ΔAOP ≅ ΔBOQ.
2. AB and PQ bisect each other.

Answer 1

In the figure, AP and BQ are equal and parallel to each other. 

∴ AP = BQ and AP || BQ. 

We need to prove that 

(i) ΔAOP≅ ΔBOQ.

(ii) AB and PQ bisect each other

(i) ∵ AP || BQ 

∴ ∠APO = ∠BOQ           ...[Alternate angles]...(1)

and ∠PAO = ∠QBO       ...[Alternate angles]...(2)

Now in ΔAOP and ΔBOQ.

∠APO = ∠BQO            ...[From (1)]

AP = BQ                      ...[Given]

∠PAO = ∠QBO            ...[From (1)]

∴ By Angel-Side-Angel criterion of congruence, we have

ΔAOP ≅ ΔBOQ

(ii) The corresponding parts of the congruent triangles are congruent.

∴ OP = OQ                ...[c. p. c. t]

OA = OB                  ...[c. p. c. t]

Hence, AB and PQ bisect each other.