Question

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

Prove that:  
(i) ΔAOP≅ ΔBOQ.
(ii) 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.