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प्रश्न
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.
उत्तर
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.
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