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Question
In the following diagram, AP and BQ are equal and parallel to each other.
Prove that:
- ΔAOP ≅ ΔBOQ.
- AB and PQ bisect each other.
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Solution
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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