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प्रश्न
ABCD is a quadrilateral in which AB || DC and AD = BC. Prove that ∠A = ∠B and ∠C = ∠D.
उत्तर
Given: ABCD is a quadrilateral such that AB || DC and AD = BC
Construction: Extend AB to E and draw a line CE parallel to AD.
Proof: Since, AD || CE and transversal AE cuts them at A and E, respectively.
∴ ∠A + ∠E = 180° ....[Since, sum of cointerior angles is 180°]
⇒ ∠A = 180° – ∠E ...(i)
Since, AB || CD and AD || CE
So, quadrilateral AECD is a parallelogram.
⇒ AD = CE
⇒ BC = CE ...[∵ AD = BC, given]
Now, in ΔBCE
CE = BC ...[Proved above]
⇒ ∠CBE = ∠CEB ...[Opposite angles of equal side are equal]
⇒ 180° – ∠B = ∠E ...[∵ ∠B + ∠CBE = 180°]
⇒ 180° – ∠E = ∠B ...(ii)
From equations (i) and (ii),
∠A = ∠B
Hence proved.
Notes
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