In the given figure, ABCD is a quadrilateral in which AD = BC and ∠ADC = ∠BCD. Show that the points A, B, C, D lie on a circle.

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Solution

ABCD is a quadrilateral in which AD = BC and ∠ADC = ∠BCD.
Draw DE ⊥ AB and CF ⊥ AB.
In ΔADE and ΔBCF, we have:
∠ADE = ∠ADC - 90° = ∠BCD - 90° = ∠BCF (Given: ∠ADC = ∠BCD)
AD = BC (Given)
and ∠AED = ∠BCF = 90°
∴ ΔADE ≅ ΔBCF (By AAS congruency)
⇒ ∠A = ∠B
Now, ∠A + ∠B + ∠C + ∠D = 360°
⇒ 2∠B + 2∠D = 360°
⇒ ∠B + ∠D = 180°
Hence, ABCD is a cyclic quadrilateral.

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