In the given figure, ABCD is a quadrilateral in which AD = BC and ∠ADC = ∠BCD.
Question:

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

 

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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