Question

In the given figure quadrilateral ABCD is cyclic quadrilateral in which angle ABC is congruent to angle BCD
to prove- side DC congruent to side AB and AD ll BC.​

Answer 1

=> It is given that ABCD is a quadrilateral in which AD = BC and ∠ADC =∠BCD

=> Construct DE⊥AB and CF⊥AB

=> Consider △ADE and △BCF

=> We know that 

=> ∠AED +∠BFC = 90⁰

=> From the figure it can be written as 

=> ∠ADE =∠ADC− 90⁰ =∠BCD − 90⁰ =∠BCF

=> It is given that 

=> AD = BC

=> By AAS congruence criterion 

=> △ADE ≃ △BCF

=> ∠A =∠B (c.p.c.t)

=> We know that the sum of all the angles of a quadrilateral is 360⁰

=> ∠A+∠B+∠C+∠D = 360⁰

By substituting the values 

=> 2∠B + 2∠D= 360⁰

=> By taking 2 as common

=> 2(∠B +∠D) = 360⁰

=> By division 

=> ∠B +∠D=180⁰

=> So, ABCD is a cyclic quadrilateral.

=> Therefore, it is proved that the points A,B,C and D lie on a circle.

I hope this will help you dear..

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