In the given figure, ABCD is a quadrilateral in which AD = BC and ∠A

1.	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.
Answer» 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 = 90^o From the figure it can be written as ∠ADE = ∠ADC – 90^o = ∠BCD – 90^o = ∠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 360^o ∠A + ∠B + ∠C + ∠D = 360^o By substituting the values 2 ∠B + 2 ∠D = 360^o By taking 2 as common 2 (∠B + ∠D) = 360^o By division ∠B + ∠D = 360/2 So we get ∠B + ∠D = 180^o So, ABCD is a cyclic quadrilateral. Therefore, it is proved that the points A, B, C and D lie on a circle.

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.

Answer»

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 = 90^o

From the figure it can be written as

∠ADE = ∠ADC – 90^o = ∠BCD – 90^o = ∠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 360^o

∠A + ∠B + ∠C + ∠D = 360^o

By substituting the values

2 ∠B + 2 ∠D = 360^o

By taking 2 as common

2 (∠B + ∠D) = 360^o

By division

∠B + ∠D = 360/2

So we get

∠B + ∠D = 180^o

So, ABCD is a cyclic quadrilateral.

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

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