Question

ABCD is a quadrilateral in which AD=BC and ∠ADC = ∠BCD show A, B, C, D lie on a circle.

Answer 1

ABCD is a quadrilateral in which AD=BC and ∠ADC = ∠BCD.

To prove : ABCD lies on a circle i.e they are concyclic.


Join AC and BD .


Now in ∆ADC and ∆BDC,

AD = BC (given)

∠ADC =∠BDC (given)

DC= DC (common)

∆ADC≅∆BDC ( SAS congruency)

∠DAC =∠DBC ( by CPCT )

Hence the angle made by a segment on the other part of the circle are equal.

Hence A,B,C,D must lie on a circle.

Answer 2

ABCD is a quadrilateral in which AD = BC and angle ADC = angle BCD .

To Prove : ABCD lies on a circle I.e they are concyclic .

Join AC and BD .

Now in ∆ADC and ∆BDC ,

AD = BC ( given )

angle ADC = angle BDC ( given )

DC = DC ( common )

∆ADC ≈ ∆BDC ( SAS congruency )

angle DAC = angle BDC ( by CPCT )

Hence the angle made by a segment on the other part of the circle are equal .

Hence , A , B , C , D must lie on a circle .