CBSE Class 9 Answered

In ABC,

ABC + BCA + CAB = 180° (Angle sum property of a triangle)

90° + BCA + CAB = 180°

BCA + CAB = 90° ... (1)

In ADC,

CDA + ACD + DAC = 180° (Angle sum property of a triangle)

90° + ∠ACD + ∠DAC = 180°

ACD + DAC = 90° ... (2)

Adding equations (1) and (2), we obtain

∠BCA + CAB + ACD + DAC = 180°

(BCA + ACD) + (CAB + DAC) = 180°

BCD + DAB = 180° ... (3)

However, it is given that

B + D = 90° + 90° = 180° ... (4)

From equations (3) and (4), it can be observed that the sum of the measures of opposite angles of quadrilateral ABCD is 180°. Therefore, it is a cyclic quadrilateral.

Consider chord CD.

CAD = CBD (Angles in the same segment)

or,

B = D = 90^o B + D = 180^o

ABCD is cyclic

So CAD and CBD are angles in same segment, so CAD = CBD