Question

In the figure, given below, ABCD is a cyclic quadrilateral in which ∠BAD = 75°; ∠ABD = 58° and ∠ADC = 77°. Find:

1. ∠BDC,
2. ∠BCD,
3. ∠BCA.

Answer 1

 i. By angle – sum property of triangle ABD,

∠BAD + ∠ABD + ∠ADB = 180°

133° + ∠ADB = 180°

∠ADB = 180° – 133°

∠ADB = 47°

∴ ∠ADC = ∠ADB + ∠BDC

∴ 77° = 47° + ∠BDC

∴ 77° – 47° = ∠BDC

∴ ∠BDC = 30°

ii. ∠BAD + ∠BCD = 180°    ...(Sum of opposite angles of a cyclic quadrilateral is 180°)

`=>` ∠BCD = 180° – 75° = 105°

∴ ∠BCD = 105°

iii. ∠BCA = ∠BDA = 47°   ...(Angle subtended by the same chord on the circle are equal)