Question

In the figure given, O is the centre of the circle. ∠DAE = 70°. Find giving suitable reasons, the measure of:

1. ∠BCD
2. ∠BOD
3. ∠OBD

Answer 1

i. ∠DAE = 70°  ...(Given)

∠BAD + ∠DAE = 180° ...(Linear pair)

${\displaystyle \Rightarrow }$ ∠BAD + 70° = 180°

${\displaystyle \Rightarrow }$ ∠BAD = 110°

Since ABCD is a cyclic quadrilateral, sum of the measures of the opposite angles are supplementary.

So, ∠BCD + ∠BAD = 180°

${\displaystyle \Rightarrow }$ ∠BCD + 110° = 180°

${\displaystyle \Rightarrow }$ ∠BCD = 70°

ii. ∠BOD = 2∠BCD  ...(Inscribed angle theorem)

${\displaystyle \Rightarrow }$ ∠BOD = 2(70°) = 140°

iii. In ΔOBD

OB = OD  ...(Radii of same circle)

${\displaystyle \Rightarrow }$ ∠OBD = ∠ODB

By angle sum property,

∠OBD + ∠ODB + ∠BOD = 180°

${\displaystyle \Rightarrow }$ 2∠OBD + ∠BOD = 180°

${\displaystyle \Rightarrow }$ 2∠OBD + 140° = 180°

${\displaystyle \Rightarrow }$ 2∠OBD = 40°

${\displaystyle \Rightarrow }$ ∠OBD = 20°