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Question
In the figure given, O is the centre of the circle. ∠DAE = 70°. Find giving suitable reasons, the measure of:
- ∠BCD
- ∠BOD
- ∠OBD
Solution
i. ∠DAE = 70° ...(Given)
∠BAD + ∠DAE = 180° ...(Linear pair)
`=>` ∠BAD + 70° = 180°
`=>` ∠BAD = 110°
Since ABCD is a cyclic quadrilateral, sum of the measures of the opposite angles are supplementary.
So, ∠BCD + ∠BAD = 180°
`=>` ∠BCD + 110° = 180°
`=>` ∠BCD = 70°
ii. ∠BOD = 2∠BCD ...(Inscribed angle theorem)
`=>` ∠BOD = 2(70°) = 140°
iii. In ΔOBD
OB = OD ...(Radii of same circle)
`=>` ∠OBD = ∠ODB
By angle sum property,
∠OBD + ∠ODB + ∠BOD = 180°
`=>` 2∠OBD + ∠BOD = 180°
`=>` 2∠OBD + 140° = 180°
`=>` 2∠OBD = 40°
`=>` ∠OBD = 20°
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