O is the Centre of a Circle and AB a Diameter. ABCD is a Cyclic Quadrilateral. ∠ABC=65°, ∠DAC=40°, then the Measure of ∠BCD is ______. a)75° b)105° c)115° d)80°

O is te centre of a circle and AB a diameter ABCD is a cyclic quadrilateral. ∠ABC=65°, ∠DAC=40°, then the measure of ∠BCD is ______.

a)75° b)105° c)115° d)80°

Answer : 115°

Explaination:

Given:

∠ABC=65°

∠DAC=40°

∠BCD= ??

∵ ∠ACB is angle in a semicircle.

∴ ∠ACB = 90°

∵ ∠ABC and ∠ADC are supplementary angles ( as the sum of opposite angles of a cyclic quadrilateral is 180° ).

∴ ∠ABC + ∠ADC = 180°

⇒ 65°+∠ADC = 180°

⇒ ∠ADC = 180°- 65°

⇒ ∠ADC = 115°

In Δ ADC,

∠ADC = 115°, ∠DAC = 40°

∴ ∠DAC = 180°-∠ADC-∠DAC

= 180°- 115°- 40°

= 65°- 40°

= 25°

∴ ∠BCD = ∠ACB+∠DCA

= 90°+25°

= 115°