Question

ABCD is a cyclic quadrilateral in which BC is parallel to AD, angle ADC = 110° and angle BAC = 50°. Find angle DAC and angle DCA.

Answer 1

ABCD is a cyclic quadrilateral in which AD || BC

∠ADC =  110°, ∠BAC = 50°

∠B + ∠D = 180°

(Sum of opposite angles of a quadrilateral)

`=>` ∠B + 110° = 180°

`=>` ∠B = 70°

Now in ΔABC,

∠BAC + ∠ABC + ∠ACB = 180°

`=>` 50° + 70° + ∠ACB = 180°

`=>` ∠ACB = 180° – 120° = 60°

∵ AD || BC

∴ ∠DAC = ∠ACB = 60°  ...(Alternate angles)

Now in ΔADC,

∠DAC + ∠ADC + ∠DCA = 180°

`=>` 60° + 110° + ∠DCA = 180°

`=>` ∠DCA = 180° – 170° = 10°