Mathematics
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
Given, ABCD is a cyclic quadrilateral in which AD || BC
With, ∠ADC = 110°, ∠BAC = 50°.
We know that,
⇒ ∠B + ∠D = 180° [Sum of opposite angles of a cyclic quadrilateral = 180°]
⇒ ∠B + 110° = 180°
⇒ ∠B = 180° - 110°
⇒ ∠B = 70°.
Now in ∆ABC, we have
⇒ ∠BAC + ∠ABC + ∠ACB = 180° [By angle sum property of triangle]
⇒ 50° + 70° + ∠ACB = 180°
⇒ ∠ACB = 180° - 120° = 60°
As, AD || BC we have
∠DAC = ∠ACB = 60° [Alternate angles]
Now in ∆ADC,
⇒ ∠DAC + ∠ADC + ∠DCA = 180°
⇒ 60° + 110° + ∠DCA = 180°
⇒ ∠DCA = 180° - 170° = 10°
Hence, ∠DAC = 60° and ∠DCA = 10°.
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