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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. - ICSE Class 10 - Mathematics

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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.

Solution

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 ll BC
∴ `∠`DAC= `∠`ACB = 60° (alternate angles)
Now in  ΔADC,
`∠`DAC + `∠`ADC +`∠`DCA = 180°
 ⇒ 60° + 110° ` +∠`DCA = 180°
⇒ `∠`DCA = 180°  - 170° = 10°

 

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Solution 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. Concept: Cyclic Properties.
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