Question

ABCD is a cyclic quadrilateral in which : 

(i) BC||AD, ∠ADC = 110° and ∠BAC = 50°. Find ∠DAC. 

(ii) ∠DBC = 80° and ∠BAC = 40° Find ∠BCD. 

(iii) ∠BCD = 100° and ∠ABD = 70°.Find ∠ADB.

Answer 1

(i) Since, 

ABCD is a cyclic quadrilateral 

Then, 

∠ABC + ∠ADC = 180° 

∠ABC + 110° = 180° 

∠ABC = 70° 

Since, 

AD ǁ BC 

Then, 

∠DAB + ∠ABC = 180°

(Co. interior angle) 

∠DAC + 50° + 70° = 180° 

∠DAC = 180° – 120° 

= 60° 

(ii) ∠BAC = ∠BDC = 40° 

(Angle in the same segment) 

In by angle sum property, 

∠DBC + ∠BCD + ∠BDC = 180° 

80° + ∠BCD + 40° = 180° 

∠BCD = 60° 

(iii) Given that, 

Quadrilateral ABCD is a cyclic quadrilateral 

Then, 

∠BAD + ∠BCD = 180° 

∠BAD = 80° 

In triangle ABD, 

by angle sum property 

∠ABD + ∠ADB + ∠BAD = 180° 

70° + ∠ADB + 80° = 180° 

∠ADB = 30°