Question

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC = 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.

Answer 1

∠DBC = 70°, ∠BAC = 30°, then ∠BCD =? 

AB = BC, then ∠ECD = ? 

∠DAC and ∠DBC are angles in same segment. 

∴ ∠DAC = ∠DBC = 70° 

∴ ∠DAC = 70° 

ABCD is a cyclic quadrilateral. 

∴ Sum of opposite angles is 180°. 

∠DAB + ∠DCB = 180 

100 + ∠DCB = 180 

[∵ ∠DAC + ∠BAC = ∠DAB 70 + 30 = 100] 

∠DCB = 180 – 100 

∴ ∠DCB = 80 

∠DCB = ∠BCD = 80 

∴ ∠BCD = 80 

In ∆ABC, AB = AC, 

∴ ∠BAC = ∠BCA = 30° 

∠BCA = 30° 

∠ECD = ∠BCD – ∠BCA = 80 – 30 

∴ ∠ECD = 50°.