(b) In the figure given below, O is the center of the circle. ∠AOE =150°, ∠DAO = 51°. Calculate the sizes of ∠BEC and ∠EBC.

Solution:

(a) In the given figure, ABCD is a cyclic quadrilateral

∠ADC = 80° and ∠ACD = 52°

To find the measure of ∠ABC and ∠CBD

ABCD is a Cyclic Quadrilateral

∠ABC + ∠ADC = 180^o

(Sum of opposite angles = 180^o)

∠ABC + 80^o = 180^o

∠AOE = 150^o, ∠DAO = 51^o

To find ∠BEC and ∠EBC

ABED is a cyclic quadrilateral

Ext. ∠BEC = ∠DAB = 51^o

∠ AOE = 150^o

Ref ∠AOE = 360^o – 150^o = 51^o

∠AOE = 150^o

Ref ∠AOE = 360^o – 150^o = 210^o

Now arc ABE subtends ∠AOE at the Centre

And ∠ADE at the remaining part of the circle.

∠ADE = ½ ref ∠AOE = ½ × 210^o = 105^o

But Ext ∠EBC = ∠ADE = 105^o

Hence ∠BEC = 51^o and ∠EBC = 105^o