Question

In the given figure, AC is the diameter of the circle with center O.

CD is parallel to BE.

∠AOB = 80° and ∠ACE = 20°

Calculate:

1. ∠BEC
2. ∠BCD
3. ∠CED

Answer 1

Join AE.

a. The angle subtended by a chord at the center is twice the angle subtended on the circumference.

∴ ∠AOB = 2∠AEB

⇒ 80° = 2∠AEB

∠AEB = `(80°)/2`

∠AEB = 40°

We know that an angle in a semicircle is a right angle.

⇒ ∠AEC = 90°

From figure,

∠BEC = ∠AEC − ∠AEB

= 90° − 40°

= 50°

b. From figure,

∠ECD = ∠CEB = 50°   .....(Alternate angles are equal.)

We know that the angle subtended by a chord at the centre is twice the angle subtended on the circumference.

∠AOB = 2∠BCA

80° = 2∠BCA

∠BCA = `80/2`​

∠BCA = 40°

From figure,

∠BCD = ∠BCA + ∠ACE + ∠ECD

= 40° + 20° + 50°

= 110°

c. The sum of opposite angles of a cyclic quadrilateral is 180°.

∠BED + ∠BCD = 180°

∠BED = 180° − ∠BCD

= 180° − 110°

= 70°

From figure,

∠BED = ∠BEC + ∠CED

70° = 50° + ∠CED

∠CED =  70° − 50°

∠CED = 20°