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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:
- ∠BEC
- ∠BCD
- ∠CED
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Solution
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°
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