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Question
ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.
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Solution
For chord CD,
∠CBD = ∠CAD ...(Angles in the same segment)
∠CAD = 70°
∠BAD = ∠BAC + ∠CAD = 30° + 70° = 100°
∠BCD + ∠BAD = 180° ...(Opposite angles of a cyclic quadrilateral)
∠BCD + 100° = 180°
∠BCD = 80°
In ΔABC,
AB = BC ...(Given)
∴ ∠BCA = ∠CAB ...(Angles opposite to equal sides of a triangle)
⇒ ∠BCA = 30°
We have, ∠BCD = 80°
⇒ ∠BCA + ∠ACD = 80°
30° + ∠ACD = 80°
⇒ ∠ACD = 50°
⇒ ∠ECD = 50°
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