ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC=70∘,∠BAC is 30∘ and AB = BC, find ∠ECD.
A
50∘
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B
70∘
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C
40∘
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D
60∘
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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∘