ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC=70∘ and ∠BAC=30∘, then find ∠BCD. Further, if AB = BC, then find ∠ECD.
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Solution
Since the angles in the same segment of a circle are equal, ∠CDB=∠BAC=30∘...(i)
Given, ∠DBC=70∘...(ii)
Using angle sum property inΔBCD, ∠BCD+∠DBC+∠CDB=180∘.
Using (i) and (ii), ∠BCD+70∘+30∘=180∘. ∠BCD=180∘−100∘=80∘.....(iii)
InΔABC,AB=BC
Since angles opposite to equal sides of a triangle are equal, ∴∠BCA=∠BAC=30∘...(iv) Wehave∠BCD=80∘ ⇒∠BCA+∠ECD=80∘
i.e., 30∘+∠ECD=80∘ ⇒∠ECD=80∘−30∘=50∘