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.
[3 Marks]
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Solution
Since the angles in the same segment of a circle are equal, ∠CDB=∠DAC=70∘...(i)[1Mark]
Given, ∠BAC=30∘...(ii)
Using property of sum of opposite angles in a cyclic quadrilateral, we get ∠DAB+∠BCD=180∘. ∠DAC+∠CAB+∠BCD=180∘.
Using (i) and (ii), ∠70∘+30∘+∠BCD=180∘. ∠BCD=180∘−100∘=80∘.....(iii)[1Mark]
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∘[1Mark]