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Question

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=180100=80.....(iii)

In ΔABC, AB=BC
Since angles opposite to equal sides of a triangle are equal,
BCA=BAC=30 ...(iv)
We have BCD=80
BCA+ECD=80
i.e., 30+ECD=80
ECD=8030=50

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