ABCD is a cyclic quadilateral whose diagonals intersect at a point E. If ∠DBC=70∘ and ∠BAC=30∘, find ∠BCD. Further if AB=BC, find ∠ECD.
80∘, 50∘
∠CDB=∠BAC=30∘ ...(i) [Angles in the same segment of a circle are equal]
∠DBC=70∘ .....(ii)
In △BCD,
∠BCD+∠DBC+∠CDB=180∘ [Sum of all the angles of a triangle is 180∘]
∠BCD+70∘+30∘=180∘ [Using (i) and (ii)]
∠BCD+100∘=180∘
∠BCD=180∘−100∘
∠BCD=80∘ .....(iii)
In △ABC, AB=BC
∴∠BCA=∠BAC=30∘ .....(iv) [Angle opposite to equal sides of a triangle are equal]
∠BAC=30∘ [given]
Now, ∠BCD=80∘
⇒∠BCA+∠ECD=80∘
⇒30∘+∠ECD=80∘
⇒∠ECD=80∘−30∘
⇒∠ECD=50∘