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19/07/2021 12:48 pm
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ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.
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19/07/2021 12:50 pm
Consider the following diagram.
Consider the chord CD
We know that angles in the same segment are equal.
∠CBD = ∠CAD
∴ ∠CAD = 70°
Now, ∠ BAD will be equal to the sum of angles BAC and CAD.
∠BAD = ∠BAC + ∠CAD
= 30° + 70°
∴ ∠BAD = 100°
We know that the opposite angles of a cyclic quadrilateral sums up to 180 degrees.
∠BCD + ∠BAD = 180°
It is known that ∠BAD = 100°
∠BCD = 80°
Now consider the ΔABC.
Here, it is given that AB = BC
Also, ∠BCA = ∠CAB (They are the angles opposite to equal sides of a triangle)
∠BCA = 30°
∠BCD = 80°
∠BCA + ∠ACD = 80°
Thus, ∠ACD = 50° and ∠ECD = 50°