Question

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. IF angle DBC 70°
BAC is 30°. find / BCD. Further, if AB=BC. find angle ECD.
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Answer 1

Answer:

The region between a chord and either of its arcs is called a segment the circle.

Angles in the same segment of a circle are equal.

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For chord CD,

We know, that Angles in same segment are equal.

∠CBD = ∠CAD

∠CAD = 70°

∠BAD = ∠BAC + ∠CAD = 30° + 70° = 100°

∠BCD+∠BAD= 180°       

 (Opposite angles of a cyclic quadrilateral)

 ∠BCD + 100° = 180°

∠BCD = 180° - 100°

∠BCD =80°

In ΔABC

AB = BC (given)

∠BCA = ∠CAB (Angles opposite to equal sides of a triangle)

∠BCA = 30°

also, ∠BCD = 80°

∠BCA + ∠ACD = 80°

30° + ∠ACD = 80°

∠ACD = 50°

∠ECD = 50°

Hence, ∠BCD = 80° & ∠ECD = 50°

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