Question

"Question 6 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.

Class 9 - Math - Circles Page 185"

Answer 1

 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°

Answer 2

See the attachment.......✨✨✨✨