Question

if diagonals of a cyclic quadrilateral whose diagonals intersect at a point E . if angle DBC=70 , angle BAC is 30 , find angle BCD. Further , if AB=BC , find angle ECD​

Answer 1

step by step

We will use the following concepts to answer the question:

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle.

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

The sum of angles in a triangle is 180°.

Angles in the same segment are equal.

Based on the data given, let's draw the figure as shown below.

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC = 30° find ∠BCD. Further if AB = BC, find ∠ECD.

In the triangles, ABD and BCD, ∠CAD = ∠CBD = 70°. (Angles in the same segment are equal)

Hence, ∠BAD = ∠CAB + ∠DAC

= 30° + 70° = 100°

Thus, ∠BAD = 100°

Since ABCD is a cyclic quadrilateral, the sum of either pair of opposite angles of a cyclic quadrilateral is 180º.

∠BAD + ∠BCD = 180°

∠BCD = 180° - 100°

= 80°

Thus, ∠BCD = 80°

Also given AB = BC.

So, ∠BCA = ∠BAC = 30° (Base angles of isosceles triangle are equal)

∠ECD = ∠BCD - ∠BCA

= 80° - 30°

= 50°

Thus, ∠ECD = 50°