Question

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.​

Answer 1

Answer:

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠ DBC = 70°, ∠ BAC is 30°, ∠BCD = 80°. If AB = BC, ∠ECD = 50°.

Step-by-step explanation:

• 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.

Angles in the same segment are equal

In the triangles, ABD and BCD,

∠CAD = ∠CBD = 70°

∠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°