Question

In the given figure, ΔABC is an isosceles triangle with AB = AC and ∠ABC = 50°. Find ∠BDC and ∠BЕС.

Answer 1

Given:

Triangle ABC is isosceles with AB = AC.

∠ABC = 50°.

Points D and E lie on sides AB and AC, respectively.

To find:

∠BDC

∠BEC

Since △ABC is isosceles with AB = AC, the base angles are equal: ∠ABC = ∠ACB = 50°.

Find ∠BAC: Sum of angles in △ABC = 180° ∠BAC + ∠ABC + ∠ACB = 180° ∠BAC + 50° + 50° = 180° ∠BAC = 180° −  100° = 80°

Points B, C, D, and E lie on a circle because D and E lie on sides AB and AC, respectively, creating cyclic quadrilateral BCDE.

By cyclic quadrilateral properties, ∠BDC = ∠BAC = 80° (Angle subtended by the same chord in a circle are equal)

Opposite angles of cyclic quadrilateral BCDE are supplementary:

∠BDC + ∠BEC = 180°

80° + ∠BEC = 180°

∠BEC = 100°