Question

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

Answer 1

Triangle ABC is isosceles with AB = AC.

∠ABC = 50°.

Points D and E lie on the circle passing through B and C, intersecting sides AB and AC, respectively.

Step 1: Since AB = AC, angles opposite these sides are equal. ∠ABC = ∠ACB = 50°.

Step 2: Find ∠BAC: Sum of angles in

ΔABC = 180° ∠ABC + ∠ACB + ∠BAC = 180° 

50° + 50° + ∠BAC = 180°

∠BAC = 80°

Step 3: In cyclic quadrilateral BCED (since B, C, D, and E lie on the circle), opposite angles sum to 180°. ∠BDC + ∠BAC = 180° ∠BDC + 80° = 180° ∠BDC = 100°

Step 4: Find ∠BEC: In ΔBEC, ∠BEC is the external angle for ΔBEC at E. By cyclic properties and congruent triangles, it can be shown that

∠BEC = 2 × ∠ABC = 100°