In the figure (ii) given below, AB = AC = CD, ∠ADC = 38°. Calculate

(i) ∠ABC

(ii) ∠BEC.

(i) In △ACD,

AC = CD

∴ ∠CAD = ∠ADC = 38° (∵ angles of equal sides in triangle are equal)

∠ACB = ∠CAD + ∠ADC = 38° + 38° = 76° (∵ exterior angle = sum of two opposite interior angles.)

In △ABC,

AB = AC

∴ ∠ABC = ∠ACB = 76° (As angles of equal sides in triangle are equal)

Hence, the value of ∠ABC = 76°.

(ii) We know that sum of angles in a triangle = 180°.

⇒ ∠BAC + ∠ABC + ∠ACB = 180°
⇒ ∠BAC + 76° + 76° = 180°
⇒ ∠BAC + 152° = 180°
⇒ ∠BAC = 180° - 152°
⇒ ∠BAC = 28°.

∠BEC = ∠BAC (∵ angles in same segment are equal.)

∠BEC = 28°.

Hence, the value of ∠BEC = 28°.