Question

ABC is an equilateral triangle. Its side BC is produced up to point E such that C is mid-point of BE. Calculate the measure of angles ACE and AEC.

Answer 1

ΔABC is an equilateral triangle.
⇒ Side AB = Side AC
⇒ ∠ABC = ∠ACB ........[If two sides of a triangle are equal, then angles opposite to them are equal]

Similarly, Side AC = Side BC
⇒ ∠CAB = ∠ABC .......[If two sides of a triangle are equal, then angles opposite to them are equal]

Hence, ∠ABC = ∠CAB = ∠ACB  = y(say)
As the sum of all the angles of the triangle is 180°.
∠ABC + ∠CAB + ∠ACB = 180°
⇒ 3y = 180°
⇒ y = 60°

∠ACB = ∠ACB = ∠ABC = 60°
Sum of two non-adjacent interior angles of a triangle is equal to the exterior angle.
⇒ ∠CAB + ∠CBA = ∠ACE
⇒  60° + 60° = ∠ACE
⇒  ∠ACE = 120°

Now ΔACE is an isosceles triangle with AC = CF
⇒ ∠EAC = ∠AEC
Sum of all the angles of a triangle is 180°
∠EAC + ∠AEC + ∠ACE = 180°
⇒ 2∠AEC + 120° = 180°
⇒  2∠AEC = 180° − 120°
⇒  ∠AEC = 30°