In the given figure, AC is the bisector of ∠A. If AB = AC, AD = CD and ∠ABC = 75°, find the values of x and y.

In △ABC,

AB = AC

⇒ ∠ABC = ∠ACB = 75° (Angles opposite to equal sides in a triangle are equal)

By angle sum property of triangle,

⇒ ∠ABC + ∠ACB + ∠BAC = 180°

⇒ 75° + 75° + x° = 180°

⇒ 150° + x° = 180°

⇒ x° = 180° - 150°

⇒ x° = 30°

⇒ x = 30.

Given,

AC is the bisector to ∠A

⇒ ∠DAC = ∠BAC = x° = 30°

In △ADC,

AD = CD

⇒ ∠DAC = ∠DCA = 30° (Angles opposite to equal sides are equal)

By angle sum property of triangle,

⇒ ∠DAC + ∠DCA + ∠ADC = 180°

⇒ 30° + 30° + y° = 180°

⇒ 60° + y° = 180°

⇒ y° = 180° - 60°

⇒ y° = 120°

⇒ y = 120.

Hence, the values of x = 30 and y = 120.