Question

In the given figure ∠x = 520, ∠XYZ = 480. if YO and ZO are the bisectors of ∠XYZ and ∠XZY respectively, then find the measure of ∠OZY.​

Answer 1

Correct question:

In the figure, ∠X = 62°, ∠XYZ = 54°. If YO and ZO are the bisectors of ∠XYZ and ∠XZY respectively of ∆XYZ, find ∠OZY and ∠YOZ.

Answer:

The measures of ∠OZY and ∠YOZ are 32° and 121° respectively.

Explaination:

In the ∆XYZ :

➨ ∠ZXY = 62°

➨ ∠XYZ = 54°

YO is the bisector of ∠XYZ :

➨ ∠XYO = ∠OYZ = ½ ∠XYZ

➨ ∠XYO = ∠OYZ = ½ × 54°

➨ ∠XYO = ∠OYZ = 27°

ZO is the bisector of ∠YZX

➨ ∠XZO = ∠OZY = ½ ∠YZX . . . . . (eqⁿ I)

We know that, all the angles in a triangle add upto 180°.

➨ ∠YZX + ∠YXZ + ∠XYZ = 180°

➨ ∠YZX + 62° + 54° = 180°

➨ ∠YZX + 116° = 180°

➨ ∠YZX = 180° – 116°

➨ ∠YZX = 64°

From eqⁿ I, we get :

➨ ½ ∠YZX = ∠XZO = ∠OZY

➨ ½ × 64° = 32°

➨ ∠XZO = ∠OZY = 32°

In the ∆OYZ, according to the angle sum property of a triangle :

➨ ∠OZY + ∠OYZ + ∠YOZ = 180°

➨ 32° + 27° + ∠YOZ = 180°

➨ 59° + ∠YOZ = 180°

➨ ∠YOZ = 180° – 59°

➨ ∠YOZ = 121°

∴ The angle ∠YOZ = 121° and ∠OZY = 32°.