Question

In Fig. 6.40, X = 62°, XYZ = 54°. If YO and ZO are the bisectors of XYZ and XZY respectively of Δ XYZ, find OZY and YOZ.​

Answer 1

To Find :-

∠OZY and ∠YOZ

Solution :-

First we will find ∠XZY

In ΔXYZ

↬ ∠XYZ = 54° (Given)

↬ ∠X = 62° (Given)

↬ ∠XYZ + ∠X + ∠XZY = 180° __(Angle Sum Property)

↬ 54° + 62° + ∠XZY = 180°

↬ 116° + ∠XZY = 180°

↬ ∠XZY = 180° - 116°

↬ ∠XZY = 64°

So ∠XZY is 64°

Now we will find the ∠OZY

\bold{\bigstar}★ As we known that ZO is the bisector of ∠XZY

\bold{\bigstar}★ By dividing the ∠XZY by 2 will we get ∠OZY

↬ ∠XZY = 64° __(Solved)

↬ ∠OZY = ∠XZY / 2

↬ ∠OZY = 64°/2

↬ ∠OZY = 32°

So ∠OZY is 32°___(1)

Now we will find the ∠OYZ

\bold{\bigstar}★ As we known that YO is the bisector of ∠XYZ.

\bold{\bigstar}★ By dividing the ∠XYZ by 2 will we get ∠OYZ

∠XYZ = 54° ___(Given)

↬ ∠OYZ = ∠XYZ/2

↬ ∠OYZ = 54°/2

↬ ∠OYZ = 27°

So ∠OYZ is 27° ___(2)

Now we will find the ∠YOZ

In Δ OZY

↬ ∠OYZ = 27° ___(1)

↬ ∠OZY = 32°.____(2)

Here we will use Angle sum property.

↬ ∠OYZ + ∠OZY + ∠YOZ = 180°__(Angle Sum Property)

↬ 27° + 32° + ∠YOZ = 180°

↬ 59° + ∠YOZ = 180°

↬ ∠YOZ = 180° - 59°

↬ ∠YOZ = 121°

So ∠YOZ is 121°

Answer :-

∠OZY = 32°

∠YOZ = 121°

Answer 2

Step-by-step explanation:

X +XYZ +XZY = 180°

Putting the values as given in the question we get,

62°+54° +XZY = 180°

Or, XZY = 64°

Now, we know that ZO is the bisector so,

OZY = ½ XZY

∴ OZY = 32°

Similarly, YO is a bisector and so,

OYZ = ½ XYZ

Or, OYZ = 27° (As XYZ = 54°)

Now, as the sum of the interior angles of the triangle,

OZY +OYZ +O = 180°

Putting their respective values, we get,

O = 180°-32°-27°

Hence, O = 121°