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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.

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In the given figure, 

∠X = 62° and ∠XYZ = 54°

∠XYZ + ∠XZY + ∠YXZ = 180° …(i) 

[Angle sum property of a triangle] 

⇒ 54° + ∠XZY + 62° = 180° 

⇒ ∠XZY + 116° = 180° 

⇒ ∠XZY = 180° – 116° = 64° 

Now, ∠OZY = 1/ 2 × ∠XZY [∵ZO is bisector of ∠XZY] 

= 1 /2 × 64° = 32° 

Similarly, ∠OYZ = 1/2 × 54° = 27° 

Now, in ∆OYZ, we have 

∠OYZ + ∠OZY + ∠YOZ = 180° Angle sum property of a triangle] 

⇒ 27° + 32° + ∠YOZ = 180° 

⇒ ∠YOZ = 180° – 59° = 121° 

Hence, ∠OZY = 32° and ∠YOZ = 121° Ans.

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