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Mathematics

In the given figure, the bisectors of ∠B and ∠C intersect each other at O and ∠BAC = 50°. The measure of ∠BOC is :

In the given figure, the bisectors of ∠B and ∠C intersect each other at O and ∠BAC = 50°. The measure of ∠BOC is. R.S. Aggarwal Mathematics Solutions ICSE Class 9.

  1. 100°

  2. 115°

  3. 130°

  4. 140°

Triangles

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Answer

In △ABC,

By angle sum property of triangle,

∠A + ∠B + ∠C = 180°

⇒ 50° + ∠B + ∠C = 180°

⇒ ∠B + ∠C = 180° - 50°

⇒ ∠B + ∠C = 130° …….(1)

From figure,

As, OB is bisector of angle B.

∠B = ∠ABO + ∠OBC = ∠OBC + ∠OBC = 2∠OBC

⇒ ∠OBC = ∠B2\dfrac{∠B}{2}

As, OC is bisector of angle C.

∠C = ∠ACO + ∠OCB = ∠OCB + ∠OCB = 2∠OCB

⇒ ∠OCB = ∠C2\dfrac{∠C}{2}

In △OBC,

By angle sum property of triangle,

⇒ ∠OBC + ∠BOC + ∠OCB = 180°

⇒ ∠B2\dfrac{∠B}{2} + ∠BOC + ∠C2\dfrac{∠C}{2} = 180°

⇒ ∠BOC + ∠B+∠C2\dfrac{∠B + ∠C}{2} = 180°

⇒ ∠BOC + 130°2\dfrac{130°}{2} = 180° [Substituting from eq.(1)]

⇒ ∠BOC + 65° = 180°

⇒ ∠BOC = 180° - 65°

⇒ ∠BOC = 115°.

Hence, option 2 is the correct option.

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