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Question
If one of the angles of a triangle is 110°, then the angle between the bisectors of the other two angles is ______.
Options
70°
110°
35°
145°
Solution
If one of the angles of a triangle is 110°, then the angle between the bisectors of the other two angles is 145°.
Explanation:
In ΔABC, ∠A = 110° .....[Given]
We know that, ∠A + ∠B + ∠C = 180° ......[Angle sum property of a triangle]
⇒ ∠B + ∠C = 180° – ∠A
⇒ ∠B + ∠C = 180° – 110°
⇒ ∠B + ∠C = 70° ......(i)
⇒ `1/2 ∠B + 1/2 ∠C = 70/2` = 35° ......[∵ Equation (i) is divided by 2]
⇒ `1/2 (∠B + ∠C)` = 35°
Now, In ΔBOC,
∠BOC + ∠OBC + ∠OCB = 180° [Angle sum property of a triangle] ......(ii)
⇒ `∠BOC + 1/2 (∠B + ∠C)` = 180° ......[∵ OB and OC are the bisectors of ∠B and ∠C, then ∠OBC = `1/2`∠B and ∠OCB = `1/2`∠C]
⇒ ∠BOC + 35° = 180°
⇒ ∠BOC = 180° – 35°
⇒ ∠BOC = 145°
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