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Question
If one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles can be ______.
Options
50°
65°
145°
155°
Solution
If one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles can be 155°.
Explanation:
Let angles of a triangle be ∠A, ∠B and ∠C.
In ΔABC,
∠A + ∠B +∠C = 180° ...[Sum of all interior angles of a triangle is 180°]
⇒ `1/2 ∠A + 1/2 ∠B + 1/2 ∠C = (180^circ)/2 = 90^circ` ...[Dividing both sides by 2]
⇒ `1/2 ∠B + 1/2 ∠C = 90^circ - 1/2 ∠A` ...[∵ In ΔOBC, ∠OBC + ∠BCO + ∠COB = 180°]
`["Since", (∠B)/2 + (∠C)/2 + ∠BOC = 180^circ "as" BO and OC "are the angle bisectors of" ∠ABC "and" ∠BCA, "respectively"]`
⇒ `180^circ - ∠BOC = 90^circ - 1/2 ∠A`
∴ `∠BOC = 180^circ - 90^circ + 1/2 ∠A`
= `90^circ + 1/2 ∠A`
= `90^circ + 1/2 xx 130^circ` ...[∴ ∠A = 130° (given)]
= 90° + 65°
⇒ 155°
Hence, the required angle is 155°.
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