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Question

In the given figure, BI is the bisector of ∠ABC and CI is the bisector of ∠ACB. Find ∠BIC


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Solution

Step1: Find ∠ABC and ∠ACB

In ∆ABC, AB=AC

[ From the figure]

∴ ∠B = ∠C

[∵ Angles opposite to equal sides]

And ∠A + ∠B + ∠C = 180

[∵ Angle sum property of a triangle]

40 + ∠B + ∠B = 180

2∠B = 180 - 40

2∠B = 140

∴ ∠B = 70

∴ ∠ABC = ∠ACB = 70

Step2: Find ∠IBC and ∠ICB

Now, BI and CI are the bisector of ∠ABC and ∠ACB, respectively.

∴ ∠IBC = 12∠ABC = 12 × 70

∠IBC = 35

Similarly, ∠ICB = 35

Step 3: Find ∠BIC using angle sum property of a triangle

In ∆IBC,

∠BIC + ∠IBC + ∠ICB = 180

[∵ Angle sum property of a triangle]

∠BIC + 35 + 35 = 180

∠BIC + 70 = 180

∠BIC = 180 - 70

= 110

Hence, the value of ∠BIC is 110.


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