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Question
In Δ ABC, bisectors of ∠A and ∠B intersect at point O. If ∠C = 70°. Find the measure of ∠AOB.
Solution
∠OAB ≅ ∠OAC ...(Seg AO bisects ∠BAC) ...(i)
∠OBA ≅ ∠OBC ...(Seg RO bisects ∠ABC) ...(ii)
In ΔABC,
∠BAC + ∠ABC + ∠ACB = 180° ...[Sum of the measures of the angles of a triangle is 180°]
∴ ∠BAC + ∠ABC + 70° = 180°
∴ ∠BAC + ∠ABC = 180° - 70°
∴ ∠BAC + ∠ABC = 110°
∴ `1/2 ("∠BAC") + 1/2 ("∠ABC")`
= `1/2 xx 110° ...["MuItiplying both sides by" 1/2]`
∴ ∠OAB + ∠OBA = 55° ...(iii)...[From (i) and (ii)]
In ΔOAB,
∠OAB + ∠OBA + ∠AOB = 180° ...[Sum of the measures of the angles of a triangle is 180°]
∴ 55° + ∠AOB = 180° ...[From (iii)]
∴ ∠AOB = 180° - 55°
∴ ∠AOB = 125°
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