Question

In Δ ABC, bisectors of ∠A and ∠B intersect at point O. If ∠C = 70°. Find the measure of ∠AOB. 

Answer 1

∠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°