Question

If the measure of one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles is _________.

Answer 1

In ∆ABC, ∠A = 130º.

Also, OB and OC are the bisectors of ∠B and ∠C, respectively.

$\therefore \angle \mathrm{OBC}=\frac{\angle \mathrm{B}}{2}.....\left(1\right)$

Similarly, $\angle \mathrm{OCB}=\frac{\angle \mathrm{C}}{2}.....\left(2\right)$

Now,

$\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180°$ (Angle sum property of triangle)

$\Rightarrow 130°+\angle \mathrm{B}+\angle \mathrm{C}=180°$

$\Rightarrow \angle \mathrm{B}+\angle \mathrm{C}=180°-130°=50°$ .....(3)

In ∆BOC,

∠OBC + ∠OCB + ∠BOC = 180º (Angle sum property of triangle)

$$\begin{gathered} \Rightarrow \frac{\angle \mathrm{B}}{2}+\frac{\angle \mathrm{C}}{2}+\angle \mathrm{BOC}=180°\left[\mathrm{Using}\left(1\right)\mathrm{and}\left(2\right)\right] \\ \Rightarrow \frac{\angle \mathrm{B}+\angle \mathrm{C}}{2}+\angle \mathrm{BOC}=180° \\ \Rightarrow \frac{50°}{2}+\angle \mathrm{BOC}=180°\left[\mathrm{Using}\left(3\right)\right] \\ \Rightarrow \angle \mathrm{BOC}=180°-25°=155° \end{gathered}$$

If the measure of one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles is ___155º___.