Question

If one of the angles of a triangle is $130°$, then the angle between the bisectors of the other two angles can be:

• A. $50°$
• B. $65°$
• C. $145°$
• D. $155°$

Answer 1

The correct option is D

$155°$

Applying the angle-sum property of a triangle:

In triangle ABC,

$$\begin{gathered} \angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180° \\ \Rightarrow 130°+\angle \mathrm{B}+\angle \mathrm{C}=180° \\ \Rightarrow \angle \mathrm{B}+\angle \mathrm{C}=180°-130°=50° \\ Dividingbothsidesby2, \\ \Rightarrow \frac{\angle B}{2}+\frac{\angle C}{2}=25° \end{gathered}$$

Now, in triangle BOC,

$$\begin{gathered} \angle \mathrm{BOC}+\angle OBC+\angle OCB=180° \\ \Rightarrow \angle \mathrm{BOC}=180°-\left(\angle OBC+\angle OCB\right) \\ \Rightarrow \angle \mathrm{BOC}=180°-\left(\frac{\angle \mathrm{B}}{2}+\frac{\angle \mathrm{C}}{2}\right) \\ \Rightarrow \angle \mathrm{BOC}=180°-25°=155° \end{gathered}$$

Therefore, the angle between the bisectors of the other two angles measures $155°$.

Hence, the correct option is (D).