Question

In $∆$ABC, the measure of angle A is equal to the sum of the measures of $\angle$ B and $\angle$ C. Also the ratio of measures of $\angle$ B and $\angle$C is 4 : 5. Then find the measures of angles of the triangle.

Answer 1

$\angle$A = $\angle$B + $\angle$C .....(I)
$$\begin{gathered} \frac{\angle B}{\angle C}=\frac{4}{5} \\ \Rightarrow 5\angle B=4\angle C.....\left(\mathrm{II}\right) \end{gathered}$$
We know by angle sum property,
$\angle$A + $\angle$B + $\angle$C = 180$°$
From (I)
$\angle$B + $\angle$C +$\angle$B + $\angle$C = 180$°$
$$\begin{gathered} \Rightarrow 2\left(\angle \mathrm{B}+\angle \mathrm{C}\right)=180° \\ \Rightarrow 2\left(\angle \mathrm{B}+\frac{5\angle \mathrm{B}}{4}\right)=180° \\ \Rightarrow \angle \mathrm{B}+\frac{5\angle \mathrm{B}}{4}=90° \\ \Rightarrow 4\angle \mathrm{B}+5\angle \mathrm{B}=360° \\ \Rightarrow 9\angle \mathrm{B}=360° \\ \Rightarrow \angle \mathrm{B}=40° \end{gathered}$$
$$\begin{gathered} 5\times 40°=4\angle \mathrm{C} \\ \Rightarrow \angle \mathrm{C}=50° \end{gathered}$$
And $\angle$A = $\angle$B + $\angle$C = $40°+50°=90°$.