Question

The difference between the two acute angles of a right-angled triangle is $\frac{2\mathrm{\pi }}{5}$ radians. Express the angles in degrees.

Answer 1

Given:
Difference between two acute angles of a right-angled triangle = $\frac{2\mathrm{\pi }}{5}$ rad
$\because 1\mathrm{rad}={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\right)}^{°}$

$$\begin{gathered} \therefore \frac{2\mathrm{\pi }}{5}\mathrm{rad}={\left({\displaystyle \frac{180}{\mathrm{\pi }}}\times {\displaystyle \frac{2\mathrm{\pi }}{5}}\right)}^{°} \\ ={\left(36\times 2\right)}^{°} \\ ={72}^{°} \end{gathered}$$

Now, let one acute angle of the triangle be x$°$.
Therefore, the other acute angle will be 90$°$$-$x$°$.
Now,
$$\begin{gathered} x°-\left(90°-x°\right)=72° \\ \Rightarrow x-90+x=72 \\ \Rightarrow 2x=162 \\ \Rightarrow x=81 \end{gathered}$$
Thus, we have:
x$°$ = 81$°$
And,
90$°$$-$x$°$ = 90$°$$-$81$°$ = 9$°$