Answer: [tex]18^{\circ}[/tex] and [tex]72^{\circ}[/tex]
Step-by-step explanation:
Let the acute angles of the triangle be [tex]\alpha[/tex] and [tex]\beta[/tex], where [tex]\alpha > \beta[/tex].
- The acute angles of a right triangle add to [tex]\frac{\pi}{2}[/tex], so [tex]\alpha+\beta=\frac{\pi}{2}[/tex].
- It is also given that [tex]\alpha-\beta=\frac{3\pi}{10}[/tex].
Adding these equations yields:
[tex]2\alpha=\frac{4\pi}{5} \implies \alpha=\frac{2\pi}{5}[/tex]
Substituting this back into the first equation,
[tex]\frac{2\pi}{5}+\beta=\frac{\pi}{2} \implies \beta=\frac{\pi}{10}[/tex]
Converting these to degrees (by multiplying the radian measure by [tex]\frac{180}{\pi}[/tex]),
[tex]\alpha=\frac{2\pi}{5} \cdot \frac{180}{\pi}=72^{\circ}\\\\\beta=\frac{\pi}{10} \cdot \frac{180}{\pi}=18^{\circ}[/tex]
Therefore, the acute angles of the triangle are [tex]18^{\circ}[/tex] and [tex]72^{\circ}[/tex].
