Answer
$\angle A=\dfrac{3\pi}{10}$
$h\approx180.34$ $;$ $a\approx145.9$
Work Step by Step
The triangle is shown in the attached image below.
Two angles are known. Let $\angle A$ be the unknown angle, $\angle B$ be angle marked as $\dfrac{\pi}{5}$ and $\angle C$ be the right angle. Since $\angle A+\angle B+\angle C=\pi$, substitute the known angles into the formula and solve for $\angle A$:
$\angle A=\pi-\dfrac{\pi}{2}-\dfrac{\pi}{5}=\dfrac{3\pi}{10}$
Let $h$ be the hypotenuse of the triangle. Use the sine trigonometric ratio of the angle marked as $\dfrac{\pi}{5}$ to obtain it:
$\sin\dfrac{\pi}{5}=\dfrac{106}{h}$
$h=\dfrac{106}{\sin\dfrac{\pi}{5}}\approx180.34$
Use the Pythagorean Theorem to obtain the unknown cathetus $a$:
$a=\sqrt{180.34^{2}-106^{2}}\approx145.9$
