Question

solve the right triangle.
write your answers in simplified, rationalized form. do not round.
hj =
( mangle j = )
( mangle h = )

Explanation:

Step1: Calculate hypotenuse HJ

Use Pythagorean theorem: $HJ = \sqrt{HI^2 + IJ^2}$
$$HJ = \sqrt{(\sqrt{15})^2 + (\sqrt{5})^2} = \sqrt{15 + 5} = \sqrt{20} = 2\sqrt{5}$$

Step2: Find $m\angle J$

Use $\tan(J) = \frac{HI}{IJ}$
$$\tan(J) = \frac{\sqrt{15}}{\sqrt{5}} = \sqrt{3}$$
$m\angle J = \arctan(\sqrt{3}) = 60^\circ$

Step3: Find $m\angle H$

Right triangle angles sum to $180^\circ$
$$m\angle H = 180^\circ - 90^\circ - 60^\circ = 30^\circ$$

Answer:

$HJ = 2\sqrt{5}$
$m\angle J = 60^\circ$
$m\angle H = 30^\circ$