Question

solve the right triangle. write your answers in simplified, rationalized form. do not round. $ik =$ $ij =$ $m\angle k =$ $\degree$

Explanation:

Step1: Find $m\angle K$

The sum of angles in a triangle is $180^\circ$. For right triangle $IJK$, $\angle I=90^\circ$, $\angle J=45^\circ$.
$m\angle K = 180^\circ - 90^\circ - 45^\circ = 45^\circ$

Step2: Identify triangle type

Since $\angle J=\angle K=45^\circ$, $\triangle IJK$ is an isosceles right triangle, so $IK=IJ$.

Step3: Calculate $IK$ and $IJ$

Use Pythagorean theorem: $IJ^2 + IK^2 = JK^2$. Let $IK=IJ=x$, $JK=4\sqrt{2}$.
$$x^2 + x^2 = (4\sqrt{2})^2$$
$$2x^2 = 16\times2$$
$$2x^2 = 32$$
$$x^2 = 16$$
$$x = 4$$

Answer:

$IK = 4$
$IJ = 4$
$m\angle K = 45^\circ$