Question

find the measures of angles 1 through 5 in the figure shown.
$m\angle1=\square^{\circ}$

Explanation:

Step1: Calculate $\angle 1$ via triangle sum

The right triangle has angles $90^\circ$, $39^\circ$, and $\angle 1$. The sum of angles in a triangle is $180^\circ$.
$\angle 1 = 180^\circ - 90^\circ - 39^\circ$

Step2: Compute $\angle 1$ value

$\angle 1 = 51^\circ$

Step3: Find $\angle 2$ via linear pair

$\angle 1$ and $\angle 2$ form a straight line ($180^\circ$).
$\angle 2 = 180^\circ - 51^\circ = 129^\circ$

Step4: Find $\angle 3$ via vertical angles

$\angle 1$ and $\angle 3$ are vertical angles (congruent).
$\angle 3 = 51^\circ$

Step5: Find $\angle 4$ via vertical angles

$\angle 2$ and $\angle 4$ are vertical angles (congruent).
$\angle 4 = 129^\circ$

Step6: Calculate $\angle 5$ via triangle sum

The top triangle has angles $81^\circ$, $\angle 3$, and $\angle 5$.
$\angle 5 = 180^\circ - 81^\circ - 51^\circ = 48^\circ$

Answer:

$\angle 1 = 51^\circ$, $\angle 2 = 129^\circ$, $\angle 3 = 51^\circ$, $\angle 4 = 129^\circ$, $\angle 5 = 48^\circ$