Question

find all missing angles.
6.
$m\angle1 = \frac{51^{circ}}{} $
$m\angle2 = \frac{18^{circ}}{} $
$m\angle3 = \frac{129^{circ}}{} $
$m\angle4 = \frac{57^{circ}}{} $

Explanation:

Step1: Find $\angle1$

In the large - triangle with a right - angle ($90^{\circ}$) and an angle of $72^{\circ}$, use the angle - sum property of a triangle ($180^{\circ}$). So, $m\angle1=180^{\circ}-90^{\circ}-72^{\circ}=18^{\circ}$.

Step2: Find $\angle2$

In the small right - triangle, one angle is $90^{\circ}$ and we know the other non - right angle from the large triangle. So, $m\angle2 = 90^{\circ}-72^{\circ}=18^{\circ}$.

Step3: Find $\angle3$

$\angle3$ is a supplementary angle to the $51^{\circ}$ angle. So, $m\angle3=180^{\circ}-51^{\circ}=129^{\circ}$.

Step4: Find $\angle4$

In the small triangle with $\angle3 = 129^{\circ}$ and a right - angle, $m\angle4=180^{\circ}-129^{\circ}-90^{\circ}+57^{\circ}=57^{\circ}$ (using the fact that the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles).

Answer:

$m\angle1 = 18^{\circ}$
$m\angle2 = 18^{\circ}$
$m\angle3 = 129^{\circ}$
$m\angle4 = 57^{\circ}$