Question

given (mangle2 = 41^{circ}), (mangle5=94^{circ}), and (mangle10 = 108^{circ}) find the measure of each missing angle.
a. (mangle1=) d. (mangle6=) g. (mangle9=)
b. (mangle3=) e. (mangle7=)
c. (mangle4=) f. (mangle8=)

Explanation:

Step1: Use linear - pair property

$\angle1$ and $\angle2$ form a linear - pair. So, $m\angle1 + m\angle2=180^{\circ}$. Given $m\angle2 = 41^{\circ}$, then $m\angle1=180^{\circ}-41^{\circ}=139^{\circ}$.

Step2: Use angle - sum property of a triangle

In the left - hand triangle, if we consider the triangle with angles $\angle2$, $\angle3$, and $\angle4$. Let's assume the third angle of this triangle (not labeled) is $x$.
We know that $\angle5$ is an exterior angle of this triangle. By the exterior - angle property of a triangle, $m\angle5=m\angle2 + m\angle4$. Given $m\angle5 = 94^{\circ}$ and $m\angle2 = 41^{\circ}$, then $m\angle4=m\angle5 - m\angle2=94^{\circ}-41^{\circ}=53^{\circ}$.
Also, using the angle - sum property of a triangle ($m\angle2 + m\angle3 + m\angle4=180^{\circ}$), $m\angle3=180^{\circ}-m\angle2 - m\angle4=180^{\circ}-41^{\circ}-53^{\circ}=86^{\circ}$.

Step3: Use vertical - angles and linear - pairs

$\angle6$ and $\angle2$ are vertical angles, so $m\angle6 = m\angle2=41^{\circ}$.
$\angle7$ and $\angle3$ are vertical angles, so $m\angle7 = m\angle3=86^{\circ}$.
$\angle8$ and $\angle4$ are vertical angles, so $m\angle8 = m\angle4=53^{\circ}$.
$\angle9$ and $\angle10$ form a linear - pair. Given $m\angle10 = 108^{\circ}$, then $m\angle9=180^{\circ}-108^{\circ}=72^{\circ}$.

Answer:

a. $m\angle1 = 139^{\circ}$
b. $m\angle3 = 86^{\circ}$
c. $m\angle4 = 53^{\circ}$
d. $m\angle6 = 41^{\circ}$
e. $m\angle7 = 86^{\circ}$
f. $m\angle8 = 53^{\circ}$
g. $m\angle9 = 72^{\circ}$