Question

example 7 if m∠2 = 41°, m∠5 = 94°, and m∠10 = 109°, find each measure. a. m∠1 = b. m∠3 = c. m∠4 = d. m∠6 = e. m∠7 = f. m∠8 = g. m∠9 = © gina wilson (all things algebra®, llc), 2014 - 2019

Explanation:

Step1: Use linear - pair property

$\angle1$ and $\angle2$ form a linear - pair. Since a linear - pair of angles is supplementary ($m\angle1 + m\angle2=180^{\circ}$) and $m\angle2 = 41^{\circ}$, then $m\angle1=180^{\circ}-41^{\circ}=139^{\circ}$.

Step2: Use linear - pair property for $\angle3$

$\angle3$ and $\angle2$ form a linear - pair. So $m\angle3 = 180^{\circ}-41^{\circ}=139^{\circ}$.

Step3: Use angle - sum property of a triangle

In the triangle with angles $\angle4$, $\angle5$ and the angle adjacent to $\angle3$. The sum of angles in a triangle is $180^{\circ}$. Let the angle adjacent to $\angle3$ be $x$. Since $\angle3 = 139^{\circ}$, then $x = 180^{\circ}-139^{\circ}=41^{\circ}$. So $m\angle4=180^{\circ}-(41^{\circ}+94^{\circ}) = 45^{\circ}$.

Step4: Use vertical - angle property for $\angle6$

$\angle6$ and $\angle4$ are vertical angles. Vertical angles are equal. So $m\angle6 = m\angle4=45^{\circ}$.

Step5: Use linear - pair property for $\angle7$

$\angle7$ and $\angle5$ form a linear - pair. So $m\angle7=180^{\circ}-94^{\circ}=86^{\circ}$.

Step6: Use angle - sum property of a triangle for $\angle8$

In the triangle with angles $\angle8$, $\angle9$ and $\angle10$. Let's first find the angle adjacent to $\angle10$. The angle adjacent to $\angle10$ is $180^{\circ}-109^{\circ}=71^{\circ}$. In the triangle, $m\angle8 = 180^{\circ}-(71^{\circ}+45^{\circ})=64^{\circ}$.

Step7: Use linear - pair property for $\angle9$

$\angle9$ and $\angle8$ form a linear - pair. So $m\angle9=180^{\circ}-64^{\circ}=116^{\circ}$.

Answer:

a. $m\angle1 = 139^{\circ}$
b. $m\angle3 = 139^{\circ}$
c. $m\angle4 = 45^{\circ}$
d. $m\angle6 = 45^{\circ}$
e. $m\angle7 = 86^{\circ}$
f. $m\angle8 = 64^{\circ}$
g. $m\angle9 = 116^{\circ}$