Question

1. if (mangle8 = 23^{circ}), find each measure. give your reasoning.

a. (mangle1=)
b. (mangle2=)
c. (mangle3=)
d. (mangle4=)
e. (mangle5=)
f. (mangle6=)
g. (mangle7=)

Explanation:

Step1: Vertical - angle property

Vertical angles are equal. $\angle8$ and $\angle6$ are vertical angles, so $m\angle6 = m\angle8=23^{\circ}$

Step2: Linear - pair property

$\angle8$ and $\angle5$ form a linear - pair. Since the sum of angles in a linear - pair is $180^{\circ}$, $m\angle5 = 180^{\circ}-m\angle8=180 - 23=157^{\circ}$

Step3: Vertical - angle property

$\angle5$ and $\angle7$ are vertical angles. So $m\angle7 = m\angle5 = 157^{\circ}$

Step4: Corresponding - angles property

Assuming the two lines are parallel, $\angle8$ and $\angle4$ are corresponding angles. So $m\angle4 = m\angle8 = 23^{\circ}$

Step5: Vertical - angle property

$\angle4$ and $\angle2$ are vertical angles. So $m\angle2 = m\angle4 = 23^{\circ}$

Step6: Linear - pair property

$\angle4$ and $\angle3$ form a linear - pair. So $m\angle3=180 - m\angle4 = 157^{\circ}$

Step7: Vertical - angle property

$\angle3$ and $\angle1$ are vertical angles. So $m\angle1 = m\angle3 = 157^{\circ}$

Answer:

a. $m\angle1 = 157^{\circ}$
b. $m\angle2 = 23^{\circ}$
c. $m\angle3 = 157^{\circ}$
d. $m\angle4 = 23^{\circ}$
e. $m\angle5 = 157^{\circ}$
f. $m\angle6 = 23^{\circ}$
g. $m\angle7 = 157^{\circ}$