Question

questions 5 through 8 refer to the following: in the diagram below, $overline{ab}paralleloverline{cd}$. 5) if $mangle3 = 60^{circ}$, what is $mangle7$? 6) if $mangle4 = 120^{circ}$, what is $mangle5$? 7) if $mangle5 = 120^{circ}$, what is $mangle3$? 8) if $mangle8 = 110^{circ}$, what is $mangle1$?

Explanation:

Step1: Recall corresponding - angles property

Since $\overline{AB}\parallel\overline{CD}$, corresponding angles are equal. $\angle3$ and $\angle7$ are corresponding angles.

Step2: Find the measure of $\angle7$

Given $m\angle3 = 60^{\circ}$, then $m\angle7=m\angle3 = 60^{\circ}$.

Step3: Recall alternate - interior angles property

$\angle4$ and $\angle5$ are alternate - interior angles. Since $\overline{AB}\parallel\overline{CD}$, alternate - interior angles are equal. Given $m\angle4 = 120^{\circ}$, then $m\angle5=m\angle4 = 120^{\circ}$.

Step4: Recall corresponding - angles property

$\angle3$ and $\angle5$ are corresponding angles. Given $m\angle5 = 120^{\circ}$, then $m\angle3=m\angle5 = 120^{\circ}$.

Step5: Recall vertical - angles and corresponding - angles property

$\angle1$ and $\angle3$ are vertical angles, so $m\angle1 = m\angle3$. Also, $\angle3$ and $\angle7$ are corresponding angles, and $\angle7$ and $\angle8$ are vertical angles. Given $m\angle8 = 110^{\circ}$, then $m\angle7=m\angle8 = 110^{\circ}$, and since $m\angle1 = m\angle3$ and $m\angle3=m\angle7$, $m\angle1 = 110^{\circ}$.

Answer:

1. $60^{\circ}$
2. $120^{\circ}$
3. $120^{\circ}$
4. $110^{\circ}$