Question

questions 1 through 3 refer to the following: in the diagram below, $overline{ab}paralleloverline{cd}$. 1) if $mangle2 = 70^{circ}$, what is $mangle7$? 2) if $mangle4 = 120^{circ}$, what is $mangle5$? 3) if $mangle8 = 120^{circ}$, what is $mangle1$?

Explanation:

Step1: Identify vertical - angle relationship

$\angle2$ and $\angle4$ are vertical angles, so $m\angle4 = m\angle2=70^{\circ}$. Since $\overline{AB}\parallel\overline{CD}$, $\angle4$ and $\angle7$ are alternate - exterior angles. Alternate - exterior angles are congruent when two parallel lines are cut by a transversal. So $m\angle7 = 70^{\circ}$.

Step2: Identify corresponding - angle relationship

$\angle4$ and $\angle5$ are same - side interior angles. When two parallel lines are cut by a transversal, same - side interior angles are supplementary, i.e., $m\angle4 + m\angle5=180^{\circ}$. Given $m\angle4 = 120^{\circ}$, then $m\angle5=180^{\circ}-120^{\circ}=60^{\circ}$.

Step3: Identify vertical - angle and corresponding - angle relationships

$\angle8$ and $\angle6$ are vertical angles, so $m\angle6 = m\angle8 = 120^{\circ}$. Since $\overline{AB}\parallel\overline{CD}$, $\angle6$ and $\angle1$ are corresponding angles. Corresponding angles are congruent when two parallel lines are cut by a transversal. So $m\angle1 = 120^{\circ}$.

Answer:

1. $m\angle7 = 70^{\circ}$
2. $m\angle5 = 60^{\circ}$
3. $m\angle1 = 120^{\circ}$