Question

the figure shows a pair of parallel lines cut by a transversal.
which three angles are congruent to ∠2? choose all three correct angles. ∠4 ∠6 ∠8
what argument can you make to explain why m∠2 = m∠8? complete the statement.
when a transversal cuts parallel lines, the ? angles are congruent.

Explanation:

Step1: Recall Vertical Angles

Vertical angles are congruent. $\angle 2$ and $\angle 4$ are vertical angles, so $\angle 4 \cong \angle 2$.

Step2: Recall Corresponding Angles

When parallel lines are cut by a transversal, corresponding angles are congruent. $\angle 2$ and $\angle 6$ are corresponding angles, so $\angle 6 \cong \angle 2$.

Step3: Recall Vertical Angles for $\angle 6$

$\angle 6$ and $\angle 8$ are vertical angles, so $\angle 8 \cong \angle 6$. Since $\angle 6 \cong \angle 2$, then $\angle 8 \cong \angle 2$.

Step4: Identify Congruent Angles

From steps 1 - 3, $\angle 4$, $\angle 6$, and $\angle 8$ are congruent to $\angle 2$.

For the second part (why $m\angle 2 = m\angle 8$):

Step1: Identify Angle Relationships

$\angle 2$ and $\angle 6$ are corresponding angles (parallel lines, transversal), so $m\angle 2 = m\angle 6$. $\angle 6$ and $\angle 8$ are vertical angles, so $m\angle 6 = m\angle 8$. By transitivity, $m\angle 2 = m\angle 8$. The missing term in "When a transversal cuts parallel lines, the [blank] angles are congruent" for $\angle 2$ and $\angle 8$ (which are alternate exterior angles? Wait, no—$\angle 2$ and $\angle 8$: let's check positions. $\angle 2$ is upper - right at the first intersection, $\angle 8$ is lower - left at the second intersection. Wait, actually, $\angle 2$ and $\angle 6$ are corresponding, $\angle 6$ and $\angle 8$ are vertical. Alternatively, $\angle 2$ and $\angle 8$: maybe alternate exterior? Wait, no, let's re - examine. The two parallel lines are the vertical lines, and the transversal is the slanted line. $\angle 2$ is on the upper side of the transversal, right of the first vertical line. $\angle 8$ is on the lower side of the transversal, left of the second vertical line. Wait, maybe the term is "alternate exterior" or "vertical/corresponding chain". But from the earlier steps, since $\angle 2$ corresponds to $\angle 6$ (corresponding), $\angle 6$ is vertical to $\angle 8$, so the angles here (for $\angle 2$ and $\angle 8$) can be related through corresponding and vertical. But the blank: when a transversal cuts parallel lines, the alternate exterior (or through corresponding and vertical) angles? Wait, no, let's see: $\angle 2$ and $\angle 8$: if we consider the two parallel lines, the transversal, $\angle 2$ and $\angle 8$ are alternate exterior angles? Wait, $\angle 2$ is exterior to the two parallel lines (above the first parallel line, right of the transversal), $\angle 8$ is exterior (below the second parallel line, left of the transversal). So alternate exterior angles are congruent. So the blank is "alternate exterior" (or through the chain of corresponding and vertical). But the main three angles congruent to $\angle 2$ are $\angle 4$, $\angle 6$, $\angle 8$.

Answer:

For the first question (three angles congruent to $\angle 2$): $\angle 4$, $\angle 6$, $\angle 8$.

For the second question (why $m\angle 2 = m\angle 8$): When a transversal cuts parallel lines, the alternate exterior (or through corresponding and vertical angle relationships) angles are congruent. The argument is that $\angle 2\cong\angle 6$ (corresponding angles), $\angle 6\cong\angle 8$ (vertical angles), so $\angle 2\cong\angle 8$ (transitive property), hence $m\angle 2 = m\angle 8$. The blank in the statement "When a transversal cuts parallel lines, the [blank] angles are congruent" for $\angle 2$ and $\angle 8$ is "alternate exterior" (or if we follow the chain, but more accurately, through corresponding and vertical, but the standard term for $\angle 2$ and $\angle 8$: let's check the positions again. The two parallel lines are $l_1$ (first vertical) and $l_2$ (second vertical), transversal $t$. $\angle 2$: outside $l_1$ and $l_2$? No, $\angle 2$ is between the transversal and $l_1$? Wait, no, the vertical lines are parallel. The transversal crosses them. $\angle 1$, $\angle 2$, $\angle 3$, $\angle 4$ are at the intersection of $t$ and $l_1$; $\angle 5$, $\angle 6$, $\angle 7$, $\angle 8$ at $t$ and $l_2$. $\angle 2$ and $\angle 6$: same position relative to $l_1$, $l_2$ and $t$ (corresponding). $\angle 6$ and $\angle 8$: vertical. So $\angle 2$ and $\angle 8$: $\angle 2$ is above $t$, right of $l_1$; $\angle 8$ is below $t$, left of $l_2$. So they are alternate exterior angles. So the blank is "alternate exterior".