Question

in this figure, \\( \overline { a b } \parallel \overline { c d } \\) and \\( m \angle 4 = 30 ^ { \circ } \\).
what is \\( m \angle 8 \\)?
enter your answer in the box.

Explanation:

Step1: Identify vertical angles

$\angle 4$ and $\angle 2$ are vertical angles, but we need to relate to $\angle 8$. Wait, $\angle 4$ and $\angle 8$: since $AB \parallel CD$, and the transversal, $\angle 4$ and $\angle 8$ are corresponding? Wait no, let's check the positions. Wait, $\angle 4$ and $\angle 8$: actually, $\angle 4$ and $\angle 8$ are alternate exterior? Wait no, let's see the lines. $AB \parallel CD$, transversal crosses them. $\angle 4$ and $\angle 8$: $\angle 4$ is on $AB$, below the transversal, $\angle 8$ is on $CD$, below the transversal? Wait no, the diagram: $AB$ is top line, $CD$ is bottom line. Transversal crosses them. $\angle 4$ is between $AB$ and transversal, below $AB$? Wait, $\angle 4$ and $\angle 8$: are they corresponding angles? Wait, no, maybe vertical angles or alternate interior. Wait, $\angle 4$ and $\angle 8$: let's see, $\angle 4$ and $\angle 8$: if $AB \parallel CD$, then $\angle 4$ and $\angle 8$ are... Wait, $\angle 4$ and $\angle 8$: actually, $\angle 4$ and $\angle 8$ are equal? Wait, no, wait $\angle 4$ is 30 degrees. Wait, maybe $\angle 4$ and $\angle 8$ are corresponding? Wait, no, let's check the angles. Wait, $\angle 4$ and $\angle 8$: looking at the diagram, $\angle 4$ is adjacent to $\angle 3$, and $\angle 8$ is adjacent to $\angle 7$. Wait, maybe $\angle 4$ and $\angle 8$ are alternate interior angles? Wait, $AB \parallel CD$, transversal, so alternate interior angles are equal. $\angle 4$ and $\angle 8$: are they alternate interior? Let's see: $\angle 4$ is between $AB$ and $CD$, on the left side of the transversal? Wait, no, the transversal is a line crossing $AB$ and $CD$. $\angle 4$ is below $AB$, to the right of the transversal? Wait, maybe I made a mistake. Wait, the problem says $m\angle 4 = 30^\circ$. $\angle 4$ and $\angle 8$: are they vertical angles? No, vertical angles are opposite. Wait, $\angle 4$ and $\angle 8$: maybe $\angle 4$ and $\angle 8$ are equal because of parallel lines. Wait, no, wait $\angle 4$ and $\angle 8$: let's think again. Wait, $\angle 4$ and $\angle 8$: if $AB \parallel CD$, then $\angle 4$ and $\angle 8$ are corresponding angles? Wait, no, maybe $\angle 4$ and $\angle 8$ are equal. Wait, the answer is 30 degrees? Wait, no, wait maybe I messed up. Wait, no, let's check: $\angle 4$ and $\angle 8$: are they alternate interior angles? Yes, because $AB \parallel CD$, and the transversal, so alternate interior angles are equal. So $\angle 4 = \angle 8$. Since $m\angle 4 = 30^\circ$, then $m\angle 8 = 30^\circ$. Wait, is that right? Wait, maybe I confused the angles. Wait, let's confirm: alternate interior angles are equal when lines are parallel. So if $AB \parallel CD$, and the transversal, then $\angle 4$ (alternate interior to $\angle 8$) would be equal. So $m\angle 8 = 30^\circ$.

Step2: Confirm the angle relationship

Since $AB \parallel CD$ and the transversal intersects them, $\angle 4$ and $\angle 8$ are alternate interior angles. By the Alternate Interior Angles Theorem, alternate interior angles are congruent when two parallel lines are cut by a transversal. Therefore, $m\angle 8 = m\angle 4$.

Step3: Substitute the given angle measure

Given $m\angle 4 = 30^\circ$, so $m\angle 8 = 30^\circ$.

Answer:

$30$