Question

which angle is not congruent to $\angle 4$? diagram: two parallel lines cut by a transversal, with angles labeled 1, 2, 3, 4, 5, 6, 7, 8. options: $\angle 2$, $\angle 7$, $\angle 8$, $\angle 6$

Explanation:

Step1: Recall Angle Congruence Rules

When two parallel lines are cut by a transversal, corresponding angles, alternate interior angles, and alternate exterior angles are congruent. Vertical angles are also congruent.

Step2: Analyze Each Angle

- $\angle 2$: Alternate exterior angle to $\angle 4$ (or corresponding/vertical chain), so congruent.
- $\angle 7$: Let's see, $\angle 4$ and $\angle 7$: $\angle 4$ and $\angle 5$ are supplementary, $\angle 5$ and $\angle 7$ are corresponding? Wait, no. Wait, $\angle 4$ and $\angle 3$ are vertical? No, $\angle 4$ and $\angle 3$: Wait, the lines are parallel. $\angle 4$ and $\angle 2$: alternate exterior. $\angle 4$ and $\angle 8$: $\angle 8$ is vertical to $\angle 2$? Wait, $\angle 8$ and $\angle 2$ are vertical? No, $\angle 1$ and $\angle 8$ are vertical? Wait, maybe better to check each:
- $\angle 6$: $\angle 4$ and $\angle 6$: $\angle 4$ and $\angle 3$ are adjacent, $\angle 3$ and $\angle 6$ are same - side interior? No, wait, $\angle 4$ and $\angle 6$: Let's use vertical angles and corresponding angles. $\angle 4$ and $\angle 3$ are adjacent (supplementary? No, $\angle 4$ and $\angle 5$ are supplementary. $\angle 3$ and $\angle 6$: same - side interior? Wait, no, the two horizontal lines are parallel. The transversal cuts them. $\angle 4$: let's find its congruent angles. $\angle 4$ is congruent to $\angle 2$ (alternate exterior), $\angle 8$ (since $\angle 2$ and $\angle 8$ are vertical, so $\angle 4\cong\angle 8$), $\angle 6$? Wait, $\angle 4$ and $\angle 6$: $\angle 4$ and $\angle 3$ are adjacent, $\angle 3$ and $\angle 6$: same - side interior? No, $\angle 3$ and $\angle 6$ are supplementary? Wait, no, the two horizontal lines are parallel, so same - side interior angles are supplementary. Wait, maybe I made a mistake. Wait, $\angle 4$: vertical angle with $\angle 3$? No, $\angle 4$ and $\angle 5$ are supplementary. $\angle 2$: alternate exterior to $\angle 4$, so $\angle 4\cong\angle 2$. $\angle 8$: $\angle 2$ and $\angle 8$ are vertical angles, so $\angle 2\cong\angle 8$, so $\angle 4\cong\angle 8$. $\angle 6$: $\angle 6$ and $\angle 2$: same - side interior? No, $\angle 6$ and $\angle 2$ are adjacent? Wait, no, the transversal cuts the two parallel lines. Let's list congruent angles to $\angle 4$:
- Corresponding angles: $\angle 4$ and $\angle 2$ (alternate exterior), $\angle 4$ and $\angle 8$ (since $\angle 2\cong\angle 8$), $\angle 4$ and $\angle 6$? Wait, $\angle 6$: $\angle 6$ and $\angle 3$ are vertical? No, $\angle 3$ and $\angle 6$: $\angle 3$ and $\angle 6$ are same - side interior? Wait, no, the two horizontal lines are parallel, so $\angle 3$ and $\angle 6$ are supplementary? Wait, maybe I messed up. Wait, $\angle 4$: let's look at $\angle 7$. $\angle 7$: $\angle 7$ and $\angle 3$: corresponding? No, $\angle 7$ and $\angle 5$: corresponding? $\angle 5$ and $\angle 7$: are they congruent? $\angle 5$ and $\angle 7$: $\angle 5$ is above the top parallel line, $\angle 7$ is below the bottom parallel line. $\angle 4$ and $\angle 7$: let's check the measures. $\angle 4$ and $\angle 5$ are supplementary ($\angle 4+\angle 5 = 180^{\circ}$), $\angle 5$ and $\angle 7$: are they equal? No, because $\angle 5$ is a corresponding angle to $\angle 7$? Wait, no, the two parallel lines: the top and bottom. The transversal. $\angle 5$ and $\angle 7$: are they same - side exterior? No, $\angle 5$ and $\angle 7$: let's use vertical angles and linear pairs. $\angle 1$ and $\angle 8$ are vertical, $\angle 1$ and $\angle 7$ are supplementary. $\angle 4$ and $\angle 5$ are supplementary. $\angle 5$ and $\angle 3$ are vertical? No, $\angle 5$ and $\angle 3$: $\angle 5$ is adjacent to $\angle 3$? Wait, maybe a better approach: congruent angles to $\angle 4$ are those with the same measure. $\angle 4$ is congruent to $\angle 2$ (alternate exterior), $\angle 8$ (vertical to $\angle 2$), $\angle 6$ (let's see, $\angle 4$ and $\angle 6$: $\angle 4$ and $\angle 3$ are adjacent, $\angle 3$ and $\angle 6$: same - side interior? No, wait, $\angle 3$ and $\angle 6$ are supplementary? No, the two parallel lines, so same - side interior angles are supplementary. Wait, maybe I was wrong about $\angle 6$. Wait, $\angle 4$ and $\angle 6$: $\angle 4$ and $\angle 3$ are vertical? No, $\angle 4$ and $\angle 3$: $\angle 4+\angle 3+\angle 6+\angle 2 = 360^{\circ}$? No, the two parallel lines, so the sum of interior angles on the same side is $180^{\circ}$. Wait, maybe the key is that $\angle 7$ is not congruent to $\angle 4$. Let's check:
- $\angle 2$: alternate exterior to $\angle 4$ → congruent.
- $\angle 8$: vertical to $\angle 2$ → congruent to $\angle 4$.
- $\angle 6$: $\angle 6$ and $\angle 2$: same - side interior? No, $\angle 6$ and $\angle 2$: $\angle 6$ is vertical to $\angle 8$? No, $\angle 6$ and $\angle 7$ are supplementary. Wait, maybe $\angle 7$: $\angle 7$ and $\angle 4$: $\angle 4$ is acute (assuming the transversal is not perpendicular), $\angle 7$: let's see, $\angle 7$ and $\angle 1$ are supplementary, $\angle 1$ and $\angle 8$ are vertical, $\angle 8$ and $\angle 2$ are vertical, $\angle 2$ and $\angle 4$ are congruent. So $\angle 7$ and $\angle 4$: $\angle 7$ is supplementary to $\angle 1$, $\angle 1$ is equal to $\angle 8$ (vertical), $\angle 8$ is equal to $\angle 2$ (vertical), $\angle 2$ is equal to $\angle 4$ (alternate exterior). So $\angle 7$ is supplementary to $\angle 4$, not congruent (unless they are right angles, but the diagram doesn't show that). So $\angle 7$ is not congruent to $\angle 4$.

Answer:

$\boldsymbol{\angle 7}$