Question

intersects two parallel lines. none of the angles are right ang which angles are congruent to ∠3? select all that apply. ∠1 ∠2 ∠4 ∠5 ∠6 ∠7 ∠8 remember, corresponding angles are congruent and vertical angles are congruent. diagram with angles 1,2,3,4 on one intersection and 5,6,7,8 on another, with two parallel lines and a transversal

Explanation:

Step1: Identify Vertical Angles of $\angle3$

Vertical angles are congruent. $\angle3$ and $\angle1$ are vertical angles, so $\angle1\cong\angle3$.

Step2: Identify Corresponding Angles of $\angle3$

Since the two lines cut by the transversal are parallel, corresponding angles are congruent. $\angle3$ and $\angle7$ are corresponding angles, so $\angle7\cong\angle3$. Also, $\angle3$ and $\angle5$? Wait, no, let's re - examine. Wait, $\angle3$ and $\angle1$ are vertical angles (vertical angles are congruent). $\angle3$ and $\angle7$ are corresponding angles (because the two lines are parallel, corresponding angles are congruent). Also, $\angle7$ and $\angle5$? No, wait, $\angle3$: let's look at the parallel lines. The first transversal creates $\angle3$, and the second transversal: $\angle3$ and $\angle7$ are corresponding (same position relative to the parallel lines and transversal). Also, $\angle1$ is vertical to $\angle3$, $\angle7$ is corresponding to $\angle3$, and $\angle5$? Wait, no, maybe I made a mistake. Wait, the vertical angle of $\angle3$ is $\angle1$ (so $\angle1\cong\angle3$). Then, since the two lines are parallel, $\angle1$ and $\angle5$ are corresponding angles? Wait, no, the first transversal and the second transversal: $\angle1$ and $\angle5$ are corresponding? Wait, the two parallel lines are the ones with arrows, and the transversal is the horizontal line. Wait, no, the two lines with arrows are parallel, and the horizontal line is the transversal? Wait, no, the two lines with arrows (the non - horizontal ones) are parallel? Wait, the diagram: there are two lines with arrows (let's say line $a$ and line $b$) which are parallel, and a transversal (the horizontal line) intersecting them. Wait, no, in the diagram, the horizontal line is intersected by two other lines (with arrows) which are parallel. So, for $\angle3$: vertical angle is $\angle1$ (so $\angle1\cong\angle3$). Corresponding angles: $\angle3$ and $\angle7$ (since line $a$ and line $b$ are parallel, and the transversal is the horizontal line, $\angle3$ and $\angle7$ are in the same position relative to the parallel lines and transversal). Also, $\angle7$ and $\angle5$? No, $\angle5$ is adjacent to $\angle7$. Wait, maybe I messed up. Wait, the problem says "corresponding angles are congruent and vertical angles are congruent". So vertical angles of $\angle3$: $\angle1$ (so $\angle1\cong\angle3$). Then, corresponding angles: $\angle3$ and $\angle7$ (because the two parallel lines, so $\angle3$ and $\angle7$ are corresponding). Also, $\angle7$ and $\angle5$? No, $\angle5$ is on the other side. Wait, maybe $\angle3$ and $\angle5$? No, let's look at the labels. The angles: $\angle1$, $\angle2$, $\angle3$, $\angle4$ are formed by the intersection of the first slanted line (with arrows) and the horizontal line. $\angle5$, $\angle6$, $\angle7$, $\angle8$ are formed by the intersection of the second slanted line (with arrows, parallel to the first) and the horizontal line. So, vertical angles: $\angle3$ and $\angle1$ (vertical angles, congruent). Corresponding angles: $\angle1$ and $\angle5$ (corresponding, so $\angle1\cong\angle5$), so by transitivity, $\angle3\cong\angle5$. Also, $\angle3$ and $\angle7$ (corresponding, since $\angle3$ is below the horizontal line, left of the first slanted line; $\angle7$ is below the horizontal line, left of the second slanted line, and the two slanted lines are parallel). So $\angle3\cong\angle1$ (vertical), $\angle3\cong\angle7$ (corresponding), $\angle3\cong\angle5$ (since $\angle1\cong\angle5$ and $\angle3\cong\angle1$). Wait, but in the initial selection, $\angle1$, $\angle5$, $\angle7$ were selected? Wait, the user's initial selection had $\angle1$, $\angle5$, $\angle7$. Let's re - check:

1. Vertical angles: $\angle3$ and $\angle1$ are vertical angles, so $\angle1\cong\angle3$.
2. Corresponding angles: Since the two slanted lines (with arrows) are parallel, $\angle3$ and $\angle7$ are corresponding angles (same position relative to the parallel lines and the transversal (horizontal line)), so $\angle7\cong\angle3$. Also, $\angle1$ and $\angle5$ are corresponding angles (same position relative to the parallel lines and transversal), so $\angle1\cong\angle5$. Since $\angle3\cong\angle1$, then $\angle3\cong\angle5$ by transitivity. So the angles congruent to $\angle3$ are $\angle1$, $\angle5$, $\angle7$.

Answer:

$\angle1$, $\angle5$, $\angle7$