Question

$\angle 5 \cong \angle 10$
a) $w \parallel x$; alternate exterior angles converse
b) $w \parallel x$; converse of corresponding angles thm.
c) $y \parallel z$; alternate interior angles converse
d) $y \parallel z$; converse of corresponding angles thm

Explanation:

Step1: Identify Angle Relationship

$\angle 5$ and $\angle 10$ are alternate interior angles formed by transversal intersecting lines $y$, $z$ and the other transversal? Wait, no—wait, lines $y$ and $z$? Wait, no, let's check the lines. Wait, $\angle 5$ and $\angle 10$: when a transversal crosses two lines, alternate interior angles are inside the two lines and on opposite sides of the transversal. So if $\angle 5 \cong \angle 10$, by Alternate Interior Angles Converse, the lines cut by the transversal are parallel. So which lines? The transversal here is the line that crosses $y$ and $z$? Wait, no, let's see the angles. $\angle 5$ is between line $x$ (or another vertical line) and line $z$? Wait, no, the angles $\angle 5$ and $\angle 10$: let's see the positions. $\angle 5$ is at the intersection of the lower transversal (the one with $z$) and the middle vertical line (maybe $x$? No, the vertical lines: $w$, then another, then $x$, then the bottom one. Wait, the transversal is the line with $y$ and $z$? Wait, $y$ and $z$ are the two slanted lines (the ones with arrows going down-right). So the transversal is the middle vertical line (the one with $x$? No, the vertical lines: $w$ (top left vertical), then the next vertical (with angles 3,7,11,15), then $x$ (vertical with 6,10,14), then the bottom vertical (with 1,5,9,13). Wait, the transversal is the line that goes through $\angle 5$ and $\angle 10$: the transversal is the vertical line (maybe the one with $x$? No, $\angle 5$ is on the bottom vertical, $\angle 10$ is on the middle vertical (with $x$? Wait, $\angle 10$ is on the line with $x$? Wait, no, $\angle 10$ is between the middle vertical (with 3,7,11,15) and the transversal (the slanted line with $y$ and $z$)? Wait, I think I messed up. Let's recall: Alternate Interior Angles Converse states that if alternate interior angles are congruent, then the two lines cut by the transversal are parallel. So $\angle 5$ and $\angle 10$: are they alternate interior angles for lines $y$ and $z$? Let's see: lines $y$ and $z$ are the two slanted lines (the ones with arrows going down-right). The transversal is the vertical line (the one with $x$? No, the vertical line that intersects both $y$ and $z$: the middle vertical line (with angles 3,7,11,15) or the one with $x$ (angles 6,10,14) or the bottom vertical (angles 1,5,9,13). Wait, $\angle 5$ is on the bottom vertical (angle 5) and $\angle 10$ is on the middle vertical (angle 10). The transversal is the slanted line (the one with $y$ and $z$? No, $y$ and $z$ are the slanted lines. Wait, no—$y$ and $z$ are the two slanted lines (the ones with arrows: $y$ is the upper slanted line, $z$ is the lower slanted line). The transversal is the vertical line that crosses both $y$ and $z$. So $\angle 5$ is between $z$ and the bottom vertical line, $\angle 10$ is between $y$ and the middle vertical line? No, wait, maybe I got the lines wrong. Let's re-express: the two lines being cut are $y$ and $z$ (the slanted lines), and the transversal is the vertical line (the one with $x$? No, the vertical line that passes through $\angle 5$ and $\angle 10$: the vertical line with angles 1,5,9,13 (bottom vertical) and the vertical line with angles 6,10,14 (middle vertical)? No, that can't be. Wait, no—$\angle 5$ and $\angle 10$: $\angle 5$ is at the intersection of $z$ (lower slanted line) and the bottom vertical line, $\angle 10$ is at the intersection of $y$ (upper slanted line) and the middle vertical line? No, that's not right. Wait, maybe the transversal is the slanted line (the one with $y$ and $z$? No, $y$ and $z$ are the slanted lines. Wait, I think I made a mistake. Let's look at the options: option C is $y \parallel z$ by Alternate Interior Angles Converse. So if $\angle 5 \cong \angle 10$, and they are alternate interior angles for lines $y$ and $z$ cut by a transversal (the vertical line), then by Alternate Interior Angles Converse, $y \parallel z$. So that's option C.

Step2: Match with Option

Check the options:
- A: $w \parallel x$ with Alternate Exterior Angles Converse: $\angle 5$ and $\angle 10$ are not exterior angles for $w$ and $x$.
- B: $w \parallel x$ with Corresponding Angles Converse: $\angle 5$ and $\angle 10$ are not corresponding angles for $w$ and $x$.
- C: $y \parallel z$ with Alternate Interior Angles Converse: $\angle 5$ and $\angle 10$ are alternate interior angles (inside the two lines $y$ and $z$, on opposite sides of the transversal (the vertical line)), so if they are congruent, $y \parallel z$ by Alternate Interior Angles Converse.
- D: $y \parallel z$ with Corresponding Angles Converse: $\angle 5$ and $\angle 10$ are not corresponding angles (corresponding angles are in the same position relative to the transversal and the two lines), so D is wrong.

Answer:

C) $y \parallel z$; Alternate Interior Angles Converse