Question

$\angle 16 \cong \angle 3$
a) $y \parallel z$: alternate exterior angles converse
b) $y \parallel z$: converse of corresponding angles thm.
c) $w \parallel x$: alternate interior angles converse
d) $w \parallel x$: consecutive exterior angles converse

Explanation:

Step1: Identify Angle Relationship

$\angle 16$ and $\angle 3$: $\angle 16$ is on line $w$, $\angle 3$ on line (let's see transversal and parallel candidates). Wait, lines $w$ and $x$? No, transversal is the slant line (let's call it $t$), cutting $w$ and $x$. Wait, $\angle 16$ is outside (exterior) relative to $w$ and $x$, and $\angle 3$ is inside? No, wait, alternate interior? Wait, no: $\angle 16$ and $\angle 3$: when transversal $t$ cuts $w$ and $x$, $\angle 16$ is above $w$, $\angle 3$ is below $w$ but inside $w$ and $x$? Wait, no, let's check the lines: $w$ and $x$ are the two vertical (or near-vertical) lines, and the slant line is the transversal. $\angle 16$ is on the left of $w$, above the transversal, and $\angle 3$ is on the right of $w$, below the transversal? No, wait, $\angle 16$ and $\angle 3$: alternate interior angles? Wait, no, the correct lines: if we consider $w$ and $x$ as the two parallel candidates, and the slant line as transversal. $\angle 16$ and $\angle 3$: are they alternate interior? Wait, $\angle 16$ is exterior to $w$ and $x$? No, $\angle 16$ is on the outer side of $w$ (left), $\angle 3$ is on the inner side between $w$ and $x$? Wait, no, let's re-express: the two lines are $w$ (left vertical) and $x$ (right vertical), transversal is the slant line. $\angle 16$: formed by $w$ and transversal, above the transversal, left of $w$. $\angle 3$: formed by $w$ and transversal? No, $\angle 3$ is formed by $x$ and transversal? Wait, no, the labels: $\angle 16$ is at the intersection of $w$ and transversal (top left), $\angle 3$ is at the intersection of $x$ and transversal (bottom right of $w$). Wait, no, the lines: $w$ and $x$ are parallel? Wait, the answer options: C is $w \parallel x$ by Alternate Interior Angles Converse. Let's check: Alternate Interior Angles Converse states that if alternate interior angles are congruent, then the lines are parallel. $\angle 16$ and $\angle 3$: are they alternate interior? Wait, $\angle 16$ and $\angle 3$: when transversal cuts $w$ and $x$, $\angle 16$ is above $w$, $\angle 3$ is below $w$ but between $w$ and $x$? Wait, no, maybe I mixed up. Wait, $\angle 16$ and $\angle 3$: let's see their positions. $\angle 16$ is at the top left of the transversal and $w$, $\angle 3$ is at the bottom right of the transversal and $x$? No, $\angle 3$ is at the intersection of $x$ and transversal, below the transversal, between $w$ and $x$. $\angle 16$ is at the intersection of $w$ and transversal, above the transversal, outside $w$ and $x$? Wait, no, alternate interior angles are between the two lines (interior) and on alternate sides of the transversal. So $\angle 16$: is it interior? No, $\angle 16$ is above $w$, left of $w$, so exterior to $w$ and $x$? Wait, maybe the correct lines are $w$ and $x$, and the transversal is the slant line. $\angle 15$ and $\angle 10$? No, the given congruence is $\angle 16 \cong \angle 3$. Wait, $\angle 16$ and $\angle 3$: let's check the angles. $\angle 16$: formed by $w$ and transversal, upper left. $\angle 3$: formed by $x$ and transversal, lower right, between $w$ and $x$. Wait, maybe I made a mistake. Let's look at the options: Option C is $w \parallel x$ by Alternate Interior Angles Converse. So if $\angle 16 \cong \angle 3$, are they alternate interior angles? Let's see: the two lines are $w$ and $x$, transversal is the slant line. $\angle 16$ is on one side of the transversal (left), above the transversal, and $\angle 3$ is on the other side (right), below the transversal, between $w$ and $x$ (interior). So yes, alternate interior angles. So by Alternate Interior Angles Converse, if they are congruent, then $w \parallel x$. So the correct option is C.

Step2: Evaluate Options

- Option A: $y \parallel z$? But $\angle 16$ and $\angle 3$ are not related to $y$ and $z$. Eliminate.
- Option B: Corresponding Angles? $\angle 16$ and $\angle 3$ are not corresponding (same position). Eliminate.
- Option C: $w \parallel x$ by Alternate Interior Angles Converse. $\angle 16$ and $\angle 3$ are alternate interior angles (between $w$ and $x$, on alternate sides of transversal). If congruent, lines are parallel. This matches.
- Option D: Consecutive Exterior Angles? Not a standard theorem, and $\angle 16$ and $\angle 3$ are not consecutive exterior. Eliminate.

Answer:

C) $w \parallel x$; Alternate Interior Angles Converse