Question

in the diagram below, $overleftrightarrow{vy}$ intersects $overleftrightarrow{wz}$ and $overrightarrow{oz}$ at point $o$.

use the tiles to correctly complete the statements. a tile may be used once, more than once, or not at all.
$angle woy$ and $\boxed{}$ are supplementary angles.
$angle woy$ and $\boxed{}$ are vertical angles.

Explanation:

Step1: Recall Supplementary Angles

Supplementary angles sum to \(180^\circ\). \(\angle WOY\) and \(\angle VOY\) form a linear pair? Wait, no. Wait, \(\overleftrightarrow{VY}\) intersects \(\overleftrightarrow{WZ}\) at \(O\). Wait, \(\angle WOY\) and \(\angle YOZ\)? No, wait. Wait, \(\angle WOY\) and \(\angle VOY\) – no, wait, \(\overleftrightarrow{VY}\) is a straight line? Wait, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle WOY + \angle VOW\)? No, wait, let's look at the diagram. \(\overleftrightarrow{WZ}\) is horizontal, \(\overleftrightarrow{VY}\) is a line through \(O\), and \(OX\) is perpendicular to \(\overleftrightarrow{WZ}\) (right angle).

For supplementary angles: \(\angle WOY\) and \(\angle VOY\)? No, wait, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle WOY + \angle YOZ\)? Wait, no, \(\overleftrightarrow{WZ}\) is a straight line? Wait, \(\overleftrightarrow{WZ}\) is horizontal, so \(\angle WOZ\) is straight? Wait, no, \(\overleftrightarrow{WZ}\) is a straight line, so \(\angle WOW\) no. Wait, \(\angle WOY\) and \(\angle YOZ\) – no, wait, \(\overleftrightarrow{VY}\) intersects \(\overleftrightarrow{WZ}\) at \(O\). So \(\angle WOY\) and \(\angle VOZ\)? No, vertical angles. Wait, supplementary angles: two angles that add to \(180^\circ\). So \(\angle WOY\) and \(\angle VOY\) – no, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle WOY + \angle VOW\) – no, I'm confused. Wait, let's re-express. \(\overleftrightarrow{VY}\) is a straight line, so \(\angle VOW + \angle WOY = 180^\circ\)? No, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle VOW + \angle WOY = 180^\circ\)? Wait, no, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle VOW + \angle WOY = 180^\circ\)? Wait, maybe \(\angle WOY\) and \(\angle YOZ\) – no, wait, \(\overleftrightarrow{WZ}\) is a straight line, so \(\angle WOZ = 180^\circ\), but \(\angle WOY + \angle YOZ = \angle WOZ = 180^\circ\)? Wait, no, \(\overleftrightarrow{VY}\) intersects \(\overleftrightarrow{WZ}\) at \(O\), so \(\angle WOY\) and \(\angle VOZ\) are vertical angles? Wait, no, vertical angles are opposite each other when two lines intersect. So when \(\overleftrightarrow{VY}\) and \(\overleftrightarrow{WZ}\) intersect at \(O\), the vertical angles are \(\angle WOY\) and \(\angle VOZ\)? Wait, no, \(\angle WOV\) and \(\angle YOZ\)? Wait, maybe I made a mistake. Let's start over.

Supplementary angles: sum to \(180^\circ\). So \(\angle WOY\) and \(\angle VOY\) – no, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle WOY + \angle VOW = 180^\circ\)? Wait, no, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle VOW + \angle WOY = 180^\circ\)? Wait, maybe \(\angle WOY\) and \(\angle YOZ\) – no, \(\overleftrightarrow{WZ}\) is a straight line, so \(\angle WOZ = 180^\circ\), so \(\angle WOY + \angle YOZ = 180^\circ\)? Wait, no, \(\angle WOY\) and \(\angle VOY\) – no, I think I messed up. Wait, the diagram: \(W\) is left, \(Z\) is right, \(V\) is down-left, \(Y\) is up-right, \(X\) is up. So \(\overleftrightarrow{VY}\) is a line from \(V\) (down-left) to \(Y\) (up-right), passing through \(O\). \(\overleftrightarrow{WZ}\) is horizontal, left \(W\), right \(Z\), through \(O\). \(OX\) is vertical up, perpendicular to \(\overleftrightarrow{WZ}\) (right angle).

So for supplementary angles: \(\angle WOY\) and \(\angle VOY\)? No, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle VOW + \angle WOY = 180^\circ\)? Wait, \(\angle VOW\) and \(\angle WOY\) – no, \(\angle VOW\) is adjacent to \(\angle WOY\) on line \(\overleftrightarrow{VY}\)? Wait, no, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle VOW + \angle WOY = 180^\circ\)? Wait, no, \(\angle VOW\) is at \(O\), between \(V\) and \(W\), and \(\angle WOY\) is between \(W\) and \(Y\). So together, they make \(\angle VOY\), which is a straight line? Wait, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle VOY = 180^\circ\), so \(\angle VOW + \angle WOY = 180^\circ\)? No, that can't be. Wait, maybe \(\angle WOY\) and \(\angle YOZ\) – no, \(\overleftrightarrow{WZ}\) is a straight line, so \(\angle WOZ = 180^\circ\), so \(\angle WOY + \angle YOZ = 180^\circ\)? Wait, \(\angle YOZ\) is between \(Y\) and \(Z\). So \(\angle WOY + \angle YOZ = 180^\circ\), so they are supplementary. Wait, but also, \(\angle WOY\) and \(\angle VOW\) – no, maybe I'm wrong. Wait, another approach: supplementary angles are two angles that form a linear pair (adjacent and sum to \(180^\circ\)) or non-adjacent but sum to \(180^\circ\). Since \(\overleftrightarrow{WZ}\) is a straight line, \(\angle WOZ = 180^\circ\), but \(\angle WOY\) and \(\angle YOZ\) are adjacent on \(\overleftrightarrow{WZ}\)? No, \(\overleftrightarrow{WZ}\) is horizontal, \(\angle WOY\) is between \(W\) (left) and \(Y\) (up-right), \(\angle YOZ\) is between \(Y\) (up-right) and \(Z\) (right). So together, they make \(\angle WOZ\), which is a straight line (180°), so \(\angle WOY + \angle YOZ = 180^\circ\), so they are supplementary. Wait, but also, \(\overleftrightarrow{VY}\) is a straight line, so \(\angle VOW + \angle WOY = 180^\circ\)? Wait, \(\angle VOW\) is between \(V\) (down-left) and \(W\) (left), \(\angle WOY\) is between \(W\) (left) and \(Y\) (up-right). So together, they make \(\angle VOW + \angle WOY = \angle VOY\), which is a straight line (180°), so they are supplementary. Wait, so which one? Wait, the problem says "use the tiles" – but the tiles aren't shown. Wait, maybe the options are \(\angle YOZ\) and \(\angle VOZ\) or something. Wait, maybe I made a mistake. Let's think about vertical angles. Vertical angles are opposite each other when two lines intersect. So when \(\overleftrightarrow{VY}\) and \(\overleftrightarrow{WZ}\) intersect at \(O\), the vertical angles are \(\angle WOY\) and \(\angle VOZ\), because they are opposite each other. So \(\angle WOY\) and \(\angle VOZ\) are vertical angles.

For supplementary angles: \(\angle WOY\) and \(\angle VOY\) – no, \(\overleftrightarrow{VY}\) is straight, so \(\angle VOW + \angle WOY = 180^\circ\)? Wait, no, \(\angle VOW\) is adjacent to \(\angle WOY\) on \(\overleftrightarrow{VY}\), so they are supplementary. Wait, but also, \(\angle WOY\) and \(\angle YOZ\) – on \(\overleftrightarrow{WZ}\), they are adjacent, so supplementary. Wait, maybe the tiles include \(\angle YOZ\) and \(\angle VOZ\). Wait, maybe the first blank is \(\angle YOZ\) (supplementary) and the second is \(\angle VOZ\) (vertical).

Wait, let's confirm vertical angles: when two lines intersect, vertical angles are equal and opposite. So \(\overleftrightarrow{VY}\) and \(\overleftrightarrow{WZ}\) intersect at \(O\), so the vertical angles are \(\angle WOY\) and \(\angle VOZ\) (since \(\angle WOV\) and \(\angle YOZ\) are the other pair). Wait, \(\angle WOY\) is between \(W\)-\(O\)-\(Y\), and \(\angle VOZ\) is between \(V\)-\(O\)-\(Z\). Yes, those are vertical angles.

For supplementary angles: \(\angle WOY\) and \(\angle YOZ\) – because they are adjacent on \(\overleftrightarrow{WZ}\), forming a linear pair, so sum to \(180^\circ\). Or \(\angle WOY\) and \(\angle VOW\) – adjacent on \(\overleftrightarrow{VY}\), sum to \(180^\circ\). But since \(\overleftrightarrow{WZ}\) is a straight line, \(\angle WOZ = 180^\circ\), so \(\angle WOY + \angle YOZ = 180^\circ\), so they are supplementary.

Step2: Recall Vertical Angles

Vertical angles are opposite when two lines intersect. So \(\overleftrightarrow{VY}\) and \(\overleftrightarrow{WZ}\) intersect at \(O\), so \(\angle WOY\) and \(\angle VOZ\) are vertical angles (opposite each other).

So putting it together:

First blank (supplementary): \(\angle YOZ\) (or \(\angle VOW\), but likely \(\angle YOZ\) as it's on the horizontal line)

Second blank (vertical): \(\angle VOZ\)

Wait, but maybe the tiles are \(\angle YOZ\) and \(\angle VOZ\). Let's assume the tiles are \(\angle YOZ\) (supplementary) and \(\angle VOZ\) (vertical).

Answer:

First blank: \(\angle YOZ\) (supplementary with \(\angle WOY\) as they form a linear pair on \(\overleftrightarrow{WZ}\), summing to \(180^\circ\))
Second blank: \(\angle VOZ\) (vertical angles with \(\angle WOY\) as they are opposite when \(\overleftrightarrow{VY}\) and \(\overleftrightarrow{WZ}\) intersect at \(O\))

(Note: If tiles include these angles, these are the correct ones. The key is supplementary angles sum to \(180^\circ\) (linear pair) and vertical angles are opposite at intersection.)