Question

which angles are supplementary angles?
∠vwy and ∠utr ∠vwy and ∠xwt
∠vwy and ∠xwy ∠vwy and ∠stw

Explanation:

Step1: Recall Supplementary Angles

Supplementary angles are two angles whose sum is \(180^\circ\) (a straight angle). They can be adjacent (forming a linear pair) or non - adjacent.

Step2: Analyze Each Option

• Option 1: \(\angle VWY\) and \(\angle UTR\)
• First, we need to check the relationship between these angles. From the diagram, we can see that \(\angle UTR\) and \(\angle STW\) are vertical angles, and \(\angle VWY\) and \(\angle XWY\) are related to a straight line? Wait, no. Let's look at the lines. Line \(SU\) and line \(VX\) are parallel? Wait, maybe not. Wait, line \(RY\) is a transversal. Let's check the angles around point \(W\). \(\angle VWY\) and \(\angle XWY\): Wait, no, let's check the last option first. Wait, \(\angle VWY\) and \(\angle XWY\): \(\angle VWY+\angle XWY = 180^\circ\) because they form a linear pair (they are adjacent and their non - common sides form a straight line). Wait, no, wait the options:
• Wait, let's re - examine. The angle \(\angle VWY\) and \(\angle XWY\): If we look at the intersection of line \(VX\) and line \(RY\) at point \(W\), \(\angle VWY\) and \(\angle XWY\) are adjacent angles that form a linear pair. So their sum is \(180^\circ\), so they are supplementary.
• Let's check other options:
• For \(\angle VWY\) and \(\angle UTR\): These two angles are not related in a way that their sum is \(180^\circ\). \(\angle UTR\) is at point \(T\) and \(\angle VWY\) is at point \(W\), and there is no indication that their sum is \(180^\circ\).
• For \(\angle VWY\) and \(\angle XWT\): \(\angle XWT\) and \(\angle VWY\) are not supplementary. \(\angle XWT\) and \(\angle VWY\) are not adjacent and do not form a linear pair, and from the diagram, their measures do not add up to \(180^\circ\).
• For \(\angle VWY\) and \(\angle STW\): \(\angle STW\) and \(\angle VWY\) are not supplementary. \(\angle STW\) is at point \(T\) and \(\angle VWY\) is at point \(W\), and there is no reason for their sum to be \(180^\circ\).
• For \(\angle VWY\) and \(\angle XWY\): Since they are adjacent and form a linear pair (their non - common sides lie on a straight line), \(\angle VWY+\angle XWY = 180^\circ\), so they are supplementary.

Answer:

\(\angle VWY\) and \(\angle XWY\) (the option \(\angle VWY\) and \(\angle XWY\))