Question

4 determine the value of x and y in the following figure. (y +23)° x° 63° x = and y =

Explanation:

Step1: Find x using vertical angles

Vertical angles are equal. The angle \( x^\circ \) and \( 63^\circ \) are adjacent supplementary angles? Wait, no, actually, the angle \( (y + 23)^\circ \) and \( 63^\circ \) are vertical angles? Wait, no, let's look again. Wait, when two lines intersect, vertical angles are equal, and adjacent angles are supplementary (sum to \( 180^\circ \)). Wait, the angle \( x^\circ \) and \( 63^\circ \): are they adjacent? Wait, no, the angle \( (y + 23)^\circ \) and \( 63^\circ \) – wait, actually, the angle \( x^\circ \) and \( 63^\circ \): wait, no, let's correct. When two lines intersect, adjacent angles are supplementary (sum to \( 180^\circ \)), and vertical angles are equal. Wait, the angle \( (y + 23)^\circ \) and \( 63^\circ \): are they vertical angles? Wait, no, the angle \( x^\circ \) and \( 63^\circ \): wait, maybe I made a mistake. Wait, the angle \( x^\circ \) and \( 63^\circ \) – actually, the angle \( (y + 23)^\circ \) and \( 63^\circ \) are vertical angles? No, wait, the angle \( x^\circ \) and \( 63^\circ \): let's see, the two lines intersect, so the angle opposite to \( 63^\circ \) is \( (y + 23)^\circ \)? Wait, no, the angle \( x^\circ \) and \( 63^\circ \) are adjacent, forming a linear pair, so they should be supplementary? Wait, no, wait, the angle \( (y + 23)^\circ \) and \( x^\circ \) are adjacent? Wait, maybe I need to re-examine.

Wait, the figure shows two intersecting lines. So, the angle labeled \( 63^\circ \) and the angle labeled \( (y + 23)^\circ \) – are they vertical angles? Wait, no, vertical angles are opposite each other. Wait, the angle \( x^\circ \) and \( 63^\circ \): if \( x \) and \( 63^\circ \) are adjacent, then \( x + 63 = 180 \)? No, that can't be. Wait, no, maybe \( x \) and \( 63^\circ \) are vertical angles? Wait, no, vertical angles are equal. Wait, maybe the angle \( (y + 23)^\circ \) and \( 63^\circ \) are vertical angles? Wait, no, the angle \( (y + 23)^\circ \) and \( x^\circ \) are adjacent, forming a linear pair, so they sum to \( 180^\circ \). Wait, no, let's start over.

When two lines intersect, the vertical angles are equal, and adjacent angles (forming a linear pair) are supplementary (sum to \( 180^\circ \)). So, looking at the figure: the angle \( 63^\circ \) and the angle \( x^\circ \) – are they adjacent? Wait, no, the angle \( (y + 23)^\circ \) and \( 63^\circ \) are vertical angles? Wait, no, the angle \( x^\circ \) and \( 63^\circ \) are vertical angles? Wait, maybe the angle \( (y + 23)^\circ \) and \( 63^\circ \) are vertical angles, so \( y + 23 = 63 \)? No, that would make \( y = 40 \), but then what about \( x \)? Wait, no, maybe the angle \( x^\circ \) and \( 63^\circ \) are adjacent, so \( x + 63 = 180 \)? No, that would make \( x = 117 \), but then what about \( y \)? Wait, no, maybe the angle \( (y + 23)^\circ \) and \( x^\circ \) are vertical angles, and \( 63^\circ \) and \( x^\circ \) are adjacent. Wait, I think I messed up. Let's correct.

Wait, the two lines intersect, so there are two pairs of vertical angles. Let's denote the angles: when two lines intersect, the opposite angles are equal. So, the angle labeled \( 63^\circ \) and the angle opposite to it (which is \( (y + 23)^\circ \)) – wait, no, the angle \( (y + 23)^\circ \) and the angle \( 63^\circ \): are they opposite? Wait, the angle \( x^\circ \) and the angle \( 63^\circ \): maybe \( x \) is supplementary to \( 63^\circ \)? No, that doesn't make sense. Wait, no, let's look at the figure again. The angle \( (y + 23)^\circ \) and \( x^\circ \) are adjacent, forming a linear pair, so they sum to \( 180^\circ \). The angle \( 63^\circ \) and \( x^\circ \) – wait, no, the angle \( 63^\circ \) and \( (y + 23)^\circ \) are vertical angles, so they are equal. Wait, that must be it. So, vertical angles are equal, so \( y + 23 = 63 \)? No, that would make \( y = 40 \), but then what about \( x \)? Wait, no, maybe the angle \( x^\circ \) and \( 63^\circ \) are vertical angles? Wait, no, the angle \( x^\circ \) and \( (y + 23)^\circ \) are adjacent, so \( x + (y + 23) = 180 \). And the angle \( 63^\circ \) and \( x^\circ \) are vertical angles? No, vertical angles are equal. Wait, I think I made a mistake. Let's try again.

Wait, when two lines intersect, the vertical angles are equal. So, the angle labeled \( 63^\circ \) and the angle labeled \( x^\circ \) – are they vertical angles? No, because \( x \) is adjacent to \( (y + 23) \). Wait, maybe the angle \( (y + 23)^\circ \) and \( 63^\circ \) are vertical angles, so \( y + 23 = 63 \), so \( y = 63 - 23 = 40 \). Then, the angle \( x^\circ \) and \( 63^\circ \) are supplementary? No, wait, \( x \) and \( (y + 23) \) are supplementary? Wait, no, adjacent angles on a straight line sum to \( 180^\circ \). So, if \( (y + 23)^\circ \) and \( x^\circ \) are adjacent, then \( (y + 23) + x = 180 \). But if \( (y + 23) = 63 \) (vertical angles with \( 63^\circ \)), then \( 63 + x = 180 \), so \( x = 180 - 63 = 117 \). Wait, that makes sense. So:

Step1: Find y using vertical angles

Vertical angles are equal, so \( y + 23 = 63 \).
\( y = 63 - 23 \)
\( y = 40 \)

Step2: Find x using supplementary angles

Adjacent angles on a straight line sum to \( 180^\circ \), so \( x + 63 = 180 \).
\( x = 180 - 63 \)
\( x = 117 \)

Answer:

\( x = 117 \) and \( y = 40 \)