Question

20 numeric 1 point what is the value of y in the following drawing? transversal k transversal l n 132° (2y - 24) m (2x +18) 62° n||m answer submit

Explanation:

Step1: Find x using supplementary angles

Since \( n \parallel m \) and transversal \( k \) intersects them, the angle \( 132^\circ \) and \( (2x + 18)^\circ \) are supplementary (they form a linear pair). So:
\( 132 + (2x + 18) = 180 \)
\( 2x + 150 = 180 \)
\( 2x = 180 - 150 \)
\( 2x = 30 \)
\( x = 15 \) (But we don't need x for y, so let's focus on transversal \( l \))

Step2: Find the angle related to \( (2y - 24)^\circ \)

The \( 62^\circ \) angle and the angle corresponding to \( (2y - 24)^\circ \) (since \( n \parallel m \) and transversal \( l \) intersects them) should be equal to the angle we can find from the other transversal. Wait, actually, the angle \( (2y - 24)^\circ \) and the angle adjacent to \( 132^\circ \) (which is \( 180 - 132 = 48^\circ \)? No, wait, let's look at the parallel lines. Wait, the angle \( (2y - 24)^\circ \) and the angle that is equal to \( 180 - 132 = 48^\circ \)? No, maybe alternate interior or corresponding. Wait, actually, the angle \( (2y - 24)^\circ \) should be equal to the angle that is supplementary to \( 132^\circ \)? Wait, no, let's re-examine.

Wait, the two parallel lines \( n \) and \( m \), with transversal \( k \), the angle \( 132^\circ \) and \( (2x + 18)^\circ \) are supplementary, so \( 2x + 18 = 48^\circ \) (since \( 180 - 132 = 48 \)). Wait, earlier calculation: \( 132 + (2x + 18) = 180 \) => \( 2x + 18 = 48 \) => \( 2x = 30 \) => \( x = 15 \), so \( 2x + 18 = 48^\circ \). Now, looking at transversal \( l \), the \( 62^\circ \) angle and the angle \( (2y - 24)^\circ \): wait, no, maybe the angle \( (2y - 24)^\circ \) is equal to \( 48 + 62 \)? Wait, no, maybe the two angles \( 48^\circ \) (from transversal \( k \)) and \( 62^\circ \) (from transversal \( l \)) add up to \( (2y - 24)^\circ \)? Wait, no, let's think about the triangle or the parallel lines. Wait, actually, the angle \( (2y - 24)^\circ \) should be equal to \( 48 + 62 = 110 \)? No, wait, maybe the angle \( (2y - 24)^\circ \) is equal to \( 180 - (180 - 132 + 62) \)? No, this is confusing. Wait, let's start over.

Wait, the key is that \( n \parallel m \), so the angle \( (2y - 24)^\circ \) and the angle formed by \( 132^\circ \) and \( 62^\circ \)? Wait, no, the angle \( (2y - 24)^\circ \) should be equal to \( 132 - 62 = 70 \)? No, that's not right. Wait, maybe the angle \( (2y - 24)^\circ \) is equal to \( 180 - 132 + 62 \)? No, \( 180 - 132 = 48 \), \( 48 + 62 = 110 \). Wait, let's set \( 2y - 24 = 110 \)? No, that would give \( y = 67 \), but let's check.

Wait, another approach: the angle \( (2y - 24)^\circ \) and the angle that is \( 180 - 132 = 48^\circ \) and \( 62^\circ \) form a triangle? No, the lines are parallel, so the angle \( (2y - 24)^\circ \) should be equal to \( 48 + 62 = 110 \)? Wait, no, maybe the angle \( (2y - 24)^\circ \) is equal to \( 180 - (132 - 62) \)? No, \( 132 - 62 = 70 \), \( 180 - 70 = 110 \). Wait, let's solve \( 2y - 24 = 110 \): \( 2y = 134 \), \( y = 67 \). No, that's not right. Wait, maybe the angle \( (2y - 24)^\circ \) is equal to \( 132 - 62 = 70 \)? Then \( 2y - 24 = 70 \), \( 2y = 94 \), \( y = 47 \). No, that's not. Wait, maybe I made a mistake in the first step.

Wait, the angle \( 132^\circ \) and \( (2x + 18)^\circ \) are same-side interior angles, so they are supplementary. So \( 132 + (2x + 18) = 180 \), so \( 2x + 18 = 48 \), so \( 2x = 30 \), \( x = 15 \), so \( 2x + 18 = 48^\circ \). Now, the angle \( 62^\circ \) and the angle \( (2y - 24)^\circ \): since \( n \parallel m \), the angle \( (2y - 24)^\circ \) should be equal to \( 48 + 62 = 110^\circ \)? Wait, no, that's not a standard angle relationship. Wait, maybe the angle \( (2y - 24)^\circ \) is equal to \( 180 - (180 - 132 + 62) \)? No, \( 180 - 132 = 48 \), \( 48 + 62 = 110 \), \( 180 - 110 = 70 \). No, this is confusing. Wait, let's look at the diagram again. The line \( n \) has a rightward arrow, and \( m \) also has a rightward arrow. The transversal \( k \) makes \( 132^\circ \) with \( n \), and transversal \( l \) makes \( (2y - 24)^\circ \) with the rightward part of \( n \). The \( m \) line has a \( 62^\circ \) angle with transversal \( l \) downward. So, since \( n \parallel m \), the angle \( (2y - 24)^\circ \) and the angle formed by \( 132^\circ \) and \( 62^\circ \) should be equal? Wait, maybe the angle \( (2y - 24)^\circ \) is equal to \( 132 - 62 = 70 \)? No, \( 132 - 62 = 70 \), then \( 2y - 24 = 70 \), \( 2y = 94 \), \( y = 47 \). No, that's not. Wait, maybe the angle \( (2y - 24)^\circ \) is equal to \( 180 - 132 + 62 = 110 \), so \( 2y - 24 = 110 \), \( 2y = 134 \), \( y = 67 \). But let's check with the supplementary angle. Wait, the angle \( 132^\circ \) and the angle adjacent to \( (2y - 24)^\circ \): if \( (2y - 24)^\circ = 110^\circ \), then the adjacent angle would be \( 70^\circ \), but that doesn't match. Wait, maybe I made a mistake in the angle relationship.

Wait, another way: the angle \( (2y - 24)^\circ \) and the angle \( 132^\circ \) are related by the two transversals. Since \( n \parallel m \), the sum of the angles formed by the two transversals should be equal to \( 180^\circ \)? No, maybe the angle \( (2y - 24)^\circ \) is equal to \( 180 - 132 + 62 = 110 \), so solving \( 2y - 24 = 110 \):

\( 2y = 110 + 24 \)

\( 2y = 134 \)

\( y = 67 \). Wait, but let's check with the other angle. The angle \( 62^\circ \) and the angle \( (2x + 18)^\circ = 48^\circ \): \( 48 + 62 = 110 \), which is equal to \( (2y - 24)^\circ \). So that makes sense. So \( 2y - 24 = 110 \), so \( y = 67 \).

Wait, let's verify: \( 2y - 24 = 2*67 - 24 = 134 - 24 = 110 \). Now, the angle \( 110^\circ \) and the angle formed by \( 48^\circ \) (from transversal \( k \)) and \( 62^\circ \) (from transversal \( l \)): \( 48 + 62 = 110 \), which matches. So that's correct.

Answer:

67