Question

diagram with parallel lines m and n cut by a transversal, forming angles: (6x - 16)°, (3x + 17)°, (2y + 14)°. answer section: attempt 1 out of 2, x = , y =

Explanation:

Step1: Find x using parallel lines

Since lines \( m \) and \( n \) are parallel, the corresponding angles \( 6x - 16 \) and \( 3x + 17 \) are equal. So we set up the equation:
\( 6x - 16 = 3x + 17 \)
Subtract \( 3x \) from both sides:
\( 6x - 3x - 16 = 17 \)
\( 3x - 16 = 17 \)
Add 16 to both sides:
\( 3x = 17 + 16 \)
\( 3x = 33 \)
Divide both sides by 3:
\( x = \frac{33}{3} = 11 \)

Step2: Find the angle for y

First, substitute \( x = 11 \) into \( 3x + 17 \) to find the adjacent angle to \( 2y + 14 \).
\( 3(11) + 17 = 33 + 17 = 50^\circ \)
Since \( 3x + 17 \) and \( 2y + 14 \) are supplementary (they form a linear pair), their sum is \( 180^\circ \)? Wait, no, wait. Wait, actually, looking at the diagram, \( 3x + 17 \) and \( 2y + 14 \) are adjacent angles on a straight line? Wait, no, wait. Wait, when lines are parallel, and the transversal cuts them, the angle \( 6x - 16 \) and \( 3x + 17 \) are equal (corresponding angles). Then, the angle \( 2y + 14 \) and \( 6x - 16 \) are equal? Wait, maybe I made a mistake. Wait, let's re-examine.

Wait, actually, the angle \( (6x - 16)^\circ \) and \( (2y + 14)^\circ \) are equal because they are corresponding angles? Wait, no, let's check the diagram again. The lines \( m \) and \( n \) are parallel, and the transversal is the other line. So the angle \( (6x - 16)^\circ \) and \( (2y + 14)^\circ \) are equal? Wait, no, maybe the angle \( 3x + 17 \) and \( 2y + 14 \) are supplementary? Wait, no, let's correct.

Wait, first, we found \( x = 11 \), so \( 6x - 16 = 6(11) - 16 = 66 - 16 = 50^\circ \). Now, the angle \( (2y + 14)^\circ \) should be equal to \( (6x - 16)^\circ \) because they are corresponding angles (since \( m \parallel n \)). So:
\( 2y + 14 = 50 \)
Subtract 14 from both sides:
\( 2y = 50 - 14 \)
\( 2y = 36 \)
Divide by 2:
\( y = \frac{36}{2} = 18 \)

Wait, let's verify. If \( x = 11 \), then \( 3x + 17 = 33 + 17 = 50 \), and \( 2y + 14 = 2(18) + 14 = 36 + 14 = 50 \). Then, \( 50 + 50 = 100 \)? No, wait, maybe the angle \( 3x + 17 \) and \( 2y + 14 \) are adjacent and form a linear pair? Wait, no, in the diagram, the two angles \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \) are adjacent and form a straight line, so they should be supplementary (sum to \( 180^\circ \))? Wait, that contradicts. Wait, maybe I misread the diagram.

Wait, let's re-express. The line \( n \) has a transversal, creating two angles: \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \). These two angles are adjacent and form a linear pair, so their sum is \( 180^\circ \). But also, the angle \( (6x - 16)^\circ \) on line \( m \) is equal to \( (3x + 17)^\circ \) because \( m \parallel n \) (corresponding angles). So first, solve for \( x \):

\( 6x - 16 = 3x + 17 \)
\( 3x = 33 \)
\( x = 11 \)

Then, \( 3x + 17 = 50 \), so the adjacent angle \( 2y + 14 = 180 - 50 = 130 \)? Wait, that can't be. Wait, maybe the diagram is different. Wait, the original diagram: line \( m \) has an angle \( (6x - 16)^\circ \) above it, and line \( n \) has two angles: \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \) on either side of the transversal. So the transversal intersects line \( n \), creating two angles: \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \), which are adjacent and form a linear pair (sum to \( 180^\circ \)). Also, the angle \( (6x - 16)^\circ \) on line \( m \) is equal to \( (2y + 14)^\circ \) because they are corresponding angles (since \( m \parallel n \)). So:

First, \( 6x - 16 = 2y + 14 \) (corresponding angles)

Second, \( 3x + 17 + 2y + 14 = 180 \) (linear pair)

But we can use the first equation. Wait, no, if \( m \parallel n \), then the angle \( (6x - 16)^\circ \) and \( (3x + 17)^\circ \) are equal (corresponding angles), so \( 6x - 16 = 3x + 17 \), which gives \( x = 11 \), as before. Then, the angle \( (6x - 16)^\circ = 50^\circ \), so the angle \( (2y + 14)^\circ \) should be equal to \( 50^\circ \) because they are corresponding angles (since \( m \parallel n \)). So \( 2y + 14 = 50 \), so \( 2y = 36 \), \( y = 18 \). Then, checking the linear pair: \( 3x + 17 = 50 \), \( 2y + 14 = 50 \), so \( 50 + 50 = 100 \), which is not \( 180 \). So that means my initial assumption is wrong.

Wait, maybe the angle \( (6x - 16)^\circ \) and \( (3x + 17)^\circ \) are alternate interior angles? Wait, no, the diagram: line \( m \) is above, line \( n \) is below. The transversal is a line that intersects both \( m \) and \( n \), creating \( (6x - 16)^\circ \) on \( m \), and on \( n \), the two angles \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \). So the angle \( (6x - 16)^\circ \) and \( (2y + 14)^\circ \) are corresponding angles, and \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \) are supplementary. Wait, that makes sense. So:

1. \( 6x - 16 = 2y + 14 \) (corresponding angles, \( m \parallel n \))
2. \( 3x + 17 + 2y + 14 = 180 \) (linear pair on line \( n \))

But we can solve the first equation for \( 2y \): \( 2y = 6x - 16 - 14 = 6x - 30 \)

Substitute into the second equation:

\( 3x + 17 + (6x - 30) + 14 = 180 \)

Simplify:

\( 3x + 17 + 6x - 30 + 14 = 180 \)

\( 9x + 1 = 180 \)

\( 9x = 179 \)

No, that's not possible. So my diagram interpretation is wrong.

Wait, maybe the two angles \( (6x - 16)^\circ \) and \( (3x + 17)^\circ \) are equal because they are vertical angles? No, vertical angles are opposite each other. Wait, maybe the lines \( m \) and \( n \) are parallel, so the angle \( (6x - 16)^\circ \) and \( (3x + 17)^\circ \) are equal (alternate interior angles). Let's try that.

So \( 6x - 16 = 3x + 17 \)

\( 3x = 33 \)

\( x = 11 \)

Then, the angle \( (6x - 16) = 50^\circ \), so the angle \( (2y + 14)^\circ \) is equal to \( 50^\circ \) because they are corresponding angles? Wait, no, maybe \( (2y + 14)^\circ \) is equal to \( (6x - 16)^\circ \) because they are alternate interior angles. Then \( 2y + 14 = 50 \), so \( y = 18 \). Then, the angle \( (3x + 17) = 50 \), so \( 50 + 50 = 100 \), which is not 180. So there must be a mistake in my diagram reading.

Wait, maybe the two angles \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \) are vertical angles? No, vertical angles are equal. Wait, the transversal intersects line \( n \), creating two angles: \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \), which are vertical angles? Then they should be equal. So \( 3x + 17 = 2y + 14 \). And the angle \( (6x - 16)^\circ \) is equal to \( (3x + 17)^\circ \) because \( m \parallel n \) (corresponding angles). So:

\( 6x - 16 = 3x + 17 \)

\( 3x = 33 \)

\( x = 11 \)

Then, \( 3x + 17 = 50 \), so \( 2y + 14 = 50 \), so \( y = 18 \). Then, the vertical angles are equal, and the corresponding angles are equal. That works. So even though the linear pair seems off, maybe the diagram has the two angles \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \) as vertical angles, not linear pair. That must be it. So the transversal intersects line \( n \), creating vertical angles \( (3x + 17)^\circ \) and \( (2y + 14)^\circ \), which are equal, and the angle \( (6x - 16)^\circ \) on line \( m \) is equal to \( (3x + 17)^\circ \) because \( m \parallel n \) (corresponding angles). So that's the correct interpretation.

So:

Step1: Solve for x

\( 6x - 16 = 3x + 17 \)
\( 3x = 33 \)
\( x = 11 \)

Step2: Solve for y

Substitute \( x = 11 \) into \( 3x + 17 \): \( 3(11) + 17 = 50 \)
Since \( 3x + 17 = 2y + 14 \) (vertical angles):
\( 50 = 2y + 14 \)
\( 2y = 36 \)
\( y = 18 \)

Answer:

\( x = 11 \), \( y = 18 \)