Question

given $m \parallel n$, find the value of $x$ and $y$.
(there is a diagram with two parallel lines m and n, and a transversal intersecting them. on line m, the angles formed with the transversal are $(9x + 2)^\circ$ and $(5x - 18)^\circ$. on line n, the angle formed with the transversal is $(2y + 20)^\circ$.)

Explanation:

Step1: Solve for \( x \)

The angles \( (9x + 2)^\circ \) and \( (5x - 18)^\circ \) are supplementary (they form a linear pair), so their sum is \( 180^\circ \).

$$ (9x + 2) + (5x - 18) = 180 $$

Combine like terms:

$$ 14x - 16 = 180 $$

Add 16 to both sides:

$$ 14x = 196 $$

Divide by 14:

$$ x = 14 $$

Step2: Find the angle for \( y \)

First, find the measure of \( (9x + 2)^\circ \) by substituting \( x = 14 \):

$$ 9(14) + 2 = 126 + 2 = 128^\circ $$

The angle \( (2y + 20)^\circ \) and \( (9x + 2)^\circ \) are same - side interior angles? Wait, no. Wait, since \( m\parallel n \), the angle \( (5x - 18)^\circ \) and \( (2y + 20)^\circ \) are corresponding angles? Wait, no. Wait, first, let's find \( (5x - 18)^\circ \) when \( x = 14 \):

$$ 5(14)-18=70 - 18 = 52^\circ $$

Wait, actually, the angle \( (9x + 2)^\circ \) and \( (2y + 20)^\circ \): Wait, no, the angle \( (5x - 18)^\circ \) and \( (2y + 20)^\circ \) are alternate interior angles? Wait, no, let's re - examine. The angle \( (9x + 2)^\circ \) and \( (5x - 18)^\circ \) are supplementary. Now, the angle \( (5x - 18)^\circ \) (when \( x = 14 \), it's \( 52^\circ \)) and \( (2y + 20)^\circ \): Wait, no, actually, the angle \( (9x + 2)^\circ=128^\circ \), and the angle \( (2y + 20)^\circ \) and \( (9x + 2)^\circ \) are same - side interior angles? Wait, no, that can't be. Wait, no, the correct approach: Since \( m\parallel n \), the angle \( (5x - 18)^\circ \) and \( (2y + 20)^\circ \) are corresponding angles? Wait, no, let's look at the diagram again. The transversal cuts \( m \) and \( n \). The angle \( (9x + 2)^\circ \) and \( (2y + 20)^\circ \): Wait, no, the angle \( (5x - 18)^\circ \) and \( (2y + 20)^\circ \) are alternate interior angles? Wait, no, I think I made a mistake. Wait, the angle \( (9x + 2)^\circ \) and \( (2y + 20)^\circ \): Wait, no, let's calculate \( (9x + 2)^\circ \) when \( x = 14 \), we get \( 128^\circ \). Then, since \( m\parallel n \), the angle \( (2y + 20)^\circ \) and \( (5x - 18)^\circ \): Wait, no, the angle \( (5x - 18)^\circ = 52^\circ \), and if we consider that \( (2y + 20)^\circ \) and \( (5x - 18)^\circ \) are equal? No, that's not right. Wait, no, the angle \( (9x + 2)^\circ \) and \( (2y + 20)^\circ \): Wait, no, the correct relation is that the angle \( (9x + 2)^\circ \) and \( (2y + 20)^\circ \) are same - side interior angles? No, that would mean they are supplementary. Wait, \( 128+(2y + 20)=180 \)? Wait, no, that would be if they are same - side interior angles. Let's check:
If \( 128+(2y + 20)=180 \), then \( 2y+148 = 180 \), \( 2y=32 \), \( y = 16 \). Wait, but let's check with the other angle. Wait, the angle \( (5x - 18)^\circ=52^\circ \), and if \( (2y + 20)^\circ \) is equal to \( (5x - 18)^\circ \), then \( 2y+20 = 52 \), \( 2y = 32 \), \( y = 16 \). Ah, right! Because \( (5x - 18)^\circ \) and \( (2y + 20)^\circ \) are corresponding angles (since \( m\parallel n \) and the transversal cuts them, so corresponding angles are equal). So we set \( 2y+20=5x - 18 \). We know \( x = 14 \), so substitute \( x = 14 \):

$$ 2y+20=5(14)-18 $$

$$ 2y+20 = 70 - 18 $$

$$ 2y+20 = 52 $$

Subtract 20 from both sides:

$$ 2y=32 $$

Divide by 2:

$$ y = 16 $$

Answer:

\( x = 14 \), \( y = 16 \)