Question

given $m \parallel n$, find the value of $x$ and $y$.

Explanation:

Step1: Find x using supplementary angles

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 measure of \((9x + 2)^\circ\)

Substitute \(x = 14\) into \((9x + 2)^\circ\):

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

Step3: Find y using alternate interior angles

Since \(m \parallel n\), the angle \((2y + 20)^\circ\) and \((9x + 2)^\circ\) are alternate interior angles, so they are equal. We know \((9x + 2)^\circ = 128^\circ\), so:

$$ 2y + 20 = 128 $$

Subtract 20 from both sides:

$$ 2y = 108 $$

Divide by 2:

$$ y = 54 $$

Answer:

\(x = 14\), \(y = 54\)