Question

1. (9x - 2)° (10y + 6)° (5x + 54)° l m

Explanation:

Step1: Identify the relationship between angles

The angles \((9x - 2)^\circ\) and \((5x + 54)^\circ\) are alternate exterior angles (since lines \(l\) and \(m\) are parallel, as indicated by the diagram's structure), so they are equal.
\[9x - 2 = 5x + 54\]

Step2: Solve for \(x\)

Subtract \(5x\) from both sides:
\[9x - 5x - 2 = 54\]
\[4x - 2 = 54\]
Add 2 to both sides:
\[4x = 54 + 2\]
\[4x = 56\]
Divide both sides by 4:
\[x = \frac{56}{4}\]
\[x = 14\]

Step3: Find the measure of the angle with \(x\)

Substitute \(x = 14\) into \((9x - 2)^\circ\):
\[9(14) - 2 = 126 - 2 = 124^\circ\]
This angle and \((10y + 6)^\circ\) are supplementary (they form a linear pair), so:
\[10y + 6 + 124 = 180\]

Step4: Solve for \(y\)

Simplify the left side:
\[10y + 130 = 180\]
Subtract 130 from both sides:
\[10y = 180 - 130\]
\[10y = 50\]
Divide both sides by 10:
\[y = \frac{50}{10}\]
\[y = 5\]

Answer:

\(x = 14\), \(y = 5\)