Question

try this for yourself!
algebra find the value of x that makes ( m parallel n ).
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Explanation:

Problem 3:

Step1: Identify angle relationship (vertical & corresponding)

The \(120^\circ\) angle and \(3x^\circ\) are equal (vertical angles, then corresponding as lines are parallel). So \(3x = 120\).

Step2: Solve for \(x\)

Divide both sides by 3: \(x=\frac{120}{3}=40\).

Step1: Identify supplementary angles (same - side interior)

For parallel lines, same - side interior angles are supplementary. So \((2x + 15)+135 = 180\).

Step2: Simplify the equation

\(2x+150 = 180\). Subtract 150 from both sides: \(2x=180 - 150 = 30\).

Step3: Solve for \(x\)

Divide by 2: \(x=\frac{30}{2}=15\).

Step1: Identify supplementary angles (same - side interior or vertical - corresponding)

The \(150^\circ\) angle and \((3x - 15)^\circ\) are supplementary (same - side interior angles for parallel lines). So \((3x-15)+150 = 180\) or we can also consider that the angle equal to \((3x - 15)^\circ\) and \(150^\circ\) are supplementary.

Step2: Simplify the equation

\(3x+135 = 180\). Subtract 135: \(3x=180 - 135 = 45\).

Step3: Solve for \(x\)

Divide by 3: \(x=\frac{45}{3}=15\)? Wait, no, wait. Wait, actually, the angle \((3x - 15)^\circ\) and the angle supplementary to \(150^\circ\) (which is \(30^\circ\))? Wait, no, let's re - check. Wait, the \(150^\circ\) and \((3x - 15)^\circ\) are same - side interior angles? Wait, no, maybe vertical angles. Wait, no, the correct relationship: if \(m\parallel n\), then the angle \((3x - 15)^\circ\) and \(150^\circ\) are supplementary? Wait, no, let's look at the diagram. The \(150^\circ\) and \((3x - 15)^\circ\) should be supplementary (same - side interior angles sum to \(180^\circ\)). So:

Step1: Set up the equation

\(3x-15 + 150=180\)

Step2: Simplify

\(3x + 135=180\)
Subtract 135 from both sides: \(3x=180 - 135 = 45\)

Step3: Solve for \(x\)

\(x = \frac{45}{3}=15\)? Wait, no, that can't be. Wait, maybe the angle \((3x - 15)^\circ\) is equal to the angle supplementary to \(150^\circ\). The supplementary angle of \(150^\circ\) is \(180 - 150=30^\circ\). So \(3x-15 = 30\)

Step1 (corrected): Set up the correct equation

\(3x-15=30\) (because the angle \((3x - 15)^\circ\) and \(150^\circ\) are same - side interior angles, so they are supplementary, so \((3x - 15)=180 - 150 = 30\))

Step2: Solve for \(x\)

Add 15 to both sides: \(3x=30 + 15=45\)
Divide by 3: \(x = 15\)? Wait, no, \(3x-15 = 30\) gives \(3x=45\), \(x = 15\). Wait, but let's check: \(3(15)-15=45 - 15 = 30\), and \(30+150 = 180\), which works for supplementary angles. So:

Answer:

\(x = 40\)

Problem 4: