Question

1. find the value of x for which ( l parallel m ).

31
19
35
41

Explanation:

Step1: Find the supplementary angle of \(105^{\circ}\)

The supplementary angle of \(105^{\circ}\) is \(180^{\circ}- 105^{\circ}=75^{\circ}\)

Step2: Set up the equation for parallel lines

Since \(l\parallel m\), the angle \(75^{\circ}\) and \((3x - 18)^{\circ}\) are complementary (because of the right - angle in between). So \(75+(3x - 18)=90\)

Step3: Solve the equation

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Wait, there is a mistake. Let's re - analyze.

The correct approach:
The angle adjacent to \(105^{\circ}\) (on the same line) is \(180 - 105=75^{\circ}\). Since \(l\parallel m\), the sum of the angle \(75^{\circ}\) and \((3x - 18)^{\circ}\) is \(90^{\circ}\) (because of the right - angle formed by the transversal and the middle line).

Step1: Set up the equation

\(75+(3x - 18)=90\)

Step2: Simplify the left - hand side

\(3x+(75 - 18)=90\), so \(3x + 57=90\)

Step3: Isolate \(x\)

Subtract \(57\) from both sides: \(3x=90 - 57\), \(3x=33\). Then divide both sides by \(3\): \(x = 11\). But this is not in the options.

Wait, another approach:
The angle \(105^{\circ}\) and the angle \((3x-18)^{\circ}\) and the right - angle \((90^{\circ})\) are related as \(105+(3x - 18)+90 = 180\) (sum of angles on a straight line)

Step1: Set up the equation

\(105+(3x - 18)+90=180\)

Step2: Simplify the left - hand side

\(3x+(105 - 18+90)=180\), \(3x+(87 + 90)=180\), \(3x+177 = 180\)

Step3: Isolate \(x\)

Subtract \(177\) from both sides: \(3x=180 - 177\), \(3x = 3\), \(x = 1\) (wrong)

Correct approach:
The angle \(105^{\circ}\) and the angle \((3x-18)^{\circ}\) are related as \(105+(3x - 18)=180 - 90\) (because of the right - angle and parallel lines)

Step1: Set up the equation

\(105+(3x - 18)=90\) (wrong)

Wait, correct:
The angle \(105^{\circ}\) and the angle \((3x - 18)^{\circ}\) are related as \(105+(3x - 18)+90=360 - 180\) (sum of angles around the intersection of transversal and lines \(l\) and \(m\)). No, better:
Since \(l\parallel m\), the angle \(105^{\circ}\) and the angle \((3x - 18)^{\circ}\) and the right - angle \((90^{\circ})\) satisfy \(105+(3x - 18)=180\) (sum of angles on a straight line)

Step1: Set up the equation

\(3x-18=180 - 105\)

Step2: Simplify

\(3x-18 = 75\)

Step3: Solve for \(x\)

Add \(18\) to both sides: \(3x=75 + 18\), \(3x=93\), then \(x = 31\)

Answer:

\(31\)