Question

1. find x
2. find x
3. find x
4. find x

Explanation:

Step1: Use vertical - angles property

Vertical angles are equal. For the first pair of intersecting lines with angles \((2x - 6)^{\circ}\) and \((x + 18)^{\circ}\), we set up the equation \(2x-6=x + 18\).

Step2: Solve the equation for \(x\)

Subtract \(x\) from both sides: \(2x-x-6=x - x+18\), which simplifies to \(x-6 = 18\). Then add 6 to both sides: \(x=18 + 6=24\).

For the second pair of intersecting lines with angles \((x - 6)^{\circ}\) and \((x + 12)^{\circ}\), since vertical - angles are equal, we set up the equation \(x-6=x + 12\). Subtracting \(x\) from both sides gives \(-6=12\), which is a contradiction. There is an error in the problem - setup for this case.

For the third pair of intersecting lines with angles \((3x-18)^{\circ}\) and \((x + 12)^{\circ}\), set up the equation \(3x-18=x + 12\).

Step3: Solve the new equation

Subtract \(x\) from both sides: \(3x-x-18=x - x+12\), so \(2x-18 = 12\). Add 18 to both sides: \(2x=12 + 18=30\). Divide both sides by 2: \(x = 15\).

For the fourth pair of intersecting lines with angles \((2x+31)^{\circ}\) and \((x + 23)^{\circ}\), set up the equation \(2x+31=x + 23\).

Step4: Solve the equation

Subtract \(x\) from both sides: \(2x-x+31=x - x+23\), so \(x+31 = 23\). Subtract 31 from both sides: \(x=23 - 31=-8\).

We'll focus on the first non - contradictory case.

Answer:

\(x = 24\)