Question

what is the value of x?
$x = 2.25$
$x = 13$
$x = 22$
$x = 11.25$

Explanation:

Step1: Use vertical - angle and supplementary - angle relationships

We know that the angle \((2x + 10)^{\circ}\) and the angle adjacent to \((6x+1)^{\circ}\) (which is \(180-(6x + 1)=(179-6x)^{\circ}\)) are vertical angles. So \(2x+10=179 - 6x\).

Step2: Solve the equation for \(x\)

Add \(6x\) to both sides of the equation \(2x+10=179 - 6x\):
\(2x+6x+10=179-6x + 6x\), which simplifies to \(8x+10=179\).
Subtract \(10\) from both sides: \(8x+10 - 10=179 - 10\), so \(8x=169\).
Divide both sides by \(8\): \(x=\frac{169}{8}=21.125\) (This approach is wrong).

Let's use another approach.
We know that \((6x + 1)+79+(2x + 10)=180\) (sum of angles on a straight line).

Step1: Simplify the left - hand side of the equation

Combine like terms: \((6x+2x)+(1 + 79+10)=180\), so \(8x+90=180\).

Step2: Solve for \(x\)

Subtract \(90\) from both sides: \(8x+90 - 90=180 - 90\), which gives \(8x=90\).
Divide both sides by \(8\): \(x = 11.25\).

Answer:

\(x = 11.25\)