Question

what is the value of x?
○ x = 2.25
○ x = 11.25
○ x = 13
○ x = 22
(2x + 10)°
(6x + 1)°
79°

Explanation:

Step1: Identify angle relationships

The angle \((6x + 1)^\circ\) and \(79^\circ\) are alternate interior angles (assuming parallel lines and a transversal), and the angle \((2x + 10)^\circ\) and the sum of \((6x + 1)^\circ\) and \(79^\circ\) are related (since they form a straight line or vertical angles? Wait, actually, looking at the diagram, the angle \((2x + 10)^\circ\) should be equal to \((6x + 1)^\circ + 79^\circ\) because of vertical angles or linear pair? Wait, no, let's re - examine. Wait, actually, the angle \((2x + 10)^\circ\) and the angle formed by \((6x + 1)^\circ\) and \(79^\circ\) are vertical angles? Wait, no, maybe the sum of \((6x + 1)^\circ\) and \(79^\circ\) is equal to \((2x + 10)^\circ\)? Wait, no, that can't be. Wait, actually, the correct relationship: since the two horizontal lines are parallel (assuming) and the transversal, the angle \((6x + 1)^\circ\) and \(79^\circ\) are alternate interior angles? No, wait, the angle \((2x + 10)^\circ\) and the angle adjacent to \(79^\circ\) and \((6x + 1)^\circ\) form a triangle? Wait, no, let's think again.

Wait, the correct equation: The angle \((2x + 10)^\circ\) is equal to \((6x + 1)^\circ+79^\circ\) because of the exterior angle theorem or vertical angles? Wait, actually, if we look at the intersection of the lines, the angle \((2x + 10)^\circ\) and the angle that is the sum of \((6x + 1)^\circ\) and \(79^\circ\) are vertical angles, so they are equal. So:

\(2x + 10=(6x + 1)+79\)

Step2: Solve the equation

First, simplify the right - hand side:

\(2x + 10=6x+80\)

Subtract \(2x\) from both sides:

\(10 = 4x+80\)

Subtract 80 from both sides:

\(10 - 80=4x\)

\(- 70 = 4x\)? Wait, that can't be right. Wait, maybe I got the angle relationship wrong.

Wait, maybe the angle \((6x + 1)^\circ\) and \(79^\circ\) are supplementary to \((2x + 10)^\circ\)? Wait, no. Wait, let's look at the other way. Maybe the angle \((2x + 10)^\circ\) and \(79^\circ\) are related such that \((6x + 1)^\circ\) and \((2x + 10)^\circ\) are related to \(79^\circ\) as alternate interior angles. Wait, no, let's try another approach.

Wait, maybe the two angles \((6x + 1)^\circ\) and \(79^\circ\) are equal? No, because then \(6x+1 = 79\), \(6x=78\), \(x = 13\), but then let's check the other angle. If \(x = 13\), \(6x + 1=79\), and \(2x+10 = 36\), which doesn't make sense. Wait, maybe the angle \((2x + 10)^\circ\) and \(79^\circ\) are equal? Then \(2x+10 = 79\), \(2x=69\), \(x = 34.5\), which is not an option.

Wait, maybe the sum of \((6x + 1)^\circ\) and \((2x + 10)^\circ\) is equal to \(180^\circ-79^\circ\)? No, that seems complicated. Wait, let's check the options. Let's plug in \(x = 11.25\):

\(6x+1=6\times11.25 + 1=67.5 + 1 = 68.5\)

\(2x + 10=2\times11.25+10 = 22.5+10 = 32.5\)

\(68.5+79=147.5 eq32.5\)

Plug in \(x = 22\):

\(6x + 1=6\times22+1 = 132 + 1=133\)

\(2x + 10=2\times22 + 10=44 + 10 = 54\)

\(133+79 = 212 eq54\)

Plug in \(x = 13\):

\(6x+1=6\times13 + 1=78 + 1 = 79\)

\(2x+10=2\times13+10 = 26 + 10 = 36\)

\(79 + 79=158 eq36\)

Wait, maybe the angle \((2x + 10)^\circ\) is equal to \(180-(6x + 1)-79\)? Let's set up the equation:

\(2x+10=180-(6x + 1)-79\)

Simplify the right - hand side:

\(2x + 10=180 - 6x-1 - 79\)

\(2x + 10=100 - 6x\)

Add \(6x\) to both sides:

\(8x+10 = 100\)

Subtract 10 from both sides:

\(8x=90\)

\(x=\frac{90}{8}=11.25\)

Ah, that works. So the correct equation is based on the fact that the sum of the three angles ( \((6x + 1)^\circ\), \(79^\circ\), and \((2x + 10)^\circ\)) is \(180^\circ\) (since they form a straight line). So:

\((6x + 1)+79+(2x + 10)=180\)

Step3: Solve the correct equation

Combine like terms:

\(6x+2x+1 + 79+10=180\)

\(8x+90 = 180\)

Subtract 90 from both sides:

\(8x=180 - 90\)

\(8x=90\)

Divide both sides by 8:

\(x=\frac{90}{8}=11.25\)

Answer:

\(x = 11.25\) (corresponding to the option \(x = 11.25\))