Question

current skill $
8x + 68
2
3
4
5
-6x + 116
7
8
find the measure of angle 2.

Explanation:

Step1: Identify Parallel Lines and Transversal

The two slanted lines are parallel (indicated by arrow marks), and the vertical line is a transversal. So, angle \(8x + 68\) and angle \(-6x + 116\) are corresponding angles (or alternate exterior angles), hence equal.
\[8x + 68=-6x + 116\]

Step2: Solve for \(x\)

Add \(6x\) to both sides:
\[8x+6x + 68=-6x+6x + 116\]
\[14x + 68 = 116\]
Subtract 68 from both sides:
\[14x+68 - 68=116 - 68\]
\[14x = 48\]
Wait, no, 116 - 68 is 48? Wait, 116 - 68 = 48? Wait, 68 + 48 = 116? Yes. Then divide by 14:
Wait, no, 14x = 48? Wait, no, 8x + 68 and -6x + 116: let's re - calculate.
\(8x+68=-6x + 116\)
\(8x + 6x=116 - 68\)
\(14x = 48\)? Wait, 116 - 68 is 48? Wait, 68+48 = 116, yes. Then \(x=\frac{48}{14}=\frac{24}{7}\)? Wait, that can't be right. Wait, maybe I made a mistake in angle identification. Wait, maybe angle \(8x + 68\) and angle \(-6x + 116\) are alternate interior angles? Wait, no, looking at the diagram, the two parallel lines: the upper angle is \(8x + 68\) (angle 2's adjacent, maybe angle 3 or angle 4? Wait, no, the two parallel lines: the top slanted line and the bottom slanted line. The vertical line is transversal. So angle \(8x + 68\) (let's say angle at the top intersection, above the transversal) and angle \(-6x + 116\) (at the bottom intersection, above the transversal) are corresponding angles, so they should be equal. Wait, maybe I miscalculated 116 - 68. 116 - 68: 116 - 70 = 46, then +2 = 48. Yes. So \(14x = 48\), \(x=\frac{48}{14}=\frac{24}{7}\)? Wait, that seems odd. Wait, maybe the angles are alternate exterior angles? Wait, no, maybe the angles are equal because the lines are parallel, so let's re - check.

Wait, maybe angle \(8x + 68\) and angle \(-6x + 116\) are equal because they are corresponding angles. So:

\(8x+68=-6x + 116\)

\(8x + 6x=116 - 68\)

\(14x = 48\)

\(x=\frac{48}{14}=\frac{24}{7}\approx3.428\)

Wait, but then angle \(8x + 68=8\times\frac{24}{7}+68=\frac{192}{7}+68=\frac{192 + 476}{7}=\frac{668}{7}\approx95.43\)

Angle \(-6x + 116=-6\times\frac{24}{7}+116=-\frac{144}{7}+116=\frac{- 144+812}{7}=\frac{668}{7}\approx95.43\), so they are equal. Now, angle 2 and angle \(8x + 68\): are they supplementary? Wait, angle 2 and the angle \(8x + 68\): if the two lines (the slanted and the vertical) intersect, then angle 2 and the angle adjacent to \(8x + 68\) (angle 3) are vertical angles? Wait, no, the intersection of the two lines: the slanted line and the vertical line. So angle \(8x + 68\) and angle 3 are vertical angles? Wait, no, the slanted line and vertical line intersect, forming angle \(8x + 68\), angle 2, angle 3, angle 4. So angle \(8x + 68\) and angle 4 are vertical angles? Wait, maybe I should look at the linear pair. Wait, angle 2 and angle \(8x + 68\): are they supplementary? Wait, no, if the two slanted lines are parallel, then the angle \(8x + 68\) and angle \(-6x + 116\) are equal (corresponding angles). Then, angle 2 and angle \(8x + 68\): let's see, angle 2 and the angle \(8x + 68\) form a linear pair? Wait, no, the vertical line and the slanted line: angle \(8x + 68\) and angle 2 are adjacent angles, forming a linear pair? Wait, no, the slanted line and vertical line intersect, so angle \(8x + 68\) and angle 2 are adjacent, and their sum is 180? Wait, no, if the two slanted lines are parallel, then the angle \(8x + 68\) and angle \(-6x + 116\) are equal. Let's solve for \(x\) again.

Wait, maybe I made a mistake in the equation. Let's re - do:

\(8x+68=-6x + 116\)

\(8x+6x=116 - 68\)

\(14x = 48\)

\(x=\frac{48}{14}=\frac{24}{7}\approx3.428\)

Now, angle \(8x + 68=8\times\frac{24}{7}+68=\frac{192}{7}+\frac{476}{7}=\frac{668}{7}\approx95.43\)

Now, angle 2 and angle \(8x + 68\): are they vertical angles? Wait, no, the intersection of the two lines (slanted and vertical) creates vertical angles. Angle \(8x + 68\) and angle 4 are vertical angles, angle 2 and angle 3 are vertical angles. And angle \(8x + 68\) and angle 2: are they supplementary? Wait, if the two slanted lines are parallel, then the angle \(8x + 68\) and angle \(-6x + 116\) are equal. Then angle 2 and angle \(-6x + 116\): are they equal? Wait, maybe angle 2 and angle \(-6x + 116\) are corresponding angles. Wait, no, the vertical line is transversal, and the two slanted lines are parallel. So angle 2 (at the top intersection, above the transversal) and angle \(-6x + 116\) (at the bottom intersection, above the transversal) are corresponding angles, so they should be equal. So angle 2=\(-6x + 116\), and angle \(8x + 68\) is equal to angle \(-6x + 116\), so angle 2=\(8x + 68\). Wait, that would mean \(8x + 68=-6x + 116\), which is the same equation. So if angle 2 is equal to \(8x + 68\), then we can find angle 2 once we find \(x\).

Wait, maybe I made a mistake in the equation. Let's check the arithmetic again.

\(8x+68=-6x + 116\)

Add \(6x\) to both sides: \(14x+68 = 116\)

Subtract 68: \(14x=116 - 68 = 48\)

\(x=\frac{48}{14}=\frac{24}{7}\approx3.428\)

Now, angle 2=\(8x + 68=8\times\frac{24}{7}+68=\frac{192 + 476}{7}=\frac{668}{7}\approx95.43\). But this seems odd. Wait, maybe the angles are alternate interior angles. Wait, maybe the two parallel lines: the angle \(8x + 68\) and angle \(-6x + 116\) are alternate interior angles, so they are equal. Then, angle 2 and angle \(8x + 68\) are supplementary? Wait, no, if the two lines (slanted and vertical) intersect, angle 2 and angle \(8x + 68\) are adjacent, forming a linear pair, so their sum is 180. Wait, that would be a different equation. Let's try that.

If angle \(8x + 68\) and angle 2 are supplementary: \(8x + 68+\text{angle }2 = 180\), and angle 2 is equal to \(-6x + 116\) (corresponding angles). So:

\(8x + 68+(-6x + 116)=180\)

\(2x+184 = 180\)

\(2x=180 - 184=-4\)

\(x = - 2\)

Now, angle 2=\(-6x + 116=-6\times(-2)+116 = 12 + 116 = 128\)

And angle \(8x + 68=8\times(-2)+68=-16 + 68 = 52\)

And 52+128 = 180, which is a linear pair. Ah! So I made a mistake in angle identification. Angle \(8x + 68\) and angle 2 are supplementary (linear pair), and angle 2 and angle \(-6x + 116\) are corresponding angles (so equal). So the correct equation is \(8x + 68+(-6x + 116)=180\) (since angle \(8x + 68\) and angle 2 are supplementary, and angle 2=\(-6x + 116\)).

So let's re - do:

Step1: Correct Angle Relationship

Angle \(8x + 68\) and angle 2 are supplementary (linear pair), and angle 2=\(-6x + 116\) (corresponding angles, as the two parallel lines are cut by a transversal). So:

\(8x + 68+(-6x + 116)=180\)

Step2: Simplify the Equation

Combine like terms:

\((8x-6x)+(68 + 116)=180\)

\(2x+184 = 180\)

Step3: Solve for \(x\)

Subtract 184 from both sides:

\(2x=180 - 184=-4\)

Divide by 2:

\(x=-2\)

Step4: Find Angle 2

Angle 2=\(-6x + 116\). Substitute \(x = - 2\):

\(-6\times(-2)+116=12 + 116 = 128\)

Answer:

The measure of angle 2 is \(\boldsymbol{128}\) degrees.