Question

current skill $ find the measure of angle 8. (image shows intersecting lines with angles labeled 7, 8, 6, 3, 1, 4, and angle expressions 2x + 118, 2x + 98)

Explanation:

Step1: Identify Parallel Lines and Transversal

The two lines with red arrows are parallel, and the other line is a transversal. So, the angles \(2x + 118\) and \(2x + 98\) are same - side exterior angles, and they are supplementary. So, we can set up the equation \((2x + 118)+(2x + 98)=180\).

Step2: Solve for \(x\)

Combine like terms: \(4x+216 = 180\). Subtract 216 from both sides: \(4x=180 - 216=- 36\). Divide both sides by 4: \(x=\frac{-36}{4}=-9\).

Step3: Find the measure of \(2x + 118\)

Substitute \(x = - 9\) into \(2x+118\): \(2\times(-9)+118=-18 + 118 = 100\).

Step4: Identify Vertical Angles or Linear Pairs

Angle 7 and \(2x + 118\) are supplementary (linear pair), so angle 7 \(=180 - 100 = 80\)? Wait, no. Wait, angle 8 and angle 7: Wait, angle 7 and angle 8 are vertical angles? No, angle 7 and angle 8: Wait, angle 7 and angle 6 are vertical angles? Wait, no. Wait, angle 8 and the angle \(2x + 98\): Wait, maybe I made a mistake. Wait, let's re - examine the diagram. Wait, angle 8 and angle 1: No, let's look at the parallel lines. Wait, angle \(2x + 118\) and angle 3: Maybe alternate interior angles? Wait, no. Wait, let's start over.

Wait, the two angles \(2x + 118\) and \(2x + 98\) are same - side exterior angles, so they should be supplementary. So:

\((2x + 118)+(2x + 98)=180\)

\(4x+216 = 180\)

\(4x=180 - 216=-36\)

\(x=-9\)

Now, angle \(2x + 98=2\times(-9)+98=-18 + 98 = 80\).

Angle 8 and angle (the angle with measure \(2x + 98\)): Wait, angle 8 and the angle \(2x + 98\) are corresponding angles? Wait, no. Wait, angle 8 and angle 1: No, let's look at the vertical angles and linear pairs.

Wait, angle 7 and \(2x + 118\) form a linear pair, so angle 7 \(=180-(2x + 118)\). Substituting \(x=-9\), angle 7 \(=180-(2\times(-9)+118)=180-( - 18 + 118)=180 - 100 = 80\).

Angle 8 and angle 7 are vertical angles? No, angle 7 and angle 8: Wait, angle 7 and angle 8 are adjacent and form a linear pair? No, angle 7 and angle 8: Wait, in the diagram, angle 7 and angle 8 are vertical angles? Wait, no, angle 7 and angle 6 are vertical angles, angle 8 and angle 6: No, angle 8 and angle 7: Wait, maybe angle 8 is equal to angle (the angle \(2x + 98\)) because of parallel lines and transversal. Wait, angle \(2x + 98 = 80\), and angle 8 is equal to angle (the angle \(2x + 98\)) because they are corresponding angles? Wait, if the two lines with red arrows are parallel, and the transversal cuts them, then angle 8 and the angle \(2x + 98\) are corresponding angles, so angle 8 \(=2x + 98\). Since \(2x + 98 = 80\), angle 8 is 80. Wait, but let's check again.

Wait, when \(x=-9\), \(2x + 118=2\times(-9)+118 = 100\), \(2x + 98=2\times(-9)+98 = 80\).

Angle 7 and \(2x + 118\) are supplementary: \(180 - 100 = 80\), so angle 7 is 80. Angle 8 and angle 7 are vertical angles? No, angle 8 and angle 7: Wait, angle 7 and angle 8 are adjacent and form a linear pair? No, angle 7 and angle 8: Wait, in the diagram, angle 7 and angle 8 are vertical angles? Wait, no, angle 7 and angle 6 are vertical angles, angle 8 and angle 6: No, angle 8 and angle 7: Wait, maybe angle 8 is equal to angle 7 because they are vertical angles? Wait, angle 7 is 80, so angle 8 is 80.

Answer:

The measure of angle 8 is \(80^{\circ}\)