Question

current skill $
find the measure of angle 8.
label optional

Explanation:

Step1: Identify vertical angles

Angle 5 and angle 3 are vertical angles, so they are equal. Angle 5 is supplementary to angle 8, and angle 3 is labeled as $8x - 19$. Also, angle $5x + 56$ and angle $8x - 19$ are corresponding angles (since the two horizontal lines are parallel, indicated by red arrows), so they are equal.

Step2: Set up equation for x

Corresponding angles equality: $5x + 56 = 8x - 19$

Step3: Solve for x

Rearrange terms: $56 + 19 = 8x - 5x$ → $75 = 3x$ → $x = 25$

Step4: Find angle 8

Angle 8 and angle $5x + 56$ are supplementary (linear pair). Calculate $5x + 56$: $5(25) + 56 = 125 + 56 = 181$? Wait, no—correction: Angle 8 and angle 7 are vertical angles, angle 7 and angle 5 are supplementary, angle 5 is equal to angle $5x + 56$? No, correct: Angle $5x + 56$ and angle 8 are vertical angles? No, recheck: The angle labeled $5x + 56$ and angle 8 are vertical angles? Wait, the top horizontal line: left side is $5x + 56$, opposite side (vertical) is angle 7? No, the two intersecting lines: the angle $5x + 56$ and angle 7 are a linear pair? Wait, no—corresponding angles: the two horizontal lines are parallel, so the angle $5x + 56$ (on top left line, below transversal) corresponds to angle 3 (on bottom right line, above transversal), so $5x + 56 = 8x - 19$ (correct). Then angle 8 is equal to angle 5 (vertical angles), and angle 5 is supplementary to angle $5x + 56$? No, angle $5x + 56$ and angle 5 are a linear pair, so they add to 180. Angle 8 is equal to angle 5 (vertical angles), so angle 8 = 180 - (5x + 56). First, x=25, so 5x+56=181 is impossible—wait, mistake: $5x +56 =8x -19$ → 56+19=3x →75=3x→x=25. Then 8x-19=8*25-19=200-19=181, which is wrong. Oh, maybe alternate interior angles: angle $5x+56$ and angle 4 are alternate interior, angle 4 and angle 3 are supplementary, angle 3=8x-19. So $5x+56 +8x -19=180$ →13x +37=180→13x=143→x=11. That makes sense. So correct equation: since the two angles are consecutive interior angles (supplementary), $5x+56 +8x -19=180$.

Step5: Recalculate x

$13x +37=180$ →13x=143→x=11.

Step6: Find angle 8

Angle 8 is equal to angle $8x -19$ (corresponding angles). So $8*11 -19=88-19=69$.

Answer:

69