Question

1. (y - 12)° (6x - 32)° (3y - 8)° (2x - 20)°

Explanation:

Step1: Use vertical angles property

Vertical angles are equal, so \(6x - 32 = 2x - 20\) (vertical angles) and \(y - 12 + 3y - 8 = 180\) (linear pair, supplementary angles).

First, solve for \(x\):
\(6x - 32 = 2x - 20\)
Subtract \(2x\) from both sides: \(4x - 32 = -20\)
Add 32 to both sides: \(4x = 12\)
Divide by 4: \(x = 3\)

Next, solve for \(y\):
\(y - 12 + 3y - 8 = 180\)
Combine like terms: \(4y - 20 = 180\)
Add 20 to both sides: \(4y = 200\)
Divide by 4: \(y = 50\)

Step2: Verify angles (optional)

For \(x = 3\), \(6x - 32 = 6(3) - 32 = -14\) (Wait, that can't be. Wait, maybe I mixed up vertical angles. Wait, actually, the linear pair for the \(x\) angles: \(6x - 32 + 2x - 20 = 180\)? Wait, no, vertical angles are equal, but maybe I misidentified. Wait, let's re-examine.

Wait, the two angles with \(x\): \(6x - 32\) and \(2x - 20\) – maybe they are supplementary? Wait, no, vertical angles are equal, but if they are adjacent, they are supplementary. Wait, no, when two lines intersect, vertical angles are equal, and adjacent angles are supplementary.

Wait, let's correct:

For the \(x\) angles: \(6x - 32\) and \(2x - 20\) – if they are adjacent (linear pair), then \(6x - 32 + 2x - 20 = 180\)

So \(8x - 52 = 180\)

\(8x = 232\)

\(x = 29\)

Ah, I made a mistake earlier. Let's redo:

Step1 (corrected): Solve for \(x\) (adjacent angles, supplementary)

\(6x - 32 + 2x - 20 = 180\)
Combine like terms: \(8x - 52 = 180\)
Add 52: \(8x = 232\)
Divide by 8: \(x = 29\)

Step2 (corrected): Solve for \(y\) (adjacent angles, supplementary)

\(y - 12 + 3y - 8 = 180\)
Combine like terms: \(4y - 20 = 180\)
Add 20: \(4y = 200\)
Divide by 4: \(y = 50\)

Now check \(x = 29\):

\(6x - 32 = 6(29) - 32 = 174 - 32 = 142\)

\(2x - 20 = 2(29) - 20 = 58 - 20 = 38\)

142 + 38 = 180, which is supplementary, correct.

For \(y = 50\):

\(y - 12 = 50 - 12 = 38\)

\(3y - 8 = 150 - 8 = 142\)

38 + 142 = 180, correct. Also, vertical angles: \(y - 12 = 2x - 20\) (38 = 38) and \(3y - 8 = 6x - 32\) (142 = 142), which are equal, so that's correct.

Answer:

\(x = 29\), \(y = 50\)