https://file.aiquickdraw.com/mathbot/file/1725541038248926r75zq.jpg

In this image, we are given two angles formed by two parallel lines crossed by a transversal. The two angles, \( x + 138 \) and \( x + 58 \), are alternate interior angles, which means they are equal because the lines are parallel.

### Equation setup:
\[
x + 138 = x + 58
\]

### Solving for \( x \):
1. Subtract \( x \) from both sides:
\[
138 = 58
\]
This is a contradiction, which indicates that the angles given are not alternate interior angles in the correct sense. Upon closer inspection, one could interpret them as supplementary angles instead.

### Interpretation as supplementary angles:
The two angles on a straight line (formed by the transversal and one of the parallel lines) must add up to 180°:
\[
(x + 138) + (x + 58) = 180
\]

### Solving:
1. Simplify the equation:
\[
2x + 196 = 180
\]

2. Subtract 196 from both sides:
\[
2x = 180 - 196
\]
\[
2x = -16
\]

3. Divide by 2:
\[
x = -8
\]

Thus, the value of \( x \) is \( -8 \).

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How do we identify alternate interior angles in parallel lines?
2. What is the property of supplementary angles on a straight line?
3. How can contradictions in equations help us interpret geometry problems?
4. Can angles ever be negative in geometry?
5. How does the transversal affect angle relationships in parallel lines?

**Tip:** Always verify the geometric relationship (alternate, corresponding, or supplementary) before solving angles in parallel line problems.

Learn how to solve for x in a geometry problem involving supplementary angles formed by parallel lines and a transversal. The problem uses key concepts of algebra and geometry, including the theorem of supplementary angles. Suitable for students in Grades 8-10.

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