https://file.aiquickdraw.com/mathbot/file/1728045110705at5zakep.jpg

To solve these problems, we will use properties of parallel lines and angles. Specifically, when two lines are parallel, certain angle relationships arise, such as:

1. **Corresponding angles** are equal.
2. **Alternate interior angles** are equal.
3. **Same-side interior angles** are supplementary (sum to 180°).

Let's solve each one:

### 5.
- \( l \parallel m \) means that the angles are either corresponding or supplementary.
- The angle \( (9x + 2)^\circ \) and the 119° angle are **supplementary** (same-side interior angles), so:
  \[
  (9x + 2) + 119 = 180
  \]
  \[
  9x + 121 = 180
  \]
  \[
  9x = 59
  \]
  \[
  x = \frac{59}{9} \approx 6.56
  \]

### 6.
- The angles \( (12x - 8)^\circ \) and 104° are **corresponding angles**, so they are equal.
  \[
  12x - 8 = 104
  \]
  \[
  12x = 112
  \]
  \[
  x = \frac{112}{12} = 9.33
  \]

### 7.
- The angles \( (5x + 7)^\circ \) and \( (8x - 71)^\circ \) are **alternate interior angles**, so they are equal.
  \[
  5x + 7 = 8x - 71
  \]
  \[
  7 + 71 = 8x - 5x
  \]
  \[
  78 = 3x
  \]
  \[
  x = \frac{78}{3} = 26
  \]

### 8.
- The angles \( (4x - 7)^\circ \) and \( (7x - 61)^\circ \) are **corresponding angles**, so they are equal.
  \[
  4x - 7 = 7x - 61
  \]
  \[
  61 - 7 = 7x - 4x
  \]
  \[
  54 = 3x
  \]
  \[
  x = \frac{54}{3} = 18
  \]

Let me know if you'd like further details or explanations. Here are five questions to extend this topic:

1. How do you determine when angles are supplementary?
2. What are the differences between corresponding and alternate interior angles?
3. How can you check if lines are parallel based on angle relationships?
4. What happens to the angles if the lines are not parallel?
5. How can transversal lines impact angle calculations in geometric figures?

**Tip:** When solving angle problems involving parallel lines, always identify whether the angles are corresponding, alternate interior, or supplementary for accurate equation setup.

If l || m, solve for x in problems 5 through 8.

Learn how to solve for x using angle relationships in parallel lines in geometry problems. This problem set includes supplementary, corresponding, and alternate interior angles. Suitable for students in Grades 8-10.

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