https://file.aiquickdraw.com/mathbot/file/172679854204989m1runp.jpg

To solve this problem, let's analyze the information provided in the image:

1. The two lines \( r \) and \( s \) are parallel (\( r \parallel s \)).
2. \( m\angle 1 = 40 - 4x \)
3. \( m\angle 2 = 130 - 24x \)
4. We are asked to find the value of \( x \) for which \( r \parallel s \), and then determine \( m\angle 1 \) and \( m\angle 2 \).

Since \( r \parallel s \), and angles \( \angle 1 \) and \( \angle 2 \) are on opposite sides of the transversal, they form **alternate interior angles**. Alternate interior angles are congruent when the lines are parallel. Hence, we set \( m\angle 1 = m\angle 2 \).

Thus, we have the equation:

\[
40 - 4x = 130 - 24x
\]

### Solving for \( x \):

\[
40 - 4x = 130 - 24x
\]
Move all terms involving \( x \) to one side and constants to the other side:
\[
130 - 40 = 24x - 4x
\]
\[
90 = 20x
\]
\[
x = \frac{90}{20}
\]
\[
x = 4.5
\]

### Now, substitute \( x = 4.5 \) into the expressions for \( m\angle 1 \) and \( m\angle 2 \):

- For \( m\angle 1 \):
  \[
  m\angle 1 = 40 - 4(4.5) = 40 - 18 = 22^\circ
  \]

- For \( m\angle 2 \):
  \[
  m\angle 2 = 130 - 24(4.5) = 130 - 108 = 22^\circ
  \]

Thus, \( m\angle 1 = 22^\circ \) and \( m\angle 2 = 22^\circ \).

### Final Answer:
- \( x = 4.5 \)
- \( m\angle 1 = 22^\circ \)
- \( m\angle 2 = 22^\circ \)

Would you like more details or have any other questions?

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Here are five related questions to explore:
1. What are corresponding angles, and how do they relate to parallel lines?
2. How can you identify different types of angle pairs formed by a transversal?
3. What happens when the lines are not parallel—how do the angles change?
4. Can alternate exterior angles be used to solve similar problems?
5. How can you find unknown angles using supplementary or complementary angle properties?

**Tip:** In problems involving parallel lines and transversals, always look for congruent angle pairs, such as alternate interior angles or corresponding angles, to set up equations easily.

Determine the value of x for which r is parallel to s. Then find m∠1 and m∠2. Given: m∠1 = 40 - 4x, m∠2 = 130 - 24x.

Learn how to solve for x when two lines are parallel, using alternate interior angles. This math problem involves algebraic equations and parallel line theorems, suitable for Grades 9-10. The problem gives m∠1 = 40 - 4x and m∠2 = 130 - 24x, and we solve for x and find the angle measures.

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