To solve for the value of $x$ that makes lines $m$ and $n$ parallel, we need to use the fact that when two lines are parallel, the alternate interior angles are equal.

In the image, you have two angles:

• One angle is $3^\circ$.
• The other angle is expressed as $(2x + 20)^\circ$.

Since these are alternate interior angles, they must be equal when $m \parallel n$. Therefore, we set up the equation:

$2x + 20 = 3$

Now, solve for $x$:

1. Subtract 20 from both sides: $2x = 3 - 20$ $2x = -17$

2. Divide both sides by 2: $x = \frac{-17}{2}$ $x = -8.5$

Thus, $x = -8.5$.

Would you like a deeper explanation of the steps? Here are some related questions you might want to explore:

1. Why are alternate interior angles equal when lines are parallel?
2. What would happen if the angles given were supplementary instead of equal?
3. Can we verify the solution by plugging $x = -8.5$ back into the equation?
4. How do we recognize alternate interior angles in different diagrams?
5. What are other types of angle relationships that indicate parallel lines?

Tip: Always double-check the geometric configuration to ensure the correct angle relationship (like alternate interior, corresponding, etc.) is applied when dealing with parallel lines!