The problem you've shared involves two parallel lines $m$ and $n$ cut by a transversal. Given that the lines are parallel, corresponding or alternate angles will be congruent. The angles provided in the diagram are expressions in terms of $x$:

• One angle is $8x - 23^\circ$.
• The other angle is $7x - 2^\circ$.

These two angles appear to be alternate interior angles, which are equal when two lines are parallel and cut by a transversal.

So, we set the expressions for these angles equal to each other:

$8x - 23 = 7x - 2$

Now, let's solve for $x$:

1. Subtract $7x$ from both sides: $8x - 7x - 23 = -2$ $x - 23 = -2$

2. Add 23 to both sides: $x = 21$

So, the value of $x$ is $21$.

Would you like a detailed explanation of why alternate interior angles are equal, or do you have any questions about this solution?

Here are 5 related questions for further practice:

1. What would happen if the angles were corresponding angles instead of alternate interior angles?
2. How do you verify that two lines are parallel using angles formed by a transversal?
3. If one of the angles was $5x + 30^\circ$, how would the equation change?
4. What is the sum of the interior angles on the same side of a transversal?
5. What would the value of $x$ be if one of the angles was supplementary to $8x - 23^\circ$?

Tip: When dealing with parallel lines cut by a transversal, always consider which angle relationships (alternate, corresponding, or supplementary) are involved to set up the correct equation.