The diagram shows two intersecting pairs of parallel lines, creating angles at the intersection points. The angles labeled are:

• $10x + 5^\circ$ and $15x - 30^\circ$.

Since these two angles are formed by parallel lines and intersecting transversals, they are alternate interior angles, which means they are equal.

Thus, we can set up the following equation:

$10x + 5 = 15x - 30$

Solving for $x$:

1. Subtract $10x$ from both sides: $5 = 5x - 30$

2. Add $30$ to both sides: $35 = 5x$

3. Divide both sides by 5: $x = 7$

Final Answer:

$x = 7$

Would you like further details on how the equation was set up, or do you have any specific questions?

Here are 5 related questions for further practice:

1. What are alternate interior angles, and how are they identified in a diagram with parallel lines?
2. If one angle is $7x + 15^\circ$ and the other is $9x - 10^\circ$, how would you solve for $x$?
3. What are corresponding angles, and how are they different from alternate interior angles?
4. Can alternate exterior angles also be equal in parallel line situations?
5. How can we verify if two lines are truly parallel in geometric proofs?

Tip: When working with parallel lines and angles, always look for patterns like alternate interior, alternate exterior, and corresponding angles to help set up your equations.