In the diagram provided, we have two parallel lines $AB \parallel CD$ and two transversals $EF$ and $GH$. The angles formed are labeled as $3x + 15^\circ$ at two positions: one at $\angle G$ and the other at $\angle H$.

Step-by-Step Solution:

1. Identify the corresponding angles: Since $AB \parallel CD$, the angle $\angle G$ and $\angle H$ are corresponding angles, meaning they are equal due to the "Corresponding Angles Postulate" (when two parallel lines are crossed by a transversal, the corresponding angles are equal).

2. Set up the equation: Since $\angle G = \angle H$, we can set the expressions for both angles equal: $3x + 15^\circ = 3x + 15^\circ$ This is an identity, meaning the equation is true for all values of $x$. However, the diagram seems to have an implicit meaning for the sum of angles around point $G$ or $H$.

3. Interpretation via supplementary angles: Based on the visual arrangement, angles $\angle G$ and $\angle H$ might also represent a situation where these angles are supplementary. If they are on the same side of the transversal, their sum must be $180^\circ$.

4. Equation with supplementary angles: We can set up the equation assuming $\angle G$ and $\angle H$ are supplementary: $3x + 15^\circ + 3x + 15^\circ = 180^\circ$ Simplifying this: $6x + 30^\circ = 180^\circ$ Subtract $30^\circ$ from both sides: $6x = 150^\circ$ Divide by 6: $x = 25^\circ$

Thus, the value of $x$ is $\boxed{25^\circ}$.

Would you like a more detailed breakdown or explanation? Here are 5 related questions to practice:

1. What are corresponding angles, and why are they equal when lines are parallel?
2. If the angles were not corresponding, what other geometric relationships could be applied to solve for $x$?
3. How do you determine whether angles are supplementary when transversals intersect parallel lines?
4. Can you apply this reasoning to a similar diagram with different angle expressions?
5. How would the solution change if the angle expression had a different coefficient of $x$?

Tip:

Always identify whether angles are corresponding, alternate, or supplementary when working with parallel lines and transversals. This will guide you in setting up the correct equation.