This is a diagram involving parallel lines cut by a transversal, creating angles. Let's analyze it step-by-step to determine the value of $x$:

Step 1: Properties of Parallel Lines and Transversals

1. The two lines indicated by arrows are parallel.
2. The transversal intersects both parallel lines, forming two sets of angles.

Step 2: Relationships Between the Angles

• The given $115^\circ$ angle is an exterior angle on the top left.
• By the corresponding angles postulate, the angle corresponding to $115^\circ$ on the bottom intersection is also $115^\circ$.
• $x$ is supplementary to this corresponding angle, as they form a straight line.

Step 3: Supplementary Angles

Angles on a straight line sum to $180^\circ$. Therefore: $x + 115^\circ = 180^\circ$

Step 4: Solve for $x$

$x = 180^\circ - 115^\circ = 65^\circ$

Final Answer:

$x = 65^\circ$

Would you like further clarification or a deeper explanation?

Here are 5 related questions to deepen your understanding:

1. What are the different types of angle relationships formed by parallel lines and a transversal?
2. How does the corresponding angles theorem work in this case?
3. What is the definition of supplementary angles, and how can they be used in geometry?
4. Can we prove the parallel lines using angle relationships in this diagram?
5. What would $x$ be if the given angle was $120^\circ$ instead of $115^\circ$?

Tip: Always remember that corresponding angles are equal when a transversal cuts parallel lines!