Let's analyze the diagram and the corresponding statements to determine which one is true.

The diagram shows two parallel lines being intersected by a transversal. In this type of geometric setup, several angle relationships arise:

• Corresponding Angles: Angles that are in the same position relative to the parallel lines and the transversal. These angles are congruent. For example, $\angle 4 \cong \angle 8$ and $\angle 3 \cong \angle 7$.
• Alternate Interior Angles: Angles on opposite sides of the transversal but inside the parallel lines. These are also congruent. For example, $\angle 4 \cong \angle 5$ and $\angle 3 \cong \angle 6$.
• Same-Side Interior Angles (Consecutive Interior Angles): Angles on the same side of the transversal and inside the parallel lines. These angles are supplementary, meaning they sum to 180°. For example, $\angle 4 + \angle 6 = 180^\circ$.

Now, let's examine each statement in the problem:

• A. $\angle 4 \cong \angle 8$ and $\angle 5 \cong \angle 2$, so $\angle 4 + \angle 5 = 180^\circ$.
  This is incorrect. While $\angle 4 \cong \angle 8$, $\angle 5 \cong \angle 3$ (not $\angle 2$). Also, $\angle 4$ and $\angle 5$ are not supplementary.

• B. $\angle 4 \cong \angle 8$ and $\angle 3 \cong \angle 7$, so $\angle 1 \cong \angle 5$.
  This is incorrect. $\angle 4 \cong \angle 8$ and $\angle 3 \cong \angle 7$ are true, but $\angle 1$ is not congruent to $\angle 5$. They are corresponding angles to different positions on the transversal.

• C. $\angle 4 \cong \angle 8$ and $\angle 6 \cong \angle 8$, so $\angle 4 \cong \angle 6 = 180^\circ$.
  This is correct. $\angle 4$ and $\angle 6$ are same-side interior angles and are supplementary, meaning their sum is $180^\circ$.

• D. $\angle 4$ and $\angle 3$ are complementary; $\angle 6 \cong \angle 3$, so $\angle 4 + \angle 6 = 90^\circ$.
  This is incorrect. $\angle 4$ and $\angle 3$ are not complementary (they don't sum to 90°), and $\angle 4 + \angle 6 = 180^\circ$, not 90°.

The correct answer is C.

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

1. What are the properties of corresponding angles in parallel lines cut by a transversal?
2. How do alternate interior angles behave in parallel lines intersected by a transversal?
3. What is the relationship between same-side interior angles?
4. Why are supplementary angles significant in geometry problems involving parallel lines?
5. How can you identify when two angles are congruent using a transversal?

Tip: When working with parallel lines and transversals, always remember that corresponding, alternate interior, and alternate exterior angles are congruent, while same-side interior angles are supplementary.