The image shows a math problem related to angles formed by intersecting lines, with a table that lists angle measurements and reasons.

Problem Breakdown:

1. Diagram: The image depicts two intersecting lines forming four angles around the intersection point, labeled as $\angle 1$, $\angle 2$, $\angle 3$, and $\angle 4$. The angles $\angle 1$ and $\angle 3$ are on opposite sides of the intersection, and the same goes for $\angle 2$ and $\angle 4$.

2. Statements and Reasons:

   • $\angle 1 \cong \angle 3$
   • $\angle 2 \cong \angle 4$
   • $\angle 1 + \angle 2 = 180^\circ$

   Reasons:

   • Vertical angles are congruent.
   • Adjacent angles are supplementary (i.e., add up to $180^\circ$).

These statements and reasons correspond to properties of angles formed by two intersecting lines. Specifically:

• Vertical Angles: Angles opposite each other when two lines cross are congruent (equal in measure).
• Supplementary Angles: Adjacent angles that form a straight line (linear pair) sum to $180^\circ$.

Conclusion:

The problem involves identifying the relationships between angles formed by intersecting lines and justifying those relationships using geometric principles.

Do you need further explanation on this concept? Here are some related questions:

1. How do you calculate the measure of vertical angles?
2. What are linear pairs of angles, and how do they relate to supplementary angles?
3. If $\angle 1 = 120^\circ$, what is the measure of $\angle 2$?
4. How can you prove that vertical angles are always congruent?
5. Can you apply these angle relationships in real-world problems, such as in architecture or engineering?

Tip:

When working with angle relationships, drawing diagrams and labeling all known values can help visualize and solve the problem more effectively.