The image contains a geometry problem where you are asked to complete a proof using the correct reasoning.

Given:

• Line $l$ and line $m$ intersect.
• The angles $\angle 1$ and $\angle 2$ are vertical angles.

Proof:

You need to complete the proof by filling in the missing statement and reason.

The proof table shows:

Statements	Reasons
1. Lines $l$ and $m$ intersect	1. Given
2. $\angle 1$ is supplementary to $\angle 3$	2. Linear Pair Theorem
3. $\angle 2$ is supplementary to $\angle 3$	3. Linear Pair Theorem
4. $\angle 1 \cong \angle 2$	4. Congruent Supplements Theorem

The question asks:

"Which statement and reason best completes the proof?"

Answer Choices:

• A. $\angle 2$ and $\angle 4$ = 180°

  • Reason: Definition of supplementary angles

• B. $\angle 1$ is supplementary to $\angle 4$

  • Reason: Linear Pair Theorem

• C. $\angle 3$ is supplementary to $\angle 4$

  • Reason: Definition of supplementary angles

• D. $\angle 2$ and $\angle 3$ = 180°

  • Reason: Definition of supplementary angles

Solution:

To complete the proof logically, you should focus on the concept of supplementary angles and the Linear Pair Theorem.

• Correct Answer:
  D. $\angle 2$ and $\angle 3$ = 180° (Reason: Definition of supplementary angles)

This statement and reason correctly explains why $\angle 2$ and $\angle 3$ are supplementary, which is crucial to applying the Linear Pair Theorem in the proof.

Final Proof:

1. Given: Lines $l$ and $m$ intersect.
2. Statement: $\angle 1$ is supplementary to $\angle 3$.
   Reason: Linear Pair Theorem.
3. Statement: $\angle 2$ is supplementary to $\angle 3$.
   Reason: Linear Pair Theorem.
4. Statement: $\angle 1 \cong \angle 2$.
   Reason: Congruent Supplements Theorem.

This step-by-step completion follows the logic needed to prove the angles are congruent.

If you have any more questions or need further clarification, feel free to ask!

---

Related Questions:

1. What is the difference between supplementary and complementary angles?
2. How is the Linear Pair Theorem applied in geometric proofs?
3. What does it mean when angles are congruent?
4. Can you provide another example of the Congruent Supplements Theorem in action?
5. What are vertical angles, and why are they always congruent?
6. How do you know if two angles are supplementary?
7. What role does the intersection of lines play in angle relationships?
8. Can the Congruent Supplements Theorem be used with other types of angle pairs?

Tip:

Remember that the definition of supplementary angles is essential when working with linear pairs and intersecting lines. Understanding these foundational concepts will help you solve more complex geometric proofs.