The paragraph proof in the image appears to be for proving the transitivity of parallel lines based on the relationship between the angles formed when a transversal intersects parallel lines. Let me help you complete it step by step.

Given:

• $\ell \parallel k$
• $m \parallel k$

To Prove:

• $\ell \parallel m$

Proof Outline:

• Since $\ell \parallel k$, $\angle 2 \cong \angle 1$ by the Corresponding Angles Theorem (a).
• Since $m \parallel k$, $\angle 3 \cong \angle 1$ (b) by the same theorem.
• By the Transitive Property of Congruence, $\angle 2 \cong \angle 3$.
• By the Converse of the Corresponding Angles Postulate (d), $\ell \parallel m$.

Missing Details:

• a: Corresponding Angles Theorem
• b: $\angle 3$
• c: $\angle 1$
• d: Converse of the Corresponding Angles Postulate

The key steps are showing that if two lines are both parallel to the same third line, the corresponding angles formed with a transversal are congruent, which leads to the conclusion that the two lines are parallel by the Converse of the Corresponding Angles Theorem.

Do you want further details on any of the steps or explanations?

Here are five related questions you might find useful:

1. What is the Corresponding Angles Theorem and how does it apply to parallel lines?
2. Can the Transitive Property of Congruence be applied to all types of angles or only specific types?
3. What is the difference between the Corresponding Angles Postulate and its converse?
4. How can you prove that two lines are parallel using only angles formed by a transversal?
5. Are there any exceptions to the transitivity of parallel lines?

Tip: When using the Corresponding Angles Theorem, always ensure that the lines involved are parallel before concluding angle congruence.