Question

Given: r∥s, ∠1≅∠3Prove: p∥qMatch the reason for each statement in order to complete the proof. The statements are listed in order. A reason may be used once, more than once, or not at all.(2 points)Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.r∥s, ∠1≅∠3 ∠1≅∠2 ∠2≅∠3 p∥q Transitive Property of CongruenceAlternate Interior Angles Converse TheoremGivenCorresponding Angles PostulateTransitive Property of EqualityCorresponding Angles Converse PostulateAlternate Interior Angles Theorem

Answer 1

To complete the proof based on the given information that \( r \parallel s \) and \( \angle 1 \cong \angle 3 \), let's step through the statements and match them with the appropriate reasons.

Statements and Reasons:

1. Statement: \( r \parallel s \)
   Reason: Given

2. Statement: \( \angle 1 \cong \angle 3 \)
   Reason: Given

3. Statement: \( \angle 1 \cong \angle 2 \)
   Reason: Corresponding Angles Postulate (since \( r \parallel s \) and \( \angle 1 \) and \( \angle 2 \) are corresponding angles)

4. Statement: \( \angle 2 \cong \angle 3 \)
   Reason: Transitive Property of Congruence (because \( \angle 1 \cong \angle 2 \) and \( \angle 1 \cong \angle 3 \))

5. Statement: \( p \parallel q \)
   Reason: Alternate Interior Angles Converse Theorem (since \( \angle 2 \) and \( \angle 3 \) are alternate interior angles)

Complete Proof Format:

• \( r \parallel s \) — Given
• \( \angle 1 \cong \angle 3 \) — Given
• \( \angle 1 \cong \angle 2 \) — Corresponding Angles Postulate
• \( \angle 2 \cong \angle 3 \) — Transitive Property of Congruence
• \( p \parallel q \) — Alternate Interior Angles Converse Theorem

This sequence aligns with the properties related to parallel lines and congruent angles, completing the proof that \( p \parallel q \).