Question

Use the image to answer the question.Two lines are intersected by two transversals. Three angles are labeled. 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≅∠3p∥q

Answer 1

To prove that lines \( p \parallel q \) given that \( r \parallel s \) and \( \angle 1 \cong \angle 3 \), you can follow these steps with appropriate reasons:

1. Statement: \( r \parallel s \), \( \angle 1 \cong \angle 3 \)
   Reason: Given

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

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

4. Statement: \( p \parallel q \)
   Reason: Alternate Interior Angles Theorem (since \( \angle 2 \) and \( \angle 3 \) are alternate interior angles, and they are congruent)

So the complete proof would look like this:

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

You would fill in the matches as follows:

1. Given
2. Corresponding Angles Postulate
3. Transitive Property of Congruence
4. Alternate Interior Angles Theorem