Question

supply the missing reasons in the two-column proof.
given: a || b, c || d
prove: ∠3 ≅ ∠1

statements reasons

1. a || b 1) given
2. ∠3 and ∠2 are congruent. 2)
3. c || d 3) given
4. ∠2 and ∠1 are congruent. 4)
5. ∠3 ≅ ∠1 5)

Explanation:

Answer:

1. Alternate Interior Angles Theorem (or Corresponding Angles Theorem, depending on the position; since \(a \parallel b\) and \(c\) is a transversal, \(\angle 3\) and \(\angle 2\) are alternate interior angles, so they are congruent by the Alternate Interior Angles Theorem)
2. Corresponding Angles Theorem (since \(c \parallel d\) and the transversal creates \(\angle 2\) and \(\angle 1\) as corresponding angles, so they are congruent by the Corresponding Angles Theorem)
3. Transitive Property of Congruence (if \(\angle 3 \cong \angle 2\) and \(\angle 2 \cong \angle 1\), then \(\angle 3 \cong \angle 1\) by the Transitive Property of Congruence)

(Note: For step 2, it could also be Corresponding Angles if the angles are corresponding, but Alternate Interior is more likely here. The key is that for parallel lines cut by a transversal, alternate interior or corresponding angles are congruent, and then transitive property for the last step.)