Question

given: m || n, ∠1 ≅ ∠3
prove: p || q
complete the two - column proof.
statements\treasons

1. m || n\t1. given
2. \t2. alternate interior ∠s theorem
3. ∠1 ≅ ∠3\t3. given
4. ∠2 ≅ ∠3\t4.
5. p || q\t5.

options for filling: angle 1 congruent to angle 2; angle 1 congruent to angle 3; line p parallel to line q

Explanation:

Step1: Find Statement 2

Since \( m \parallel n \), by the alternate interior angles theorem, \( \angle 1 \cong \angle 2 \). So Statement 2 is \( \angle 1 \cong \angle 2 \).

Step2: Find Reason 4

We know \( \angle 1 \cong \angle 2 \) (from Step 1) and \( \angle 1 \cong \angle 3 \) (given). By the transitive property of congruence (or substitution), \( \angle 2 \cong \angle 3 \). So Reason 4 is "transitive property (or substitution)".

Step3: Find Reason 5

Since \( \angle 2 \cong \angle 3 \), by the converse of the alternate interior angles theorem, \( p \parallel q \). So Reason 5 is "converse of alternate interior angles theorem".

Answer:

1. \( \angle 1 \cong \angle 2 \)
2. transitive property (or substitution)
3. converse of alternate interior angles theorem