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.

Explanation:

Step 1: Find Statement 2

Since \( m \parallel n \) and using the alternate interior angles theorem, the alternate interior angles should be \( \angle 1 \) and \( \angle 2 \). So statement 2 is \( \angle 1 \cong \angle 2 \).

Step 2: Find Reason 4

We know \( \angle 1 \cong \angle 2 \) (from step 2) and \( \angle 1 \cong \angle 3 \) (given). By the transitive property of congruence (if \( \angle A \cong \angle B \) and \( \angle B \cong \angle C \), then \( \angle A \cong \angle C \)), we can say \( \angle 2 \cong \angle 3 \). So reason 4 is "transitive property of congruence".

Step 3: Find Reason 5

Since \( \angle 2 \cong \angle 3 \), and these are corresponding angles (or alternate interior angles depending on the lines), by the converse of alternate interior angles theorem (if alternate interior angles are congruent, then the lines are parallel), we can conclude \( p \parallel q \). So reason 5 is "converse of alternate interior angles theorem".

Answer:

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