Question

given: ( l parallel m ); ( angle 1 cong angle 3 ) prove: ( p parallel q ) complete the missing parts of the paragraph proof. we know that angle 1 is congruent to angle 3 and that line ( l ) is parallel to line ( m ) because it is given. we see that angle 2 is congruent to angle 3 by the alternate interior angles theorem. therefore, angle 1 is congruent to angle 2 by the transitive property. so, we can conclude that lines ( p ) and ( q ) are parallel by the dropdown with options: corresponding angles theorem, converse alternate exterior angles theorem, converse same side interior angles theorem.

Explanation:

Step 1: Recall the theorem for parallel lines

We have established that \( \angle 1 \cong \angle 2 \). These are corresponding angles (since they are in the same position relative to the transversal and the two lines \( p \) and \( q \)).

Step 2: Identify the correct theorem

The corresponding angles theorem (or its converse) states that if corresponding angles are congruent, then the lines are parallel. Since we have \( \angle 1 \cong \angle 2 \) (corresponding angles) and we want to prove \( p \parallel q \), we use the converse of the corresponding angles theorem, which is also called the corresponding angles theorem for parallel lines (when using the converse, it's about proving parallelism from congruent corresponding angles). So the correct option is "corresponding angles theorem".

Answer:

corresponding angles theorem