Question

proving lines parallel
assignment active
identifying missing components of a paragraph proof
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 $▼$. we see that $▼$ is congruent to $▼$ 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 $▼$.

Explanation:

1. The first blank refers to the given information provided in the problem statement.
2. For the alternate interior angles theorem, since \(l \parallel m\), the alternate interior angles formed by transversal \(q\) are \(\angle 2\) and \(\angle 3\).
3. Since we establish \(\angle 1 \cong \angle 2\), these are corresponding angles for lines \(p\) and \(q\) cut by transversal \(l\); the corresponding angles converse theorem proves lines are parallel.

Answer:

1. First blank: they are given
2. Second blank: \(\angle 2\); Third blank: \(\angle 3\)
3. Fourth blank: corresponding angles converse theorem