Question

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: The given information in the problem states \(l\parallel m\) and \(\angle1\cong\angle3\), so the reason for \(l\parallel m\) is "it is given".
2. The second and third blanks: By the alternate - interior angles theorem (which states that if two parallel lines are cut by a transversal, then alternate - interior angles are congruent), when \(l\parallel m\) (the parallel lines) and considering the transversal that forms \(\angle2\) and \(\angle3\), \(\angle2\) is congruent to \(\angle3\).
3. The fourth blank: Since \(\angle1\cong\angle2\) (from the transitive property \(\angle1\cong\angle3\) and \(\angle2\cong\angle3\)), and \(\angle1\) and \(\angle2\) are corresponding angles (if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel), the theorem used to conclude \(p\parallel q\) is the "corresponding angles converse" (which states that if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel).

Answer:

1. "it is given"
2. \(\angle2\)
3. \(\angle3\)
4. "corresponding angles converse"