Question

Given: l || m; ∠1 ∠3

Prove: p || q

Horizontal and parallel lines l and m are intersected by parallel lines p and q. At the intersection of lines l and p, the uppercase left angle is angle 1. At the intersection of lines q and l, the bottom right angle is angle 2. At the intersection of lines q and m, the uppercase left angle is angle 3.

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
.

Answer 1

We know that angle 1 is congruent to angle 3 and that line l is parallel to line m because they are given. We see that angle 3 is congruent to angle 2 by the Alternate Interior Angles Theorem (q is a transversal of l and m). 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 converse of the Corresponding Angles Postulate.