Lines m and n are cut by a transversal, as shown in the figure.

Question

Given line m is not parallel to line n, prove ∠3 is not congruent to ∠5 by contradiction.

Answer 1

Assume ∠3 and ∠5 are congruent.

line l is a transversal of line m and line n

∠3 and ∠5 are alternate interior angles.

line m and line n are parallel

Contradiction to given

Therefore the first assumption is false.

Answer 2

This is a discussion about the supplementary angles that form from a transversal. I want to describe the scenario this looks to mirror, and then the actual scenario presented:

Final Answer:

1. If line m and line n are not parallel
2. then alternate interior angles (angles across from each other) are not congruent
3. therefore angle 3, an alternate interior angle, is congruent to angle 5, an alternate interior angle created by 2 lines that are not parallel but cut by the same transversal line l

1. IF LINE M AND LINE N WERE perpendicular, Angles that share a side would be supplementary (sum to 180 degrees) that would beget the straight angles (or straight lines) m and n. This would include the pairs of angles below are supplementary...
2. 1 and 2,
3. 5 and 6
4. 4 and 3
5. 8 and 7
6. 1 and 4
7. 2 and 3
8. 8 and 5
9. 6 and 7
10. Angles that are across from each other, pairs of alternate interior angles, would be congruent because they would form the same supplemental relationship with the transversal ...IF lines m and n are parallel. In other words these pairs of alternate interior angles WOULD BE congruent:
11. 1, 3, 5 and 7
12. 2, 4, 6 and 8
13. BUT lines m and n are NOT PARALLEL, so we have a fundamentally different creation of supplemental angles for each line based on the transversal line l, therefore..

1. .1, 3, 5 and 7 = not congruent
2. 2, 4, 6 and 8 = not congruent
3. Therefore, the pairs of angles that would be supplementary are still supplementary but in different degree amounts. We can affirm the angles created hold different measure from each other because line m and line n are NOT parallel.
4. 1 and 2, 5 and 6, 4 and 3, 8 and 7 are all supplementary but based on different degree angle amounts than 1 and 4, 2 and 3, 8 and 5, 6 and 7.

Final Answer:

1. If line m and line n are not parallel
2. then alternate interior angles (angles across from each other) are not congruent
3. therefore angle 3, an alternate interior angle, is not parallel to a angle 5, an alternate interior angle.

Hope this helps guys!