Proving Lines Are Parallel
There are three ways to prove that a pair of lines are parallel by looking at the angles made by a transversal.
||If two lines are crossed by a transversal and a pair of corresponding angles are congruent then the lines are parallel.
||If two lines are crossed by a transversal and a pair of alternate interior angles are congruent then the lines are parallel.
||If two lines are crossed by a transversal and a pair of interior angles on the same side of the transversal are supplementary then the lines are parallel.
Notice that these theorems are exactly the reverse (technically the converse) of the postulate 1 and theorems 1 and 2 in the Theorem's section. This helps cut down on the amount of material you really have to remember. If you can remember the relationship a type of angles has (e.g. corresponding angles ⇔ congruent) then you have both of the theorems: if the lines are parallel then the angles have the relationship and if the angles have the relationship then the lines are parallel.
A good way to remember the relationship is to draw a quick picture: a pair of parallel lines and a transversal. It's visually obvious that the corresponding angles are congruent, etc. This doesn't break the rule that you shouldn't go by what you think you see in the picture. If you were trying to draw a conclusion, for example "The angles look congruent, therefore they are.", we'd have a problem. In this case you're just useing the picture as a memory aid.
Example 1 - Proving Lines Are Parallel
Are and parallel.
The two angles that have measures in the diagram aren't any of our "special pairs" of angles. However, ∠GEF and ∠GBC and we can calculate m∠GBC because it's supplementary with ∠ABE.
m∠ABG + m∠GBC = 180°
128° + m∠GBC = 180°
m∠GBC = 52°
m∠GEF = 42°, not 52°, so the two lines aren't parallel because a pair of their corresponding angles aren't congruent.
Example 2 - Proving Lines Are Parallel
What value of x will make lines l and m parallel?
The two marked angles in the diagram are interior angles on the same side of the transversal. For the lines to be congruent, those angles have to be supplementary which means that the sum of their angles has to be 180°.
(3x + 12) + (4x - 42) = 180
7x - 30 = 180
7x = 210
x = 30