Math 1312 Sections 2.3 Proving Lines Parallel.
Theorem 2.3.1: If two lines are cut by a transversal so that the corresponding angles are congruent, then these lines are parallel.
Example 1: If you are given a figure (see below) with congruent corresponding angles then the two lines cut by the transversal are parallel. Because each angle is 35 °, then we can state that a ll b.
35 ° a ° b 35
Theorem 2.3.2: If two lines are cut by a transversal so that the alternate interior angles are congruent, then these lines are parallel.
Example 2: If you are given a pair of alternate interior angles that are congruent, then the two lines cut by the transversal are parallel. Below the two angles shown are congruent and they are alternate interior angles; therefore, we can say that a ll b.
a 75 ° ° b 75
Theorem 2.3.3: If two lines are cut by a transversal so that the alternate exterior angles are congruent, then these lines are parallel.
Example 3: If you are given a pair of alternate exterior angles that are congruent, then the two lines cut by the transversal are parallel. For example, the alternate exterior angles below are each 105 °, so we can say that a ll b.
105 ° a
b 105 °
Theorem 2.3.4: If two lines are cut by a transversal so that the interior angles on one side of the transversal are supplementary, then these lines are parallel.
Example 3: If you are given a pair of consecutive interior angles that add up to 180 °(i.e. supplementary), then the two lines cut by the transversal are parallel. Below, the two angles shown add up to 180 ° AND they are consecutive interior angles; therefore, we can say that a ll b.
a 75 °
° b 105
Theorem 2.3.6: If two lines are each parallel to a third line, then these lines are parallel to each other.
Theorem 2.3.7: If two coplanar lines are each perpendicular to a third line, then there lines are parallel to each other.
Example 4: If you have two lines and a third line that is perpendicular to the first two lines, then the lines are parallel. Below you are given lines a and b, line c is perpendicular to both. According to this rule we can then say that a ll b.
c
a
b
Theorem 2.3.5: If two lines are cut by a transversal so that the exterior angles on the same side of the transversal are supplementary, then these lines are parallel.
Example 5: Find the value of x and the measure of each angle that will make p ll q
p
q 5x + 90
14x + 9
Example 6: If ∠1 ≅ ∠2 and ∠RAB ≅ ∠CBM, which lines must be parallel?
R •••
••• D
2 • ••• 1 • •• A C E
B •••
••• M
R •••
••• D
2 ••• • ••• A 1 C E
••• B ••• M
Example 7: