Testing for Parallel Lines
Postulate 11 and Theorems 13 through 18 tell you that if two lines are parallel, then certain other statements are also true. It is often useful to show that two lines are in fact parallel. For this purpose, you need theorems in the following form: If (certain statements are true) then (two lines are parallel). It is important to realize that the converse of a theorem (the statement obtained by switching the if and then parts) is not always true. In this case, however, the converse of postulate 11 turns out to be true. We state the converse of Postulate 11 as Postulate 12 and use it to prove that the converses of Theorems 13 through 18 are also theorems.
Postulate 12: If two lines and a transversal form equal corresponding angles, then the lines are parallel.
In Figure 1, if m ∠l = m ∠2, then l // m. (Any pair of equal corresponding angles would make l // m.)
Figure 1 A transversal cuts two lines to form equal corresponding angles.
This postulate allows you to prove that all the converses of the previous theorems are also true.
Theorem 19: If two lines and a transversal form equal alternate interior angles, then the lines are parallel.
Theorem 20: If two lines and a transversal form equal alternate exterior angles, then the lines are parallel.
Theorem 21: If two lines and a transversal form consecutive interior angles that are supplementary, then the lines are parallel.
Theorem 22: If two lines and a transversal form consecutive exterior angles that are supplementary, then the lines are parallel.
Theorem 23: In a plane, if two lines are parallel to a third line, the two lines are parallel to each other.
Theorem 24: In a plane, if two lines are perpendicular to the same line, then the two lines are parallel.
Based on Postulate 12 and the theorems that follow it, any of following conditions would allow you to prove that a // b. (Figure 2).
Figure 2 What conditions on these numbered angles would guarantee that linesa and b are parallel?
Postulate 12:
Use Theorem 19:
Use Theorem 20:
Use Theorem 21:
- ∠4 and ∠5 are supplementary
- ∠3 and ∠6 are supplementary
Use Theorem 22:
- ∠1 and ∠8 are supplementary
- ∠2 and ∠7 are supplementary
Use Theorem 23:
Use Theorem 24:
Example 1: Using Figure 3, identify the given angle pairs as alternate interior, alternate exterior, consecutive interior, consecutive exterior, corresponding, or none of these: ∠1 and ∠7, ∠2 and ∠8, ∠3 and ∠4, ∠4 and ∠8, ∠3 and ∠8, ∠3, and ∠2, ∠5 and ∠7.
Figure 3 Find the angle pairs that are alternate interior, alternate exterior,
consecutive interior, consecutive exterior, and corresponding.
∠1 and ∠7 are alternate exterior angles.
∠2 and ∠8 are corresponding angles.
∠3 and ∠4 are consecutive interior angles.
∠4 and ∠8 are alternate interior angles.
∠3 and ∠2 are none of these.
∠5 and ∠7 are consecutive exterior angles.
Example 2: For each of the figures in Figure 4, determine which postulate or theorem you would use to prove l // m.
Figure 4 Conditions guaranteeing that lines l and m are parallel.
Figure 4 (a): If two lines and a transversal form equal corresponding angles, then the lines are parallel (Postulate 12).
Figure 4 (b): If two lines and a transversal form consecutive exterior angles that are supplementary, then the lines are parallel (Theorem 22).
Figure 4 (c): In a plane, if two lines are perpendicular to the same line, the two lines are parallel (Theorem 24).
Figure 4 (d): If two lines and a transversal form equal alternate interior angles, then the lines are parallel (Theorem 19).
Example 3: In Figure 5, a // b and m ∠1 = 117°. Find the measure of each of the numbered angles.
Figure 5 When lines a and b are parallel, knowing one angle makes it possible to determine
all the others pictured here.
m ∠2 = 63°
m ∠3 = 63°
m ∠4 = 117°
m ∠5 = 63°
m ∠6 = 117°
m ∠7 = 117°
m ∠8 = 63°