3.2 Proving that Lines are Parallel
For any two lines in space, one of the following conditions must exist: 1) the lines must intersect in a single point and therefore must be coplanar; 2) the lines do not intersect and are not coplanar, in which case they are called skew lines; 3) the lines do not intersect but are coplanar, in which case they are parallel lines.
- Definition:
Two lines are parallel if and
only if they are coplanar and do not intersect.
- Definition: A line that intersects two different lines in two distinct points is called a transversal.
In the diagram below, $\angle 1$ and $\angle 2$ are a pair of alternate interior angles; $\angle 1$ and $\angle 3$ are a pair of corresponding angles; and $\angle 1$ and $\angle 4$ are a pair of same-side interior angles. Are there other pairs of these special angles?
We are now faced with the challenge of proving that lines are parallel. As with all that we have done, we must start somewhere, with an assumption, and we will do this with the Parallel Postulate.
- Postulate: The
Parallel Postulate
Given a point not on a line, there is exactly one line through the given point and parallel to the given line.
- Theorem 3.2: If two distinct coplanar lines $m$ and $n$ are parallel to a third line $\ell$, then lines $m$ and $n$ are parallel to each other.
Given: $m \parallel
\ell$ and $n \parallel \ell$ Prove: $m \parallel n$ |
Proof: (Indirect): Assume that lines $m$ and $n$ are not parallel to each other. Since they are coplanar, they must
intersect in some point $P$. If they intersect at $P$, then there are two lines through the point $P$ that are parallel to the line $\ell$. This contradicts the Parallel Postulate (there can only be one line through point $P$ parallel to the given line $\ell$). Therefore our assumption that $m$ and $n$ are not parallel to each other is false; hence lines $m$ and $n$ must be parallel to each other.
This theorem suggests another way to prove that two coplanar lines are parallel — prove that the two lines are both perpendicular to the same line.
- Theorem 3.3:
In a plane, if two lines $m$ and $n$ are perpendicular to a third line
$\ell$, then the lines $m$ and $n$ are parallel to each other.
We will now look at using angles formed by lines and a transversal (alternate interior angles, corresponding angles, and same-side interior angles) to prove that lines are parallel. In the figure below, move the different points to investigate the relationship between angles formed by lines and a transversal.
- Theorem 3.4:
AI $\rightarrow$ P
Given two coplanar lines and a transversal, if a pair of alternate interior angles are congruent, then the lines are parallel.
Given: $\angle 1$ and
$\angle 2$ are congruent. Prove: $m \parallel n$ |
Proof (Indirect): Assume that lines $m$ and $n$ are not parallel. Then $m$ and $n$ must intersect at some point $P$. In that case, $\angle 1$ is an exterior angle of the triangle $\triangle PQR$ that is formed, and therefore must be greater than any remote interior angle. That makes $m\angle 1> m\angle 2$. This contradicts the given that $\angle 1$ and $\angle 2$ (a pair of alternate interior angles) are congruent. Therefore our assumption must be false and lines m and n must be parallel.
- Theorem 3.5:
CA $\rightarrow$ P
Given two coplanar lines and a transversal, if a pair of corresponding angles are congruent, then the lines are parallel.
Proof: This proof can be done directly using the Vertical Angle Theorem and Theorem 3.4 and will be left as an exercise.
- Theorem 3.6:
Given two coplanar lines and a transversal, if a pair of same-side
interior angles are supplementary, then the lines are parallel.
Given: $\angle 2$ and
$\angle 3$ are supplementary. Prove: $m \parallel n$ |
Proof: This theorem can be proved either directly (by using Theorem 3.4) or indirectly.
We will use an indirect proof. Assume that lines $m$ and $n$ are not parallel. Then $m$ and $n$ must intersect at some point $P$. Since $\angle 1$ and $\angle 2$ form a straight angle, they are supplementary. $\angle 2$ and $\angle 3$ are given to be supplementary; since supplements of the same angle are congruent, then $\angle 1\cong \angle 3$. However, $\angle 3$ is an exterior angle of $\triangle PQR$ and so its measure must be greater than the measure of each of its remote interior angles. Hence the measure of $\angle 3$ must be greater than the measure of $\angle 1$. We now have a contradiction in that $\angle 3$ cannot be both equal to and greater than $\angle 1$ at the same time; therefore our assumption must be false, so lines $m$ and $n$ must be parallel.