Parallel Lines Theorem

Parallel lines, StudySmarter Originals

Angles and parallel lines theorems

We can make statements regarding parallel lines based on the angles they form. In other words, we can prove that lines are parallel based on angles, and conversely, we can also prove angle congruency based on the existence of parallel lines. Before proceeding further, let's review some basic definitions and concepts regarding parallel lines. First, how can we tell the difference between parallel lines and those that are non-parallel?

Non-parallel lines, StudySmarter Originals

You may be wondering, how do parallel lines relate to angles if they never intersect? The answer is transversals: Transversal lines play an important role in determining the angles associated with parallel lines.

Transversal, StudySmarter Originals

Angle congruency based on parallel lines

First, we will take a look at the important statements to show angle congruency based on parallel lines.

Theorem 1: Alternate interior angle

If two parallel lines in a plane are cut by a transversal, then the alternate interior angles formed are congruent (the same). Note that interior angles are those that are on in the inside of the parallel lines.

Alternate interior angles, StudySmarter Originals

Theorem 2: Alternate exterior angle

If two parallel lines in a plane are cut by a transversal, then the exterior angles formed are congruent. Note that exterior angles are those that are on the inside of the parallel lines.

Alternate exterior angles, StudySmarter Originals

Theorem 3: Consecutive interior angles

If two parallel lines in a plane are cut by a transversal, then the consecutive interior angles formed on the same side are supplementary.

Consecutive interior angles, StudySmarter Originals

Theorem 4: Consecutive exterior angles

If two parallel lines in a plane are cut by a transversal, then the consecutive exterior angles formed on the same side are supplementary.

Consecutive exterior angles, StudySmarter Originals

Theorem 5: Corresponding angles

If two parallel lines in a plane are cut by a transversal, then the corresponding angles formed are congruent.

Corresponding angles, StudySmarter Originals

Proving parallel lines based on angles

Now we will take a look at the converse part of the above-mentioned theorems.

Parallel lines with consecutive exterior angles, StudySmarter Originals

Theorem 10: Corresponding angles converse

If two lines in a plane are cut by a transversal such that corresponding angles formed are congruent, then the two lines are parallel.

Parallel lines with corresponding angles, StudySmarter Originals

Angles and parallel lines: Solved examples

Here we will take a look at some examples regarding the above-mentioned theorems.

Now let's understand some additional important theorems regarding parallel lines and take a look at their proofs.

Perpendicular transversal theorem for parallel lines

The following statement presents the relationship between perpendicular transversal and parallel lines.

Theorem 11: Perpendicular transversal theorem

If two lines in a plane are cut by a perpendicular transversal, then both lines are parallel.

Parallel lines with perpendicular transversal, StudySmarter Originals

Proof: Here, transversal t is perpendicular to both line p and line q,$i.e.t\perp p,t\perp q$

Now we have to prove that p and q are parallel. As transversal t is perpendicular to p, it implies $m\angle 1=90°.$ Similarly, as transversal t is perpendicular to q, we get $m\angle 2=90°.$

$\Rightarrow m\angle 1=\hspace{0.17em}m\angle 2$

Now, using the definition of congruence, which states that if the measure of two angles are equal then both the angles are congruent to each other, we get $\angle 1\cong \angle 2.$

From the figure, we can clearly see that both the angles are corresponding angles. So by using theorem 10, the corresponding angles converse theorem, we can directly say that $p\parallel q.$ That is, both line p and line q are parallel to each other. Hence, the theorem is proved.

Transitivity of parallel lines theorem

One of the other important statements of parallel lines uses the transitivity relation.

Theorem 11: Transitivity of parallel lines

If two lines in a plane are parallel to the same line, then all the lines are parallel to each other.

Transitive property of parallel lines, StudySmarter Originals

Proof: Now let's prove that the line common to other parallel lines is parallel. That is, $p\parallel q,q\parallel r.$

Then, without any loss of generality, we can say that line q lies in between line p and line r.

Now we have to prove that line p and line r are parallel. $i.e.p\parallel r$

Here, we will use the method of contradiction to prove this result, which shows that a statement is true simply by proving that it isn't possible for it to be false. Therefore, to prove that line p and line r are parallel, we first assume that line p and line r are not parallel lines (a contradiction). That means line p and line r must intersect each other, based on the definition of non-parallel lines. Now, as line q lies between lines p and r, when these lines intersect, line p or line r would have to intersect with line q as well. However, as line q is parallel to both the line p and line r, this cannot be possible. Hence, our assumption that line p and line r are not parallel is false. By the method of contradiction, line p and line r are parallel to each other. So, we have proved that if $p\parallel q$ and $q\parallel r$, then $p\parallel r.$

We will take a look at the theorem which shows proportionality between three parallel lines.

Theorem 12: Three parallel lines theorem

If three parallel lines are cut by two transversals, then the segments formed on the transversal have equal proportion.

Three parallel lines, StudySmarter Originals

Proof: Here lines p, q, and r are parallel to each other. And these lines are cut by two transversal t and s at points A, B, C, and D, E, and F respectively.

Now we have to show that $\frac{AB}{BC}=\frac{DE}{EF}.$

To prove this, we will make use of the intercept theorem. We are given that lines p, q, and r are parallel. Then we construct a line AH from point A, which is parallel to DF.

Three parallel lines with third transversal, StudySmarter Originals

We can notice that the left part of the figure is accurately what the intercept theorem states. So from the intercept theorem, we get:

As we constructed the parallel line AH, we know that $AH\parallel DF.$

We are also given that lines p, q, and r are parallel lines. So, by the definition of a parallelogram, ADEG and EFHG are both parallelograms. Now, from the properties of a parallelogram, we know that opposite sides are equal.

$\Rightarrow AG=DE,GH=EF$

Using the transitive property, we can directly substitute and get the following result.

$\frac{AB}{BC}=\frac{AG}{GH}=\frac{DE}{EF}$

$\Rightarrow \frac{AB}{BC}=\frac{DE}{EF}$

Hence, we can say that the segments formed on both transversals are in equal proportions.

Solved examples involving parallel lines theorems

Let's apply the above theorems regarding parallel lines to some of the examples.