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Parallel Lines Theorem

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In Geometry, we have learned about the concept of lines. This fundamental concept is also seen everywhere in day-to-day life, like on the sides of doors and windows or any flat surface with a straight edge. Often, we see two similar parallel lines like a zebra crossing, rail tracks, or ladders.

Parallel lines are an important part of the study of quadrilaterals, such as in parallelograms, for example, which are four-sided figures that have parallel opposite sides. In this article, we will study theorems and postulates on parallel lines. But first, let us define parallel lines.

Coplanar straight lines which are equidistant from each other and never intersect at any points are known as **parallel lines.**

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** are two or more lines that are not at equal distance and which are intersecting at some point or which **will** intersect at some point.

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.

A line passing through two lines at different points in the same plane is called a **transversal line**.

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.

Angles formed on the opposite side of the transversal and are in the inner side of parallel lines are known as alternate angles.

__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.

__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.

Two angles are supplementary if the sum of the measure of both angles is${180}{\xb0}$.

**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.

__Theorem 5__: Corresponding angles

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

Angles formed on the matching corners of parallel lines formed by the transversal are known as corresponding angles.

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

__Theorem 6__: Alternate interior angles converse

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

__Theorem 7__: Alternate exterior angles converse

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

__Theorem 8__: Consecutive interior angles converse

If two lines in a plane are cut by a transversal such that consecutive interior angles formed have a sum of$\mathit{180}\mathit{\xb0}$, then the two lines are parallel.

__Theorem 9__: Consecutive exterior angles converse

If two lines in a plane are cut by a transversal such that consecutive exterior angles formed have a sum of $\mathit{180}\mathit{\xb0}$, then the two lines are parallel.

__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.

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

In the given figure $p\parallel q$ and $m\parallel n$. And $m\angle 3=102\xb0$.

Find (a) $m\angle 5$ (b) $m\angle 6$ (c) $m\angle 14$

__Solution__: (a) Here $m\parallel n$ and line p works as a transversal for lines m and n. Now applying the alternate interior angles theorem, we get $\angle 3\cong \angle 5.$

$\Rightarrow \angle 3=\angle 5\phantom{\rule{0ex}{0ex}}\Rightarrow \angle 5=102\xb0$

(b) Similarly to (a), p is a transversal to parallel lines m and n. We use the consecutive interior angle theorem. So $\angle 3$and $\angle 6$ are supplementary.

$\Rightarrow \angle 3+\angle 6=180\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow \angle 6=180\xb0-\angle 3\phantom{\rule{0ex}{0ex}}\Rightarrow \angle 6=180\xb0-102\xb0\phantom{\rule{0ex}{0ex}}\therefore \angle 6=78\xb0$

(c) We will use the $\angle 6$ to calculate $\angle 14$ as both lie on the same line n . Here $\angle 6$ and $\angle 14$ are on the parallel lines, p and q, respectively, and line n works as a transversal for both lines. So $\angle 6$ and $\angle 14$ form corresponding angles.

So from the corresponding angle theorem, both angles are congruent.

$\Rightarrow \angle 6\cong \angle 14\phantom{\rule{0ex}{0ex}}\therefore \angle 14=78\xb0$

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

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.

** 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 $m\angle 1=90\xb0.$ Similarly, as transversal t is perpendicular to q, we get $m\angle 2=90\xb0.$

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

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.

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.** **

__ 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.

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

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

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.

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

Each line is parallel to the next immediate line in the below figure. Then show that ${K}_{1}\parallel {K}_{4}.$

__Solution__: It is given that ${K}_{1}\parallel {K}_{2}$ and ${K}_{2}\parallel {K}_{3}.$ Then by applying the transitive property of parallel lines theorem, we get that ${K}_{1}\parallel {K}_{3.}$Now it is also given that ${K}_{3}\parallel {K}_{4}$ and we already found ${K}_{1}\parallel {K}_{3.}$So again applying the transitive property of parallel lines theorem, we know that ${K}_{1}\parallel {K}_{4.}$

In the following figure two lines, a and c are both perpendicular to line s. Also, it is given that $a\parallel b.$Then prove that $b\parallel c.$

__Solution__: Here, it is given that line s cuts line a and line c perpendicularly. So applying the perpendicular transversal theorem we get $a\parallel c.$We are given $a\parallel b$ and we already found that $a\parallel c.$ Then, from the transitivity property of parallel lines theorem, it immediately proves that $b\parallel c.$

- If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. And conversely, if two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel.
- If two parallel lines are cut by a transversal, then the exterior angles are congruent. Conversely, if two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.
- If two parallel lines are cut by a transversal, then the consecutive interior angles and consecutive exterior angles on the same side are supplementary. The converse of this also exists.
- Two lines in a plane cut by transversals are parallel if and only if the corresponding angles are congruent.
- If two lines in a plane are cut by a perpendicular transversal, then both lines are parallel.
- If two lines in a plane are parallel to the same line, then all the lines are parallel to each other.
- If three parallel lines are cut by two transversals, then the segments formed on the transversals have equal proportion.

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