Alternate Exterior Angles

What are Parallel Lines?

Before we understand what we mean by alternate interior angles, we must first recall the basic concepts and terms that are integral to the understanding of alternate interior angles. 

There can be many lines in a plane, some of which may intersect each other while some may not intersect when produced in either direction. Thus we can define parallel lines as – “Two lines l and m in the same plane are said to be parallel lines of they do not intersect when produced indefinitely in either direction.”

Symbolically, two parallel lines l and m are written as l || m.

It should be noted that if two lines are not parallel, they will intersect each other. For instance, below we have the lines l and m as intersecting lines as they are not parallel.

Now, what if we have three lines two of which may be parallel to each other while the third one intersects them? This is where we introduce the concept of a transversal.

What is a Transversal?

A transversal is defined as a line intersecting two or more given lines in a plane at different points.  For example, in the figure below, the lines l and m are parallel while the line p is intersecting both the line l and the line m. Hence, the line p is a transversal to the lines l and m.

Now, as we can see in the above figure, a transversal makes some angles with the lines it intersects. 

Angles Made by a Transversal with Two Lines

Now, we shall learn about the angles made by a transversal with two given lines. Some of these angles can be paired together by virtue of the positions they occupy.

Let l and m be two lines and let n be the transversal intersecting them at P and Q respectively as shown below – 

Exterior Angles of a Transversal

The angles whose arms do not include the line segment PQ are called exterior angles. Therefore, in the above figure, angles 1, 2, 7 and 8 are exterior angles.

Interior Angles of a Transversal

The angles whose arms include the line segment PQ are called interior angles. Therefore, in the above figure, angles 6, 4, 5 and 6 are interior angles.

Clearly, lines l and m make eight angles with the transversal n, four at P and four at Q. we have labeled them 1 to 8. There are different groups formed by these 8 angles. One of these groups is alternate exterior angles. Let us understand what we mean by alternate exterior angle. 

Definition

A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms are directed in opposite directions and do not include segment PQ is called alternate exterior angles in a transversal.

In the above figure, ∠ 2 and ∠ 8 form a pair of alternate exterior angles. Another pair of alternate exterior angles in this figure is ∠ 1 and ∠ 7.

Now, we shall learn about the alternate interior angle theorem that is also one of the basic properties of alternate interior angles.

Different pairs of angles made by a Transversal 

Other than alternate exterior angles, we have some other groups of angles formed under this arrangement. Some of these pairs are – 

Corresponding Angles in a Transversal

A pair of angles in which one arm of both the angles is on the same side of the transversal and their other arms are directed in the same sense is called a pair of corresponding angles. In the above figure, there are four pairs of corresponding angles, ∠1 and ∠ 5, ∠ 2 and ∠ 6, ∠ 3 and ∠ 7, ∠ 4 and ∠ 8.

We can also say that two angles on the same side of the transversal are known as corresponding angles if both lie either above the lines or below the two lines.

Alternate Interior Angles in a Transversal

A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms include segment PQ is called a pair of alternate interior angles. In other words, Alternate interior angles are angles formed when two parallel or non-parallel lines are intersected by a transversal. The angles are positioned at the inner corners of the intersections and lie on opposite sides of the transversal.

In the above figures, ∠ 3 and ∠ 5 form a pair of alternate interior angles. Another pair of alternate interior angles in this figure is ∠ 4 and ∠ 6.

Alternate Exterior Angles Theorem

The alternate exterior angle theorem states that if two lines are parallel and are intersected by a transversal, then the alternate exterior angles are considered as congruent angles or angles of equal measure. What does this mean? 

Let us consider the following figure – 

We can clearly see that the line l is parallel to line m. The two lines have been intersected by a transversal, p. The angles, ∠ 1 and ∠ 8, ∠ 2 and ∠ 7 are two pairs of alternate exterior angles. According to the alternate exterior angles theorem, these two pairs will be equal. This means that – 

∠ 1 = ∠ 8

∠ 2 = ∠ 7

Degrees

The angles formed by the alternate exterior angles lie on different sides of a transversal and are termed to be congruent if the lines are parallel. These angles can range from being more than 0^o to being less than 180^o.  

Properties

The following are the properties of alternate exterior angles – 

1. If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
2. Consecutive exterior angles are supplementary. Here, it is important to note that consecutive exterior angles are exterior angles that are on the same side of the transversal line.
3.  Alternate exterior angles don’t have any specific properties in the case of non – parallel lines.

Real Life Applications of Alternate Exterior Angles

Alternate exterior angles are not just used to geometrical calculations in mathematics. They are used extensively in real life situations as well. For instance, alternate exterior angles are used to design the fittings of the sofas and chair in your room. Another area where Alternate exterior angles are extensively used is for the designing of buildings, roads and bridges. Different polygon shape structures such as hexagons also make use of alternate exterior angles at their design stage. Even different angles made during different exercises and yoga form Alternate exterior angles at different times.

Let us, for example, consider the following intersections on a street – 

We can see above that the pair of alternate exterior angles formed above are equal. This implies that the street comprises of parallel lines. 

Using External Alternate Angle Theorem to find missing angles?

We have seen that eight angles are formed when two lines are intersected by a transversal. So, how can use alternate exterior angles to find the remaining angles? We shall understand this by an example.  

Suppose we have the figure below where ∠ 2 = 60^o.  We shall use this value to find the remaining seven angles formed by the intersection of the two lines by a transversal.

Let us start with each angle one by one.

First, let us find ∠ 3. We can clearly see that ∠ 2 and ∠ 3 form a supplementary pair of angles. This means that the sum of ∠ 2 and ∠ 3 should be equal to 180^o. Hence, we have,

 ∠ 2 + ∠ 3 = 180^o

⇒ 60^o + ∠ 3 = 180^o

⇒ ∠ 3 = 180^o – 60^o 

⇒ ∠ 3 = 120^o

Now, that we have found the value of ∠ 3, we will find the value of ∠ 1

We can clearly see that ∠ 2 and ∠ 1 form a supplementary pair of angles. This means that the sum of ∠ 2 and ∠ 1 should be equal to 180^o. Hence, we have,

 ∠ 2 + ∠ 1 = 180^o

⇒ 60^o + ∠ 1 = 180^o

⇒ ∠ 1 = 180^o – 60^o 

⇒ ∠ 1 = 120^o

Now, that we have found the value of ∠ 1, we will find the value of ∠ 4

We can clearly see that ∠ 3 and ∠ 4 form a supplementary pair of angles. This means that the sum of ∠ 3 and ∠ 4 should be equal to 180^o. Hence, we have,

 ∠ 3 + ∠ 4 = 180^o

⇒ 120^o + ∠ 4 = 180^o

⇒ ∠ 4 = 180^o – 120^o 

⇒ ∠ 4 = 60^o

Note that ∠ 4 and ∠ 6 form a pair of alternate interior angles. Since the lines that have been intersected by a transversal are parallel, therefore, the pair of alternate interior angles should be equal. Hence, we have,

∠ 4 = ∠ 6 =  60^o

Similarly, ∠ 3 = ∠ 5 =  120^o

Now, we can see that ∠ 6 and ∠ 7 form a supplementary pair of angles. This means that the sum of ∠ 6 and ∠ 7 should be equal to 180^o. Hence, we have,

∠ 6 + ∠ 7 = 180^o

⇒ 60^o + ∠ 7 = 180^o

⇒ ∠ 7 = 180^o – 60^o 

⇒ ∠ 7 = 120^o

Similarly, we can see that ∠ 5 and ∠ 8 form a supplementary pair of angles. This means that the sum of ∠ 5 and ∠ 8 should be equal to 180^o. Hence, we have,

∠ 5 + ∠ 8 = 180^o

⇒ 120^o + ∠ 8 = 180^o

⇒ ∠ 8 = 180^o – 120^o 

⇒ ∠ 8 = 60^o

Examples

Example 1 Find the value of x in the given figure, where the line L₁ and L₂ are parallel.

Solution We have been given that the lines L₁ and L₂ are parallel. Therefore, by the alternate external angle theorem, we can say that the angles ( 3x – 33 ) ^o and ( 2x + 26 ) ^o are equal. Hence, we have,

3x – 33 = 2x + 26

⇒ 3x – 2x = 26 + 33

⇒ x = 59

Hence, in the above figure, x = 59^o

Can we verify our answer? Let us substitute the value of x in the both the angles. We have,

( 3x – 33 ) ^o

= ( 3 x 59 – 33 ) ^o

= ( 177– 33 ) ^o

= 144 ^o    ……………………………… ( 1 )

Now,

( 2x + 26 ) ^o

= ( 2 x 59 + 26 ) ^o

= ( 118 + 26 ) ^o

= 144 ^o ……………………………… ( 2 )

From ( 1 ) and ( 2 ) we can see that both the angles are equal which satisfies the alternate external angle theorem. Hence, our answer x = 59 ^o is correct.

Example 2 Given is the following figure having a pair of parallel lines intersected by a transversal. Find the value of y and subsequently each angle formed by the transversal.

Solution We have been given a pair of parallel lines intersected by a transversal. This means that the pair of alternate exterior angles satisfy the alternate exterior angle theorem. Therefore, we have,

3y + 53 = 7y – 55

⇒ 3y – 7y = – 55 – 53

⇒ – 4y = – 108

⇒ 4y = 108

⇒  y = 1084  = 27^o

Now, we are required to find the value of all the angles formed by the transversal. 

Let us name each angle for the convenience of calculations. We now have,

So, we have angles marked from 1 to 8. Accoridng to the question, the angle that has been marked as 1 by us has been given as (3y + 53 ) ^o. We have already calculatedd the value of y. So, we have,

( 3 y + 53 ) ^o = ∠ 1

⇒ ( 3 x 27 + 53 ) ^o = ∠ 1

⇒ ( 81 + 53 ) ^o = ∠ 1

⇒ ∠ 1 = 134 ^o

Similarly, the angle that has been marked as 8 by us has been given as (7 y – 55) ^o. So we have,

( 7 y – 55 ) ^o = ∠ 8

⇒ ( 7 x 27 – 55 ) ^o = ∠ 8

⇒ ( 189 – 55 ) ^o = ∠ 8

⇒∠ 8 = 134 ^o

Now, that we have found the value of ∠ 1, we will find the value of ∠ 2

We can clearly see that ∠ 1 and ∠ 2 form a supplementary pair of angles. This means that the sum of ∠ 1 and ∠ 2 should be equal to 180^o. Hence, we have,

  ∠ 1 + ∠ 2 = 180^o

⇒ 134^o + ∠ 2 = 180^o

⇒ ∠ 2 = 180^o – 134^o 

⇒ ∠ 2 = 46^o

Now, that we have found the value of ∠ 2, we will find the value of ∠ 4

We can clearly see that ∠ 2 and ∠ 4 form a supplementary pair of angles. This means that the sum of ∠ 2 and ∠ 4 should be equal to 180^o. Hence, we have,

  ∠ 2 + ∠ 4 = 180^o

⇒ 46^o + ∠ 4 = 180^o

⇒ ∠ 4 = 180^o – 46^o 

⇒ ∠ 4 = 134^o

Now, that we have found the value of ∠ 4, we will find the value of ∠ 3

We can clearly see that ∠ 3 and ∠ 4 form a supplementary pair of angles. This means that the sum of ∠ 3 and ∠ 4 should be equal to 180^o. Hence, we have,

  ∠ 3 + ∠ 4 = 180^o

⇒ 134^o + ∠ 3 = 180^o

⇒ ∠ 3 = 180^o – 134^o 

⇒ ∠ 3 = 46^o

Note that ∠ 3 and ∠ 6 form a pair of alternate interior angles. Since the lines that have been intersected by a transversal are parallel, therefore, the pair of alternate interior angles should be equal. Hence, we have,

∠ 3 = ∠ 6 =  46^o

Similarly, ∠ 4 = ∠ 5 =  134^o

Note that ∠ 2 and ∠ 7 form a pair of alternate exterior angles. Since the lines that have been intersected by a transversal are parallel, therefore, the pair of alternate exterior angles should be equal. Hence, we have,

∠ 2 = ∠ 7 =  46^o

Key Facts and Summary

1. Two lines l and m in the same plane are said to be parallel lines of they do not intersect when produced indefinitely in either direction.
2. If two lines are not parallel, they will intersect each other.
3. A transversal is defined as a line intersecting two or more given lines in a plane at different points.  
4. The angles whose arms include the line segment that forms the transversal are called interior angles.
5. A pair of angles in which one arm of both the angles is on the same side of the transversal and their other arms are directed in the same sense is called a pair of corresponding angles.
6. A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms are directed in opposite directions and do not include segment PQ is called alternate exterior angles in a transversal.
7. The alternate exterior angle theorem states that if two lines are parallel and are intersected by a transversal, then the alternate exterior angles are considered as congruent angles or angles of equal measure.