Alternate Angles: Definition, Types, Theorem, Examples, Facts

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What Are Alternate Angles?

When a transversal cuts a pair of parallel lines (or non-parallel lines), it forms different types of angles. Alternate interior angles are a set of non-adjacent angles on either side of the transversal.

In each diagram given below, two parallel lines are cut by a transversal. All the angle pairs highlighted in the same color represent alternate angles. They are on alternate sides of the transversal. They don’t have common vertices.

Based on their position, they are further categorized as interior and exterior angles.

  • Diagram on the left: Alternate interior angles
  • Diagram on the right: Alternate exterior angles
Alternate angles

Observe that we can quickly spot a pair of alternate interior angles using the Z-shape. Take a look at the positions of alternate interior angles.

Alternate interior angles

Thus, alternate interior angles are also sometimes known as Z-angles.

Alternate angles are Z-shaped

Definition of Alternate Angles

Alternate angles are the non-adjacent angles that lie on the opposite sides of the transversal.  

Types of Alternate Angles

There are two types of alternate angles based on their position with respect to the transversal and the parallel lines. 

  • Alternate Interior Angles
  • Alternate Exterior Angles 

Let’s understand the two types of alternate angles and their properties.

Alternate Interior Angles

The pair of angles that lie on the inner side (or the interior region) of the two parallel lines, but on the opposite sides of the transversal are known as alternate interior angles. 

So, what do alternate interior angles look like? In the image given below, the angle-pairs highlighted in the same color represent the alternate interior angles.

Alternate interior angles

In the above image, the alternate interior angles are

$\angle 4$ and $\angle 5$

$\angle 3$ and $\angle 6$

Alternate Exterior Angles

The pair of angles that lie on the outside region of the two parallel lines, but on the opposite sides of the transversal are known as alternate exterior angles. 

The angle-pairs highlighted in the same color represent the alternate exterior angles.

Alternate exterior angles

In the above image, the alternate exterior angles are

$\angle 1$ and $\angle 8$

$\angle 2$ and $\angle 7$

Alternate Interior Angles Theorem

Alternate Angles Theorem Statement: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles formed are congruent.

Converse: If two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.

Alternate Exterior Angles Theorem

Alternate Exterior Angles Theorem Statement: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles formed are congruent.

Converse: If two lines are cut by a transversal so that the alternate exterior angles are congruent, then the lines are parallel.

Alternate Angles Theorem Proof

Assume that PQ and RS are the two parallel lines cut by a transversal LM. 

a, b, c, d are the angles created by the transversal.

Since the corresponding angles are congruent, we have a corresponding pair for each of these angles, which are also labeled as a, b, c, and d.

Interior and Exterior Angles

At the intersection point on the lines t and l,

$\angle a + \angle d = 180^{\circ}$ ( PQ is the straight line)—  (1)

$\angle b + \angle d = 180^{\circ}$ ( LM is the straight line)—  (2)

So, from (1) and (2), we get

$\angle a = \angle b$

Again, at the intersection point on the straight lines LM and RS,

$\angle a + \angle d = 180^{\circ}$ ( RS is the straight line)— (3)

$\angle a + \angle c = 180^{\circ}$ ( LM is the straight line)—  (4)

So, from (3) and (4), we get

$\angle d = \angle c$

Therefore, it is concluded that the alternate interior angles are congruent.

Hence, proved.

Another way:

Here, we use the fact that the corresponding angles formed when a transversal cuts a pair of parallel lines are congruent.

Parallel lines m and n cut by a transversal t

Here, $\angle 1 = \angle 5$ …corresponding angles

$\angle 1 = \angle 4$ …opposite angles

Thus, $\angle 4 = \angle 5$  …alternate interior angles

Also, $\angle 1 = \angle 4$ …opposite angles

$\angle 4 = \angle 8$ …corresponding angles

Thus, $\angle 1 = \angle 8$ …alternate exterior angles

Facts about Alternate Angles

  • Alternate angles are angles that lie on opposite sides of the transversal line and have the same size.
  • There are two different types of alternate angles, alternate interior angles and alternate exterior angles.
  • The co-interior angles OR same-side interior angles add up to 180 degrees. The rule is sometimes remembered as “C angles” because the angles make a C shape.
  • When two non-parallel lines intersect a transversal, the alternate interior angles formed will not be equal.

Conclusion

In this article, we learned about the alternate angles, types of alternate angles, and the theorems associated with it. Let’s solve some examples and practice problems based on each of these concepts.

Solved Examples on Alternate Angles

1. Use the alternate interior angles theorem to determine if the lines cut by the transversal are parallel.

Interior angles

Solution: 

Angle A and the angle measuring $60^{\circ}$ form a straight angle.

Thus, $m \angle A + 60^{\circ} = 180^{\circ}$ 

$m \angle A = 120^{\circ}$

Similarly, $\angle B$ and $120^{\circ}$ form a straight angle, so we know that 

$m \angle B + 120 = 180$

$m \angle B = 60^{\circ}$

$\angle A$ and the original $120^{\circ}$ angle are alternate interior angles and are equal. 

$\angle B$ and the original $60^{\circ}$ angle are also equal alternate interior angles. 

So, going by the alternate interior angles theorem, the lines cut by the transversal must be parallel.

2. In the diagram given below, the lines cut by the transversal are parallel. Determine the measures of the angles A, B, and C.

Interior angles

Solution:

Angle A and 155 degrees are alternate interior angles, and so since the lines cut by the transversal are parallel, the measure of angle A is also 155 degrees. 

Angle A and angle B form a straight angle, so $A + B = 180^{\circ}$.

Since $A = 155^{\circ},\; 155^{\circ} + B = 180^{\circ}$.

 $B = 25^{\circ}$

Thus the measure of angle B is $25^{\circ}$.

Now, since angle B and angle C are alternate interior angles, we know that the measure of angle C is also $25^{\circ}$.

Thus, the measures of $\angle A,\; \angle B$ and $\angle C$ are $155^{\circ},\; 25^{\circ}$ and $25^{\circ}$ respectively.

3. In the figure given below, CE is parallel to FH. Find the value of x.

Transversal intersecting two parallel lines

Solution: 

In the given figure, $\angle ADE$ and $\angle FGJ$ form a pair of alternate exterior angles.

By using alternate exterior angle theorem, we have, $\angle ADE = \angle FGJ$

So, $x^{\circ} + 50^{\circ} = 130^{\circ}$

$x = 130^{\circ} \;-\; 50^{\circ}$

$x = 80^{\circ}$

Therefore, the value of x is $80^{\circ}$.

4. Calculate the value for and find the value of each angle.

Parallel lines cut by a transversal

Solution: 

Angles measuring $(6x \;-\; 10)$ and $(3x + 20)$ are alternate angles.

We know that alternate angles are equal.

 So, $6x \;-\; 10  =  3x + 20$

       $6x \;-\; 3x = 20 + 10$

               $3x  = 30$

                 $x  = 10$

Substituting values in the equation of angles, we get,

$6x \;-\; 10  = 6 (10) \;-\; 10  =  60 \;-\; 10 = 50$

 $3x + 20  = 3 (10)  + 20  = 30 + 20 = 50$

Thus, the value of each labeled angle is $50^{\circ}$.

Practice Problems on Alternate Angles

Alternate Angles: Definition, Types, Theorem, Examples, Facts

Attend this quiz & Test your knowledge.

1

From the image given below, identify the pair of alternate interior angles.

Alternate Angles: Definition, Types, Theorem, Examples, Facts
1 and 2
2 and 4
2 and 6
3 and 6
CorrectIncorrect
Correct answer is: 3 and 6
The pair of angles on the inner side of the two parallel lines but on the other side of the transversal are known as alternate interior angles. Thus, 3 and 6 are alternate interior angles.
2

In the given diagram, angles X and Y are

alternate angles
corresponding angles
exterior angles
co-interior angles.
CorrectIncorrect
Correct answer is: corresponding angles
Angles X and Y are corresponding angles. They are formed at matching/corresponding corners with respect to the transversal.
3

Name the angle relationship in the diagram given below:

Alternate Interior Angles
Alternate exterior Angles
Corresponding Angles
Vertical Angles
CorrectIncorrect
Correct answer is: Alternate exterior Angles
The pair of angles on the outside of the two parallel lines but on the other side of the transversal are known as alternate exterior angles. Thus, the angles 1 and 2 are alternate exterior angles.
4

Angles inside a pair of parallel lines that lie on the opposite sides of a transversal and are congruent are called _____.

Vertical angles
Corresponding Angles
Supplementary Angles
Alternate Interior Angles
CorrectIncorrect
Correct answer is: Alternate Interior Angles
The pair of angles on the inner side of the two lines but on the other side of the transversal are known as alternate interior angles.

Frequently Asked Questions about Alternate Angles

When two lines are crossed by a transversal, the angles formed in the matching corners are called corresponding angles.When the two lines are parallel, corresponding angles are equal.

Take a look at the image given below:

Corresponding angles

In the above image, 

angle 3 corresponds to angle 7, 

angle 4 corresponds to angle 8, 

angle 1 corresponds to angle 5, 

angle 2 corresponds to angle 6.

Alternate angles are defined as angles in a plane figure that lie on opposite sides of a transversal line and have the same measurements. In the image given below, 1 and 7, 4 and 6, 2 and 8, 3 and 5 are alternate angles.

Corresponding and alternate angles

On the other hand, corresponding angles are angles that are in the same relative positions along the transversal line and still have the same measurements.

In the image given above, 1 and 5, 4 and 8, 2 and 6, 3 and 7 are pairs of corresponding angles.

The same side interior angles formed by a transversal cutting two parallel lines are shown in the figure below. They are supplementary.

Same side interior angles

If we have a triangle, we can always draw two parallel lines as shown in the figure.

Angle sum property of a triangle

Now, AB and AC are the transversals. We know that alternate angles are equal. Therefore, the two angles labeled as x are equal. Also, the two angles labeled y are equal.

We know that x, y, and z together add up to 180 degrees, because together, these are just angles in a linear pair or the angles around the straight line. So, $x + y + z = 180^{\circ}$