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

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

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.

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

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.

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.

In the above image, the alternate exterior angles are

$\angle 1$ and $\angle 8$

$\angle 2$ and $\angle 7$

Alternate Interior Angles Theorem

Alternate Exterior Angles Theorem

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.

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.

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

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.

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.

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.

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 x  and find the value of each angle.

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

Frequently Asked Questions about Alternate Angles