Alternate Angles: Definition, Formula, Types, Theorem, Examples (2024)

Home » Math Vocabulary » Alternate Angles: Definition, Types, Theorem, Examples, Facts

  • What Are Alternate Angles?
  • Alternate Interior Angles Theorem
  • Alternate Exterior Angles Theorem
  • Solved Examples on Alternate Angles
  • Practice Problems on Alternate Angles
  • Frequently Asked Questions about Alternate Angles

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: Definition, Formula, Types, Theorem, Examples (1)

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 Angles: Definition, Formula, Types, Theorem, Examples (2)

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

Alternate Angles: Definition, Formula, Types, Theorem, Examples (3)

Alternate Angles: Definition, Formula, Types, Theorem, Examples (4)Begin here

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Definition of Alternate Angles

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

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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 Angles: Definition, Formula, Types, Theorem, Examples (18)

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 Angles: Definition, Formula, Types, Theorem, Examples (19)

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.

Alternate Angles: Definition, Formula, Types, Theorem, Examples (20)

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.

Alternate Angles: Definition, Formula, Types, Theorem, Examples (21)

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.

Alternate Angles: Definition, Formula, Types, Theorem, Examples (22)

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.

Alternate Angles: Definition, Formula, Types, Theorem, Examples (23)

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.

Alternate Angles: Definition, Formula, Types, Theorem, Examples (24)

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.

Alternate Angles: Definition, Formula, Types, Theorem, Examples (25)

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

1

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

Alternate Angles: Definition, Formula, Types, Theorem, Examples (26)

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:

Alternate Angles: Definition, Formula, Types, Theorem, Examples (27)

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.

Alternate Angles: Definition, Formula, Types, Theorem, Examples (28)

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.

Alternate Angles: Definition, Formula, Types, Theorem, Examples (29)

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

Alternate Angles: Definition, Formula, Types, Theorem, Examples (30)

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, $x + y + z = 180^{\circ}$

Alternate Angles: Definition, Formula, Types, Theorem, Examples (2024)

FAQs

Alternate Angles: Definition, Formula, Types, Theorem, Examples? ›

Alternate Angle Definition

What are alternate angles theorems? ›

Alternate Interior Angles theorem states, if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Alternate interior angles are the angles formed on the opposite sides of the transversal.

What are alternate angles Year 7? ›

Alternate angles are angles on opposite sides of the transversal and between the parallel lines. Alternate angles on parallel lines are equal.

Do alternate angles add up to 180°? ›

Unless the alternate interior vertical angles are 90° then they will not add up to 180°. If the alternate interior angles are obtuse, then adding them together will result in a number higher than 180°. Therefore, if the alternate interior angles are acute, then adding them together will result in a number below 180°.

What are alternate angles with answer? ›

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.

What are the 4 different types of angles? ›

There are four main types of angles: right angles, acute angles, obtuse angles, and straight angles. Right angles are like corners and measure 90°.

What is the F rule in angles? ›

The easiest way to think of corresponding angles is the “F pattern”. If you draw the letter “F”, backwards or forwards, on the transversal, then the angles on top or below the horizontal lines will be congruent. Opposite angles are angles that are opposite each other when two lines cross. Opposite angles are equal.

What is the z rule in angles? ›

Theorem 1 (The "Z" Theorem)

If two lines are parallel then their alternate interior angles are equal. If the alternate interior angles of two lines are equal then the lines must be parallel.

Are vertical angles supplementary? ›

Supplementary angles are the angles which add up to 180°. Vertical angles are supplementary if and only if the lines intersect each other at a 90° angle. In simple words, if the two lines are perpendicular, the vertical angles formed are supplementary.

Are same side interior angles supplementary? ›

The same-side interior angle theorem states that when two lines that are parallel are intersected by a transversal line, the same-side interior angles that are formed are supplementary, or add up to 180 degrees.

Are corresponding angles supplementary? ›

Corresponding angles will be supplementary when the transversal intersects two parallel lines perpendicularly. When this happens, the corresponding angles will each measure 90 degrees. Otherwise, if the transversal intersects two parallel lines, then the corresponding angles will be congruent.

What are the conditions for alternate angles? ›

Alternate angle theorem states that when two parallel lines are cut by a transversal, then the resulting alternate interior angles or alternate exterior angles are congruent. To prove: If two parallel lines are cut by a transversal, then the alternate interior angles are equal.

What is the alternate triangle theorem? ›

The alternate segment theorem (also known as the tangent-chord theorem) states that in any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment.

How to identify an alternate segment theorem? ›

The alternate segment theorem is the angle that lies between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment.

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