Alternate Angles
A transversal is a line that intersects two or more parallel lines. Some pairs of angles are created when a transversal is drawn across the parallel lines. When a transversal cuts two parallel lines, four interior angles are created on the inside and four exterior angles are created on the outside.
To make the concept clearer, in the image given below, the two parallel lines EF and GH are cut by a transversal SR. Angles ∠1, ∠2, ∠7, and ∠8 are called the exterior angles. The angles ∠3, ∠4, ∠5, and ∠6 are the interior angles.
Alternate angles are ∠3 and ∠5, ∠4 and ∠6, ∠1 and ∠7, ∠2 and ∠8.
Note: When a transversal cuts two parallel lines, the alternate angles are equal.
Therefore, ∠3 = ∠ 5 and ∠4 = ∠6
∠2 = ∠8 and ∠1 = ∠7
Types of Alternate Angles
The alternate angles are divided into two categories:
1. Alternate interior angles: The pair of angles that are on the inside of two parallel lines but lie on two sides of the transversal.
Notice the angles marked and that each pair of angles are equal to each other.
2. Alternate exterior angles: The pair of angles that are on the outer side of two parallel lines but on two sides of the transversal.
Notice the alternate exterior angles and that they are equal to each other.
Alternate Exterior Angles
Alternate exterior angles are a pair of angles that are outside the two parallel lines but on either side of the transversal.
For example: ∠1, ∠2, ∠3, and ∠4 are alternate exterior angles, with ∠1 being equal to ∠4 and ∠2 is equal to ∠3.
Learn About Angles Formed by Parallel Lines Cut by a Transversal in this Video
Solved Examples
Example 1:
From the given image, mark the exterior angles and write their values. In the figure, ∠1 = 145°and ∠2 = 35°.
Solution:
The alternate exterior angles in the given image above are ∠1, ∠2, ∠8, and ∠7. ∠1 and ∠2 are given.
Since, we know that alternate exterior angles are equal.
So, ∠1 = ∠7 = 145°
∠2 = ∠8 = 35°
Example 2:
Find the value of the angles Y, X, and Z from the image.
Solution:
From the figure
∠Y = 75°
By the alternate interior angle theorem, ∠Y = ∠Z.
Therefore, the value of ∠Z must be equal to the value of ∠Y.
∠Z = 75°
Since, RS is the straight line, ∠W + ∠Z = 180° [Sum of angles on a straight line is 180°]
So, ∠W = 180° – ∠Z
∠W = 180° – 75° [Substitute the value ∠Z = 60°]
∠W = 105°
Again, from the alternate interior angle theorem, ∠W = ∠X.
Therefore, ∠X = 105°
Example 3:
Toby and Serah are on two parallel lanes. They were planning to construct a path that cut through the two paths such that it cuts the path at an angle. If ∠A = 45°, what is the value of ∠B?
Solution:
Since the two paths are cut by a transversal, The alternate angles should be equal ∠A is equal to ∠B.
Therefore, ∠B = ∠A = 45°.
The pair of angles created when a transversal cuts a pair of coplanar lines are known as alternate angles.
The two different kinds of alternate angles are:
- Alternate interior angles
- Alternate exterior angles
The co-interior angles are supplementary, however, the alternate interior angles are congruent.