Alternate Exterior Angles
Alternate exterior angles are formed when a line (transversal) intersects two or more (parallel) lines at distinct points. The term exterior means something which is situated externally. Alternate exterior angles always lie outside two lines that are intersected by the transversal and they are located on the opposite sides of the transversal. Thus the two exterior angles which form at the alternate ends of the transversals in the exterior section are considered as the pair of alternate exterior angles and they are equal. We get two such pairs of alternate exterior angles when a transversal cuts two parallel lines. Let us learn more about them.
|1.||What are Alternate Exterior Angles?|
|2.||Alternate Exterior Angles Theorem|
|3.||Converse of Alternate Exterior Angles Theorem|
|4.||Alternate Exterior Angles Examples|
|6.||FAQs on Alternate Exterior Angles|
What are Alternate Exterior Angles?
Alternate exterior angles are created when more than two lines are intersected by the transversal. These angles are generally formed opposite to transversal (a line that crosses two or more lines). Two lines that are parallel in nature, create some angles with the transversal. A transversal always divides these lines into sections. We usually name these sections as the exterior section and the interior section. Interior angles are created in the space inside the parallel lines whereas alternate exterior angles are created in the space outside the parallel lines (on alternating sides).
Exterior Angles Definition
In any given closed shape if a line is extended from its outer edge it forms an exterior angle. It is formed between the side of the shape and its adjacent extended side. An exterior angle is created when a side of the triangle is extended.The adjacent interior angle and exterior angle form a linear pair. It is an exterior angle because it is located at the exterior position of the triangle. When two lines are intersected by a transversal, the angles on the outer side of these two lines are called exterior angles.
Alternate Exterior Angles Definition
Two angles that lie on opposite sides of the transversal and are placed on two different lines, both either inside the two lines or outside, are called alternate angles. In the given figure below, the alternate exterior angle pairs are 1 and 7; 2 and 8. Angle 2 is on the right side of the transversal M and 8 is on the left side; ∠2 is above the line AB whereas ∠8 is below the line CD. The same goes for other pairs of angles (∠1 and ∠7). Now, a pair of angles that satisfy these conditions is called a pair of alternate exterior angles.
According to the figure, we can define alternate exterior angles as:
Two exterior angles that lie on two different lines cut by a transversal and are placed on the opposite sides of the transversal are called alternate exterior angles.
In the figure, we can observe that angle 1 and angle 7 are alternate exterior angles as both lie in the exterior of lines and are placed on the opposite sides of the transversal. The same is the case with angle 2 and angle 8.
When these two lines AB and CD are parallel, then the alternate angles will satisfy certain properties. Both the pairs of angles have equal measure. We will understand these properties in the coming section of this article.
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 are equal in measurement.
In the figure, we can observe that angles 1 and 7; 2 and 8 are pairs of alternate exterior angles. We will now prove that why and how they are congruent ( i.e. they have equal measure).
Given: Lines AB and CD are parallel lines on a transversal M.
[Hint: To prove the above theorem, we will be using the following axioms:
Corresponding Angle Axiom: When two lines are parallel the corresponding angles are congruent angles.
The converse of Corresponding Angle Axiom: When the corresponding angles made by two lines are congruent, then those two lines are parallel.]
To Prove: ∠1 = ∠7
To prove this result, we will consider the vertically opposite angle of ∠1. Let's denote it by ∠3.
Now, ∠1 = ∠3 as they are vertically opposite angles.
Here, line AB || CD,
∠3 = ∠7 (corresponding angles axiom)
∴∠1 = ∠7 (transitivity)
Converse of Alternate Exterior Angles Theorem
The converse of alternate exterior angle theorem states that, if the alternate exterior angles formed by two lines, which are cut by a transversal, are congruent, then the lines are parallel.
Hence, in the above figure, if it is given that ∠1 = ∠7 then line AB || line CD.
Note: Apart from alternate exterior angles, there is one more type of exterior angles pair. This pair is called pair of consecutive exterior angles. There are two pairs of consecutive exterior angles in the above figure.
∠2 and ∠7
∠1 and ∠8
Consecutive Exterior Angles are supplementary, i.e.,∠2 + ∠7 = 180° and ∠1 and ∠8 = 180°
Topics Related to Alternate Exterior Angles
Alternate Exterior Angles Examples
Example 1: Joe drew a map where the road toward town X crosses two roads A and B. The angles made between these roads are as shown in the figure below. He is not sure if roads A and B are parallel. Can you find out if these two roads are parallel using alternate exterior angles theory?
The first step is to find x.
x + 65° = 180° ⋯ linear pair
x = 180° − 65°
x = 115°
Now, x° and 125° are alternate exterior angles.
x ≠ 125°
Thus, the lines are not parallel.
∴ Roads A and B are not parallel.
Example 2: Using alternate exterior angle theorem solve the given problem: Given: Line RS || Line PQ.
a) Find x.
b) Also check if line XY || line RS.
i) Given: ∠XBA = 30° and ∠QCD = 30°.
Line RS || Line PQ,
Here, ∠QCD = ∠EDS (corresponding angles)
∴ x = 30°⋯(1)
ii) Now let's check if lines XY and RS are parallel or not.
It is given that Line RS || Line PQ
AE as a transversal bisecting three lines XY, PQ, and RS
The pair of alternate exterior angles have the same measure i.e., ∠XBA = ∠EDS = 30°
∠XBA = ∠EDS are alternate exterior angles.
If ∠XBA and ∠EDS are alternate exterior angles then, by the converse of alternate exterior angle theorem, we get that line XY is parallel to line RS
∴ a) x = 30° , b) line XY || line RS
Example 3: With the help of the alternate exterior angles theorem find the value of x in the given figure, if line p || line q.
Given : Line p || q,
So by alternate exterior angle theorem, we get,
(2x + 26)° = (3x−33)°...(1)
2x − 3x = −33° − 26°
−x = −59°
∴ x = 59°
By putting the value of x in equation (1) we can find the exact value of the angles
(2x + 26)° = (3x−33)°
(2(59) + 26)° = (3(59)−33)°
144° = 144°.
LHS = RHS
The value of alternate exterior angles is equal i.e., 144°.
FAQs on Alternate Exterior Angles
What is the Definition of Alternate Exterior Angles in Math?
We read about angles in the sub-category of mathematics, i.e., geometry. According to the angles theory, the two exterior angles that lie on two different lines cut by a transversal and are placed on the opposite sides of the transversal are called alternate exterior angles.
Which Pair of Angles are Alternate Exterior Angles?
Alternate exterior angles are created when a transversal always divides the two lines into sections i.e., the exterior section and the interior section. Alternate exterior angles are created in the space outside the parallel lines i.e. on alternating sides. Hence these angles are pair of alternate exterior angles.
What is an Alternate Exterior Angles Theorem?
The alternate exterior angle theorem is defined, when two parallel lines are intersected by a transversal, then the exterior angles formed on either side of the transversal make alternate exterior angles equal.
Do Alternate Exterior Angles Prove That the Lines are Parallel?
Alternate exterior angles prove that the lines are parallel only if the alternate exterior angles are congruent. This result is known as the converse of the alternate exterior angle theorem.
How to Prove Alternate Exterior Angles Theorem?
To prove the alternate exterior angle theorem which states that the pair of exterior angles that form on either side of a transversal are equal, we must know, if the lines are parallel and whether they are bisected by a transversal or not. After clear observation, using the corresponding angle axiom (when two lines are parallel the corresponding angles are congruent) and the converse of the corresponding angle axiom (when the corresponding angles made by two lines are congruent, then those two lines are parallel.) we can prove alternate exterior angles theorem easily.
Do Alternate Exterior Angles Add up to 180?
No, alternate exterior angles do not add up to 180°. In fact, they are congruent to each other. Only in the case where one of them is 90°, then the other angle will also measure 90°.
Hence, the total will be 90° + 90° = 180°.
Are Alternate Exterior Angles Congruent?
Alternate exterior angles are congruent when the lines are parallel. If the lines are not parallel, the alternate exterior angles are not congruent.