Alternate Exterior Angles
Alternate exterior angles are formed when a 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 part are considered as the pair of alternate exterior angles and they are always 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. | FAQs on Alternate Exterior Angles |
What are Alternate Exterior Angles?
Alternate exterior angles are created when two or more lines are intersected by a transversal. These angles are formed on the outer side of the transversal on the different sides.
When any two parallel lines are intersected by a transversal, they create some pairs of angles with the transversal. Interior angles are created in the space inside the parallel lines whereas alternate exterior angles are created in the space outside the parallel lines.
Alternate Exterior Angles Definition
Two angles that lie on the opposite sides of the transversal and are placed on two different lines are called alternate exterior angles. These pairs of angles are always equal if the two given lines are parallel. In the given figure below, Line AB || Line CD and they are intersected by the transversal MN. Here, the alternate exterior angle pairs are ∠1 and ∠7; ∠2 and ∠8. This means ∠1 = ∠7; ∠2 = ∠8
We can see that ∠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 rule applies for the other pair of angles (∠1 and ∠7). Therefore, the pair of angles that satisfy these conditions are called 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 ∠1 and ∠7 are alternate exterior angles and ∠2 and ∠8 are alternate exterior angles because:
- Both lie on the exterior side of the lines; and
- They are placed on the opposite sides of the transversal.
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 angles of equal measure.
Following the same figure given above, we can observe that ∠1 and ∠7; ∠2 and ∠8 are pairs of alternate exterior angles. Let us see how they are congruent (equal).
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
Proof:
To prove this result, we will consider the vertically opposite angle of ∠1. Here, the vertically opposite angle of ∠1 is ∠3.
So, ∠1 = ∠3 (vertically opposite angles)
Here, line AB || CD,
∠3 = ∠7 (corresponding angles axiom)
∴∠1 = ∠7 (transitivity) Hence, proved.
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 is called the 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 + ∠8 = 180°
Topics Related to Alternate Exterior Angles
Alternate Exterior Angles Examples
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Example 1: Using the alternate exterior angle theorem solve the given problem: Given: Line RS || Line PQ.
a) Find x.
b) Also check if line XY || line RS.
Solution:
i) Given: Line RS || Line PQ; ∠XBA = 30° and ∠QCD = 30°
Here, ∠QCD = ∠EDS (corresponding angles)
∴ x = 30°⋯(1)
ii) Now let us 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 to know that line XY is parallel to line RS
∴ a) x = 30° , b) line XY || line RS
FAQs on Alternate Exterior Angles
What are Alternate Exterior Angles in Math?
Alternate exterior angles are created when two or more lines are intersected by a transversal. These angles are formed on the outer side of the transversal on the different sides. In other words, 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.
What is the Alternate Exterior Angles Theorem?
The alternate exterior angle theorem states that when two parallel lines are intersected by a transversal, then the exterior angles formed on either side of the transversal are 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 are Alternate Exterior Angles Related?
According to the alternate exterior angle theorem, when two lines are parallel and are intersected by a transversal, then the alternate exterior angles are considered as congruent angles. In other words, alternate exterior angles that are formed by two parallel lines and a transversal are always equal in measure.
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 when they are 90° each, then they will sum up to 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.
What are Alternate Interior and Alternate Exterior Angles?
When any two parallel lines are intersected by a transversal, they create some pairs of angles with the transversal. Alternate Interior angles are always equal and are formed on the inner side of the parallel lines, but are located on the opposite sides of the transversal. Alternate exterior angles are those angles that have different vertices, they lie on the alternate sides of the transversal and are exterior to the lines.
When are Alternate Exterior Angles Congruent?
Alternate Exterior angles are congruent only when the two lines that are intersected by a transversal are parallel.