Introduction
Alternate exterior angles happen when a transversal line crosses two or more parallel lines at different points. The term "exterior" refers to something on the outside. These angles always sit outside the two lines crossed by the transversal and are on opposite sides of the transversal. So, you'll find these angles on the outer sides of the intersecting lines but not between them. When you have a transversal crossing two parallel lines, you get two pairs of alternate exterior angles. These pairs are always equal in measure.
What Are Alternate Exterior Angles?
Alternate exterior angles are angles formed on the outside of a transversal, located on different sides. When two parallel lines are intersected by a transversal, specific angle pairs are produced. Interior angles occur within the space between the parallel lines, while exterior angles emerge outside this space. For instance, in the illustration below where Line ‘`n`’ is parallel to Line ‘`m`’ and intersected by transversal ‘`o`’, the pairs of alternate exterior angles are `∠1` & `∠8` and `∠2` & `∠7`. These pairs of alternate exterior angles are equal in measure. This indicates that `∠1` equals `∠8` and `∠2` equals `∠7`.
We can conclude that `∠2` and `∠7` are alternate exterior angles. `∠2` lies to the right of transversal `o`, while `∠7` is positioned to the left. Thus, they are on opposite sides of the transversal, with `∠2` above line `n` and `∠7` below line `m`, outside the two lines. This logic applies similarly to the other angle pair (`∠1` and `∠8`). Hence, a pair of angles exhibiting these characteristics are termed alternate exterior angles.
Properties of Alternate Exterior Angles
- When a transversal intersects two parallel lines, the alternate exterior angles are identical in measure.
- If the alternate exterior angles formed by two lines intersected by a transversal are congruent, then the lines are parallel.
- Alternate exterior angles have the same orientation and are always positioned on opposite sides of the transversal.
- In a trapezoid, the alternate exterior angles are congruent.
- In a regular polygon, the alternate exterior angles are also congruent.
- The alternate exterior angles of a triangle are equal to the exterior angle that is not adjacent to them.
Alternate Exterior Angles Theorem
The Alternate Exterior Angles Theorem sheds light on a fascinating relationship in geometry. Specifically, the theorem states that the alternate exterior angles, which are those positioned on opposite sides of the transversal but outside the parallel lines, are congruent.
In simpler terms, if you have a transversal slicing through two parallel lines, the pairs of alternate exterior angles you get will be equal in size.
In the diagram provided, you'll see examples like `∠1` and `∠8`, as well as `∠2` and `∠7`, showcasing alternate exterior angles. Now, let's see how we prove that these pairs of angles are indeed equal.
Given:Lines `AB` and `CD` are parallel lines intersected by transversal `M`.
To Prove: `∠2 = ∠7`
Proof:
To demonstrate this result, we start by looking at the vertically opposite angle of `∠2`, which we'll call `∠3`.
So, `∠2 = ∠3` (due to the property of vertically opposite angles).
Since lines `AB` and `CD` are parallel, according to the axiom of corresponding angles, `∠3 = ∠7`.
Therefore, by transitivity, `∠2 = ∠7`.
Thus, we've successfully proven the theorem aboutcongruent alternate exterior angles.
Converse of Alternate Exterior Angles Theorem
The converse of the Alternate Exterior Angles Theorem, also calledAlternate Exterior Angles Converse Theorem provides an interesting insight into geometry. It states that if the alternate exterior angles formed by two lines intersected by a transversal have equal measures, then those two lines are parallel. For instance, in the figure provided above, if it's known that `∠2 = ∠7`, it can be concluded that line `AB` is parallel to line `CD`.
Additionally, besides alternate exterior angles, there's another type of exterior angle pair calledconsecutive exterior angles. In the given diagram, we observe two pairs of consecutive exterior angles: `∠2` and `∠1`, and `∠7` and `∠8`. It's noteworthy that consecutive exterior angles are supplementary, meaning that the sum of their measures is `180` degrees, such as `∠1 + ∠2 = 180°` and `∠7 + ∠8 = 180°`.
Alternate Exterior Angles Formed by Non-Parallel Lines
When two or more non-parallel lines are intersected by their transversal, the exterior angles that appear on opposite sides of the transversal are also called exterior alternate angles or alternate exterior angles of non-parallel lines and their transversal. Total two pairs of exterior angles are formed on opposite sides at the intersection of two nonparallel lines and their transversal. Therefore, the two pairs of exterior angles are called exterior alternate angles or alternate exterior angles.Unlike the case when the lines are parallel, alternate exterior angles formed by non-parallel lines are generally not congruent.
Total two pairs of exterior angles (`∠1 , ∠3` and `∠2, ∠4`) are formed on opposite sides at the intersection of two nonparallel lines and their transversal. Therefore, the two pairs of exterior angles are called exterior alternate angles or alternate exterior angles.
Unlike the case when the lines are parallel, alternate exterior angles formed by non-parallel lines are not congruent.
Solved Examples
Example `1`: The transversal `l` cuts through the parallel lines `a` and `b`. Find the value of `x`.
Solution:
Here `∠x` and `142°` form a pair of alternate exterior angles.
Since the lines ‘`a`’ and ‘`b`’ are parallel, the alternate exterior angles are congruent.
Hence, `∠x` and `142°`.
Example `2`: Line `BD` is parallel to line `EF`. Find the value of `x`.
Solution:
`∠ACB = x`
`∠CGF = 130°`
Here, `∠CGF + ∠FGH = 180° …` angles in a linear pair.
So, `130° + ∠FGH = 180°`
`∠FGH = 180° - 30°`
`∠FGH = 50°`
Also, `∠FGH = x …`Alternate Exterior Angles Theorem
Thus `x = 50°`
Example `3`: Two parallel lines intersected by a transversal have alternate exterior angles represented as `(3x + 40)°` and `(4x - 50)°`. Find the value of `x` and the actual measures of the alternate exterior angles using the Alternate Exterior Angles Theorem.
Solution:
Using the Alternate Exterior Angles Theorem, we know that the alternate exterior angles are congruent.
`(3x + 40)° = (4x - 50)°`
`3x + 40 = 4x - 50`
`x = 90°`
Substituting the value of `x` in the expressions:
a) `(3x + 40)° = [3(90) + 40]° = 310°`
b) `(4x - 50)° = [4(90) - 50]° = 310°`
Therefore, the value of `x` is `90°`, and both alternate exterior angles have a measure of `310°`.
Example `4`: A transversal cuts through a set of two lines. The alternate exterior angles formed by the transversal are `(2x + 25)°` and `(3x - 15)°`. Determine the value of `x` and verify if the lines cut by the transversal are parallel.
Solution:
a) `(2x + 25)° = (3x - 15)°`
`2x + 25 = 3x - 15`
`x = 40°`
b) To verify if lines cut by the transversal are parallel, we need to check if the alternate exterior angles are congruent.
`(2x + 25)° = [2(40) + 25]° = 105°`
`(3x - 15)° = [3(40) - 15]° = 105°`
Since both alternate exterior angles have the same measure, the two lines cut by the transversal are parallel.
Example `5`: Find the value of `x` that makes `m` `||` `n`.
Solution:
The angles marked in the above figure form a pair of alternate exterior angles.
For `m` to be parallel to `n`, the alternate exterior angles should be congruent.
Hence `3x + 1 = 58°`
`3x = 58° - 1`
`3x = 57°`
`x = 19°`
Therefore for `x = 19°`, the two lines `m` and `n` are parallel.
Practice Problems
Q`1`. Identify the alternate exterior angles in the given figure.
- `∠1` & `∠8`, `∠2` & `∠7`
- `∠1` & `∠4`, `∠2` & `∠6`
- `∠1` & `∠5`, `∠2` & `∠6`
- `∠1` & `∠3`, `∠2` & `∠4`
Answer: a
Q`2`. In the figure shown below, if `∠ a = 70°`, what is the measure of `∠ b`?
- `70°`
- `110°`
- `140°`
- `180°`
Answer: a
Q`3`. Given two parallel lines intersected by a transversal, if the alternate exterior angles are `(3x + 15)°` and `(5x - 25)°`, what is the value of `x`?
- `x = 10`
- `x = 20`
- `x = 15`
- `x = 25`
Answer: b
Q`4`. Calculate the value of `x` if the alternate exterior angles formed by two parallel lines intersected by a transversal are (7x - 30)° and (6x + 5)°.
- `x = 10`
- `x = 15`
- `x = 20`
- `x = 35`
Answer: d
Q`5`. Determine the value of `x` in the figure below.
- `20°`
- `60°`
- `24°`
- `80°`
Answer: c
Frequently Asked Questions
Q`1`.What are alternate exterior angles?
Answer: Alternate exterior angles are pairs of angles formed when a transversal intersects two lines. They are located on opposite sides of the transversal and the exterior of the parallel lines.
Q`2`. How do you identify alternate exterior angles?
Answer: To identify alternate exterior angles, look for pairs of angles that are situated on opposite sides of the transversal and the exterior of the parallel lines.
Q`3`. What is the relationship between alternate exterior angles?
Answer: The relationship between alternate exterior angles is that they are congruent or equal in measure when the lines intersected by the transversal are parallel.
Q`4`.Are alternate exterior angles congruent always?
Answer: No, alternate exterior angles are only congruent when the lines intersected by the transversal are parallel. This is a property described by the Alternate Exterior Angles Theorem.
Q`5`. How can alternate exterior angles be used to determine if lines are parallel?
Answer: If the alternate exterior angles formed by a transversal intersecting two lines have equal measures, then the lines are parallel. This is known as the Converse of the Alternate Exterior Angles Theorem.
