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What Are Alternate Exterior Angles? State The Theorem Related To These Angles

Referencing Styles : Harvard | Pages : 2

Alternate Exterior Angles

This is the type of angle that is created when a transversal crosses two of the parallel lines. Each of the two pairs of these angles is in the exterior position of the parallel lines and they are on the opposite side of the transversal.

Considering the above figure, the traversal AB cross both of the lines PQ and RS. It creates the intersection E and F. Including all the pairs of the exterior angles that are alternate, two of the angles are out of the parallel lines and they are also on the opposite side in case of the transversal. Two different pairs of angles are there. In the above figure, it can be seen.

There are two different cases in case of the alternate exterior angles:

  1. The parallel case

In this case, if the transversal cuts off and crosses through the lines, the alternate angles present there also measure the same as of the alternate exterior angle. As any of the points, A or B is moved it can be seen that the alternate exterior angles have no specific relationship with each other.

The theorem

As two of the lines are parallel to each other. In this case, the alternate exterior angle theorem can be derived. This states that whenever there is a pair of the various parallel line which cuts the traversal and the exterior angles are congruent. This is a useful tool in the case of the parallel line and all the angles that are created with the traversal along the parallel line.


In the above illustration both of the teal-colored line which is parallel to each other. These are the parallel lines in which the theorem tells about the alternate interior angle which is congruent. This means the angles starting from 1 and 8 that are congruent. This means the angles from 2 and 7 are also congruent. The congruent alternate exterior angles occur at the time when both of the lines are parallel to each other. If the lines are not parallel this cannot happen.

Since k ∥ l, by the Corresponding Angles Postulate,

∠ 1 ≅ ∠ 5 ∠ 1 ≅ ∠ 5.

Also, by the Vertical Angles Theorem,

∠ 5 ≅ ∠ 7 ∠ 5 ≅ ∠ 7.

Then, by the Transitive Property of Congruence,

∠ 1 ≅ ∠ 7 ∠ 1 ≅ ∠ 7.

You can prove that ∠4 and ∠6 are congruent using the same method.

Importance of Alternate exterior angles for parallel lines

Lines are explained to be parallel if they are of the same distance but they never meet each other.

When the parallel lines are crossed with other lines it can be seen that the different angles are the same. The concept of an alternate exterior angle is used in case of the intersection of the parallel lines. Different pairs of angles can be derived by using the concept of the alternate exterior angle. If both the angles generated on each of the lines that are parallel, the angle in both the case will be of the same value. It is known that whenever the lines are crossed through the traversal that is parallel the theorem of alternate exterior angle can be used which finds out the alternate exterior angles which are congruent. The problems can be solved by the identification and by the measurement of the alternate exterior angles.

Geometry sets up odd events. In the parallel lines which are crossed by the help of the traversal which hopes the astounded when the event of the improbable leads to a new alternate exterior angle. In any case, if the line is not parallel there would be the existence of the alternate exterior angles.

Example 1:


If it is known that angle measures 130 degrees, the angle of 1 is to be measured.

In this case, angle 8 is outside from the bottom of the parallel line and it is to the right of the traversal. This means that angle 1 is the alternate exterior angle. The theorem of alternate exterior angle also tells the value of the angle is 130 degrees.  

Example 2:

PQ and RS are two lines and UT is a transversal of PQ and RS. Which among the following is a pair of alternate exterior angles?


  1. ∠4 and ∠5
    B. ∠
  2. 2 and ∠ 8
    C. ∠ 1 and ∠ 2
    D. ∠ 1 and ∠ 8

Correct Answer: D


Step 1: When two lines are cut by another line transversal, the pairs of angles formed outside the two lines and on the opposite of the transversal are called Alternate Exterior Angles.
Step 2: From the figure, PQ and RS are parallel lines and UT is a transversal passing through the parallel lines.
Step 3: The pairs of alternate exterior angles are
∠ 1 and ∠ 8
∠ 2 and ∠ 7
Step 4: Hence, ∠ 1 and ∠ 8 are the alternate exterior angles in the given figure.

Example 3:

The map below shows three roads in Julio’s town.


Julio used a surveying tool to measure two angles at the intersections in this picture he drew (NOT to scale). Julio wants to know if Franklin Way is parallel to Chavez Avenue.

The 130∘ angles and ∠  are alternate exterior angles. I m ∠ a = 130∘, then the lines are parallel.

∠ a + 40∘ ∠ a = 180∘ = 140∘ by the Linear Pair Postulate

140∘ ≠ 130∘, so Franklin Way and Chavez Avenue are not parallel streets.

Example 4

Which lines are parallel if ∠ AFG ≅ ∠ IJM?


These two angles are alternate exterior angles so if they are congruent it means that CG ←→ || HK ←→.

Example 5

Find the measure of each angle and the value of y.


The angles are alternate exterior angles. Because the lines are parallel, the angles are equal.

(3y + 53)∘ 108 = 4y 27 = y = (7y−55)∘

If y = 27, then each angle is [3 (27) + 53]∘ = 134∘.

Example 6


Find the values of the angles b, c, d, e, f, g and h in the figure given below.



Step 1: b is a supplement of 45oo.

Therefore, b + 45o =180o => b = 180o - 45o = 135o

Step 2: b and c are vertical angles.

Therefore, c = b = 135o

Step 3: d and 45oo are vertical angles.

Therefore, d = 45o

Step 4: d and e are alternate interior angles.

Therefore, e = d = 45o

Step 5: f and e are supplementary angles.

Therefore, f + 45o = 180o => f = 180o - 45o = 135o

Step 6: g and f are vertical angles.

Therefore, g = f = 135o

Step 7: h and e are vertical angles.

Therefore, h = e = 45o

Example 7:

In the given figure, find the value of x.


Solution: We know that the sum of the exterior angles of a polygon is 360 degrees.

Thus, 70° + 60° + 65° + 40° + x = 360°

235° + x = 360°

X = 360° – 235° = 125°

Example 8:

 Identify the type of regular polygon whose exterior angle measures 120 degrees.

Solution: Since the polygon is regular, the measure of all the interior angles is the same. Therefore, all its exterior angles measure the same as well, that is, 120 degrees.

Since the sum of exterior angles is 360 degrees and each one measures 120 degrees, we have,

Number of angles = 360/120 = 3

Since the polygon has 3 exterior angles, it has 3 sides. Hence it is an  equilateral triangle.

Example 9:

Use the Alternate Exterior Angle theorem to find the measure of angle x, angle y, and angle z. Assume the lines are parallel.

First, we need to identify a pair of alternate interior angles. Angle y and 58 are on opposite sides of the transversal and inside the parallel lines, so they must be alternate interior. Since the lines are parallel, angle y = 58.

Next, notice that angle x and 58 form a straight angle. Since a straight angle measures 180 degrees, angle x + 58 = 180 and 180 - 58 = angle x, so angle x = 122.

Finally, angle x and angle z are alternate interior angles, and we know that alternate interior angles are equal. So, angle x = 122 then angle z = 122.

Example 10:




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