WHAT ARE ALTERNATE EXTERIOR ANGLES? (WITH EXAMPLES) - MATHS - 2025

The alternate exterior angles are the angles that are formed when two parallel lines are intercepted with a secant line. In addition to these angles, another pair is formed which are called alternate interior angles.

The difference between these two concepts are the words "external" and "internal" and as the name indicates, the alternate external angles are those that are formed outside the two parallel lines.

Graphical representation of alternate exterior angles

As can be seen in the previous image, there are eight angles formed between the two parallel lines and the secant line. The red angles are the alternate exterior angles, and the blue angles are the alternate interior angles.

characteristics

In the introduction we already explained what alternate exterior angles are. Besides being the external angles between the parallels, these angles fulfill another condition.

The condition they fulfill is that the alternate exterior angles formed on a parallel line are congruent; It has the same measure as the other two that are formed on the other parallel line.

But each alternate exterior angle is congruent with the one on the other side of the secant line.

What are the congruent alternate exterior angles?

If the image of the beginning and the previous explanation are observed, it can be concluded that the alternate exterior angles that are congruent with each other are: angles A and C, and angles B and D.

To show that they are congruent, we must use properties of angles such as: opposite angles by the vertex and alternate interior angles.

Examples

Below are a series of examples where the definition and property of congruence of alternate exterior angles should be applied.

First example

In the image below, what is the measure of angle A knowing that angle E measures 47 °?

Solution

As explained before, the angles A and C are congruent because they are alternate exteriors. Therefore, the measure of A is equal to the measure of C. Now, since the angles E and C are opposite angles by the vertex, they have the same measure, therefore, the measure of C is 47 °.

In conclusion, the measure of A is equal to 47 °.

Second example

Find the measure of angle C shown in the following image, knowing that angle B measures 30 °.

Solution

In this example, the definition supplementary angles is used. Two angles are supplementary if the sum of their measures is equal to 180 °.

The image shows that A and B are supplementary, therefore A + B = 180 °, that is, A + 30 ° = 180 ° and therefore A = 150 °. Now, since A and C are alternate exterior angles, then their measures are the same. Therefore, the measure of C is 150 °.

Third example

In the image below, the measure of angle A is 145 °. What is the measure of angle E?

Solution

The image shows that angles A and C are alternate exterior angles, therefore, they have the same measure. That is, the measure of C is 145 °.

Since angles C and E are supplementary angles, we have that C + E = 180 °, that is, 145 ° + E = 180 ° and therefore the measure of angle E is 35 °.

References

1. Bourke. (2007). An Angle on Geometry Math Workbook. NewPath Learning.
2. CEA (2003). Elements of geometry: with numerous exercises and geometry of the compass. University of Medellin.
3. Clemens, SR, O'Daffer, PG, & Cooney, TJ (1998). Geometry. Pearson Education.
4. Lang, S., & Murrow, G. (1988). Geometry: A High School Course. Springer Science & Business Media.
5. Lira, A., Jaime, P., Chavez, M., Gallegos, M., & Rodríguez, C. (2006). Geometry and trigonometry. Threshold Editions.
6. Moyano, AR, Saro, AR, & Ruiz, RM (2007). Algebra and Quadratic Geometry. Netbiblo.
7. Palmer, CI, & Bibb, SF (1979). Practical math: arithmetic, algebra, geometry, trigonometry, and slide rule. Reverte.
8. Sullivan, M. (1997). Trigonometry and analytical geometry. Pearson Education.
9. Wingard-Nelson, R. (2012). Geometry. Enslow Publishers, Inc.