WHAT ARE ALTERNATE INTERIOR ANGLES? (WITH EXERCISES) - MATHS - 2024

The alternate interior angles are those angles formed by the intersection of two parallel lines and a transverse line. When a line L1 is cut by a transverse line L2, 4 angles are formed.

The two pairs of angles that are on the same side of the line L1 are called supplementary angles, since their sum is equal to 180º.

In the previous image, angles 1 and 2 are supplementary, as are angles 3 and 4.

To be able to speak of alternate interior angles it is necessary to have two parallel lines and a transversal line; As seen before, eight angles will be formed.

When you have two parallel lines L1 and L2 cut by a transverse line, eight angles are formed, as illustrated in the following image.

In the previous image the pairs of angles 1 and 2, 3 and 4, 5 and 6, 7 and 8 are supplementary angles.

Now, the alternate interior angles are the ones between the two parallel lines L1 and L2, but they are located on opposite sides of the transverse line L2.

That is, angles 3 and 5 are alternate interiors. Similarly, angles 4 and 6 are alternate interior angles.

Opposite angles by the vertex

To know the usefulness of alternate interior angles, it is first necessary to know that if two angles are opposite each other at the vertex, then these two angles measure the same.

For example, angles 1 and 3 have the same measure when they are opposite each other at the vertex. Under the same reasoning it can be concluded that angles 2 and 4, 5 and 7, 6 and 8 measure the same.

Angles formed between a secant and two parallels

When you have two parallel lines cut by a secant or transversal line as in the previous figure, it is true that angles 1 and 5, 2 and 6, 3 and 7, 4 and 8 measure the same.

Alternate internal angles

Using the definition of angles set by the vertex and the property of the angles formed between a secant and two parallel lines, it can be concluded that the alternate interior angles have the same measure.

Exercises

First exercise

Calculate the measure of angle 6 in the following image, knowing that angle 1 measures 125º.

Solution

Since angles 1 and 5 are opposite each other at the vertex, we have that angle 3 measures 125º. Now, since angles 3 and 5 are alternate interiors, we have that angle 5 also measures 125º.

Finally, since angles 5 and 6 are supplementary, the measure of angle 6 is equal to 180º - 125º = 55º.

Second exercise

Calculate the measure of angle 3 knowing that angle 6 measures 35º.

Solution

It is known that angle 6 measures 35º, and it is also known that angles 6 and 4 are internal alternates, therefore they measure the same. In other words, angle 4 measures 35º.

On the other hand, using the fact that angles 4 and 3 are supplementary, we have that the measure of angle 3 is equal to 180º - 35º = 145º.

Observation

It is necessary that the lines be parallel so that they can fulfill the corresponding properties.

The exercises may perhaps be solved faster, but in this article we wanted to use the property of alternate interior angles.

References

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