Lesson Explainer: Interior and Exterior Angles of Triangles Mathematics • First Year of Preparatory School

In this explainer, we will learn how to complete geometric proofs using the angle sum of a triangle and find interior and exterior angles of triangles.

We know that the sum of the measures of the interior angles in any triangle is , and we can prove that this is indeed the case. First, we note that we cannot have a reflex angle in a triangle, so we do not need to check if each angle has a measure less than . To show that has interior angles whose measures sum to , we construct a line parallel to that passes through . We call this , as shown.

We now see that and are line segments drawn between two parallel lines (i.e., they are transversals); hence, their alternate interior angles will have equal measures, as demonstrated below.

Thus, and . We can then see that , , and combine to make a straight angle. So, their measures sum to .

Therefore,

Substituting in the congruent angles, we have

Thus, we have proved that the measures of all the angles of a triangle must add up to .

Let’s see a specific instance of applying the method of this proof to a triangle to determine the measures of its interior angles.

Example 1: Proving the Angle Sum of a Triangle

In the given figure, if , , and , find the measure of .

Answer

We start by noting that and are transversals between two parallel lines. This means that the alternate interior angles will be congruent.

Thus, and . We are told that , so . We are also told that , so .

Since the angles at make a straight angle, we can note that

We can now check for the sum of the angles in triangle :

This shows that the sum of the angles in this triangle is .

It is worth noting that we could have directly found the result in the previous example using the fact that the sum of the measures of the internal angles in a triangle is , rather than using the fact that a straight angle has a measure of .

In our next example, we will see a geometric property that we can prove using the measures of the interior angles in a triangle.

Example 2: Completing a Proof about Triangles

In the following triangle , if and , find .

Answer

We want to find the measure of . We can do this by recalling that the sum of the measures of the interior angles in a triangle is . Applying this to , we have

Adding in the measures of the angles to the diagram, we have the following.

We have that , , and . Substituting these into the equation above gives us

We can then simplify and solve the equation:

Finally, we can find by noting that it is the sum of the measures of the two interior angles at :

We can extend the method of the proof in the above example to show that if we have a triangle cut into two triangles by a line segment such that the line segment divides the angles as shown, then it must be a right triangle at the vertex cut by that line segment.

We want to show that . We can do this using the fact that the sum of the measures of the interior angles in a triangle is . Applying this to , we have

We note that , so

Dividing the equation through by 2 gives

It can also be useful to consider the exterior angles at each vertex of a triangle instead of only the interior angles. Let’s start by defining what we mean by the exterior angles of a triangle.

Definition: Exterior Angle of a Triangle

An exterior angle of a triangle is the angle formed outside the triangle between any side and the extension of another side.

It is worth noting that there are two exterior angles of a triangle at each vertex. For example, we can extend side or side .

We can note that both of the exterior angles at make a straight angle with , so they have equal measure. Since both exterior angles at a vertex are congruent, we often refer to either of the exterior angles as the exterior angle. We can also say that the exterior angle at a vertex of a triangle is the supplementary angle to its adjacent interior angle.

We can use this knowledge that the sum of the measures of the interior and exterior angles at a vertex of a triangle is to show a useful property. We note that the sum of the measures of the internal angles in a triangle is . Thus,

We can also see that

From the two equations, we get that

We have shown that the measure of the exterior angle at a vertex of a triangle is equal to the sum of the measures of the opposite interior angles in the triangle. We can state this formally as follows.

Property: Measure of an External Angle in a Triangle

The measure of any exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles in the triangle:

Let’s see an example of using this property to find the measure of an exterior angle in a triangle given the measures of the opposite interior angles.

Example 3: Finding the Exterior Angle of a Triangle given the Opposite Interior Angles

In the given figure, find the measure of the exterior angle .

Answer

First, we recall that the exterior angle of a triangle at a vertex is the angle between one side of the triangle at the vertex and the extension of the other side. We can answer this question using two different methods.

We recall that the measure of any exterior angle of a triangle is equal to the sum of the measures of the opposite interior angles in the triangle. The angles at are opposite the angles at and . So,

Alternatively, we can recall that the sum of the measures of the interior angles in a triangle is . Thus,

The exterior angle and combine to make a straight angle, so

In our next example, we will determine the measure of an angle in a triangle using the properties of parallel lines and exterior angles.

Example 4: Finding a Missing Angle Using External Angles of a Triangle

Find the measure of .

Answer

There are many different ways of determining the measure of . For example, we can note that is a transversal of parallel lines, so the alternate interior angles are congruent.

Thus, . We then recall that the sum of the measures of the internal angles in a triangle is . Applying this to yields

Alternatively, we can note that is also a transversal of parallel lines, so the following corresponding angles must be congruent.

We then recall that the sum of the measures of angles on a straight line is . Hence,

In our final example, we will show that the sum of the measures of the exterior angles in any triangle remains constant and will determine its value.

Example 5: The Sum of the Exterior Angles of a Triangle

What is the sum of the measures of the exterior angles of a triangle?

Answer

To answer this question, let’s start by sketching a triangle with its exterior angles marked and their measures labeled , , and , as shown.

It is worth noting here that there are two exterior angles at each vertex; however, they are congruent, so we only consider one exterior angle at each vertex.

We recall that the sum of the measures of the interior angles in the triangle is . So,

We can also note that the exterior and interior angles at a vertex are supplementary (their measures sum to ). Thus,

We can rearrange each equation to make the measure of the interior angle the subject. We get

Substituting these expressions into our equation for the sum of the measures of the internal angles in a triangle yields

Rearranging and simplifying then gives us

Hence, the sum of the measures of the exterior angles in a triangle is .

In the previous example, we showed that the sum of the measures of the exterior angles in a triangle is using the fact that the sum of the measures of the interior angles in a triangle is .

It is worth noting that we can derive these results in the opposite order. To do this, we consider the exterior angles in a triangle as rotations.

If we follow each of these rotations in turn, we see that we make exactly one full turn.

Thus, the sum of their measures must be .

If we call these measures , , and , we can use this result to show that the sum of the measures of the interior angles is . We have

Adding the left- and right-hand sides of the three equations together gives

Using the fact that , we have

Let’s finish by recapping some of the important points from this explainer.