The portal has been deactivated. Please contact your portal admin.

Lesson Explainer: Interior and Exterior Angles of Triangles Mathematics

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 180∘, 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 180∘. To show that △𝐴𝐡𝐢 has interior angles whose measures sum to 180∘, we construct a line parallel to ⃖⃗𝐴𝐡 that passes through 𝐢. We call this ⃖⃗𝐷𝐸, as shown.

We now see that 𝐴𝐢 and 𝐡𝐢 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, π‘šβˆ π΅=π‘šβˆ πΈπΆπ΅ and π‘šβˆ π΄=π‘šβˆ π·πΆπ΄. We can then see that ∠𝐷𝐢𝐴, ∠𝐴𝐢𝐡, and ∠𝐸𝐢𝐡 combine to make a straight angle. So, their measures sum to 180∘.

Therefore, π‘šβˆ π·πΆπ΄+π‘šβˆ πΈπΆπ΅+π‘šβˆ π΄πΆπ΅=180.∘

Substituting in the congruent angles, we have π‘šβˆ π΅π΄πΆ+π‘šβˆ π΄π΅πΆ+π‘šβˆ π΄πΆπ΅=180.∘

Thus, we have proved that the measures of all the angles of a triangle must add up to 180∘.

Theorem: Sum of Interior Angle Measures in a Triangle

The sum of all of the measures of the interior angles in a triangle is 180∘.

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 𝐴𝐢⫽𝐷𝐸, π‘šβˆ π΄π΅πΈ=55∘, and π‘šβˆ πΆ=75∘, find the measure of ∠𝐴𝐡𝐢.

Answer

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

Thus, π‘šβˆ π΄=π‘šβˆ πΈπ΅π΄ and π‘šβˆ πΆ=π‘šβˆ π·π΅πΆ. We are told that π‘šβˆ π΄π΅πΈ=55∘, so π‘šβˆ π΄=55∘. We are also told that π‘šβˆ πΆ=75∘, so π‘šβˆ π·π΅πΆ=75∘.

Since the angles at 𝐡 make a straight angle, we can note that π‘šβˆ π΄π΅πΆ+55+75=180π‘šβˆ π΄π΅πΆ=180βˆ’75βˆ’55=50.∘∘∘∘∘∘∘

We can now check for the sum of the angles in triangle 𝐴𝐡𝐢: π‘šβˆ π΅π΄πΆ+π‘šβˆ π΄πΆπ΅+π‘šβˆ π΄π΅πΆ=55+75+50=180.∘∘∘∘

This shows that the sum of the angles in this triangle is 180∘.

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 180∘, rather than using the fact that a straight angle has a measure of 180∘.

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 𝐴𝐡𝐢, if π‘šβˆ πΆ=π‘šβˆ πΆπ΄π·=43∘ and π‘šβˆ π΅=π‘šβˆ π΅π΄π·, find π‘šβˆ π΅π΄πΆ.

Answer

We want to find the measure of ∠𝐡𝐴𝐢. We can do this by recalling that the sum of the measures of the interior angles in a triangle is 180∘. Applying this to △𝐴𝐡𝐢, we have π‘šβˆ π΅π΄πΆ+π‘šβˆ π΅+π‘šβˆ πΆ=180.∘

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

We have that π‘šβˆ π΅π΄πΆ=43+π‘šβˆ π΅π΄π·βˆ˜, π‘šβˆ π΅=π‘šβˆ π΅π΄π·, and π‘šβˆ πΆ=43∘. Substituting these into the equation above gives us 43+π‘šβˆ π΅π΄π·+π‘šβˆ π΅π΄π·+43=180.∘∘∘

We can then simplify and solve the equation: 2π‘šβˆ π΅π΄π·+86=1802π‘šβˆ π΅π΄π·=180βˆ’86=94π‘šβˆ π΅π΄π·=942=47.∘∘∘∘∘∘∘

Finally, we can find π‘šβˆ π΅π΄πΆ by noting that it is the sum of the measures of the two interior angles at 𝐴: π‘šβˆ π΅π΄πΆ=π‘šβˆ πΆπ΄π·+π‘šβˆ π΅π΄π·=43+47=90.∘∘∘

We can extend the method of the proof in the above example to show that if we have a triangle 𝐴𝐡𝐢 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 π‘₯+𝑦=90∘. We can do this using the fact that the sum of the measures of the interior angles in a triangle is 180∘. Applying this to △𝐴𝐡𝐢, we have π‘₯+𝑦+π‘šβˆ π΄=180.∘

We note that π‘šβˆ π΄=π‘₯+𝑦, so π‘₯+𝑦+(π‘₯+𝑦)=1802π‘₯+2𝑦=180.∘∘

Dividing the equation through by 2 gives π‘₯+𝑦=90.∘

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 𝐴𝐡 or side 𝐡𝐢.

We can note that both of the exterior angles at 𝐡 make a straight angle with ∠𝐴𝐡𝐢, 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 180∘ to show a useful property. We note that the sum of the measures of the internal angles in a triangle is 180∘. Thus, π‘šβˆ π΄+π‘šβˆ π΅+π‘šβˆ πΆ=180.∘

We can also see that π‘šβˆ π΅+π‘šβˆ πΆπ΅π·=180.∘

From the two equations, we get that π‘šβˆ π΄+π‘šβˆ πΆ=π‘šβˆ πΆπ΅π·.

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: π‘₯=𝑦+𝑧.

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 ∠𝐢.

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 𝐢 are opposite the angles at 𝐴 and 𝐡. So, π‘₯=50+55=105.∘∘∘

Alternatively, we can recall that the sum of the measures of the interior angles in a triangle is 180∘. Thus, 50+55+π‘šβˆ π΅πΆπ΄=180π‘šβˆ π΅πΆπ΄=180βˆ’50βˆ’55=75.∘∘∘∘∘∘∘

The exterior angle and ∠𝐡𝐢𝐴 combine to make a straight angle, so 75+π‘₯=180π‘₯=180βˆ’75=105.∘∘∘∘∘

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 ∠𝐴𝐢𝐡.

Answer

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

Thus, π‘šβˆ π΄=65∘. We then recall that the sum of the measures of the internal angles in a triangle is 180∘. Applying this to △𝐴𝐡𝐢 yields 35+65+π‘₯=180π‘₯=180βˆ’65βˆ’35=80.∘∘∘∘∘∘∘

Alternatively, we can note that 𝐡𝐢 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 180∘. Hence, 35+65+π‘₯=180π‘₯=80.∘∘∘∘

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 𝐴𝐡𝐢 with its exterior angles marked and their measures labeled π‘₯, 𝑦, and 𝑧, 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 180∘. So, π‘šβˆ π΄+π‘šβˆ π΅+π‘šβˆ πΆ=180.∘

We can also note that the exterior and interior angles at a vertex are supplementary (their measures sum to 180∘). Thus, π‘₯+π‘šβˆ π΄=180,𝑦+π‘šβˆ π΅=180,𝑧+π‘šβˆ πΆ=180.∘∘∘

We can rearrange each equation to make the measure of the interior angle the subject. We get π‘šβˆ π΄=180βˆ’π‘₯,π‘šβˆ π΅=180βˆ’π‘¦,π‘šβˆ πΆ=180βˆ’π‘§.∘∘∘

Substituting these expressions into our equation for the sum of the measures of the internal angles in a triangle yields (180βˆ’π‘₯)+(180βˆ’π‘¦)+(180βˆ’π‘§)=180540βˆ’π‘₯βˆ’π‘¦βˆ’π‘§=180.∘∘∘∘∘∘

Rearranging and simplifying then gives us 540βˆ’180=π‘₯+𝑦+𝑧360=π‘₯+𝑦+𝑧.∘∘∘

Hence, the sum of the measures of the exterior angles in a triangle is 360∘.

In the previous example, we showed that the sum of the measures of the exterior angles in a triangle is 360∘ using the fact that the sum of the measures of the interior angles in a triangle is 180∘.

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 360∘.

If we call these measures π‘₯, 𝑦, and 𝑧, we can use this result to show that the sum of the measures of the interior angles is 180∘. We have π‘₯+π‘šβˆ π΄=180,𝑦+π‘šβˆ π΅=180,𝑧+π‘šβˆ πΆ=180.∘∘∘

Adding the left- and right-hand sides of the three equations together gives π‘₯+π‘šβˆ π΄+𝑦+π‘šβˆ π΅+𝑧+π‘šβˆ πΆ=180+180+180π‘₯+𝑦+𝑧+π‘šβˆ π΄+π‘šβˆ π΅+π‘šβˆ πΆ=540.∘∘∘∘

Using the fact that π‘₯+𝑦+𝑧=360∘, we have 360+π‘šβˆ π΄+π‘šβˆ π΅+π‘šβˆ πΆ=540.π‘šβˆ π΄+π‘šβˆ π΅+π‘šβˆ πΆ=540βˆ’360=180.∘∘∘∘∘

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

Key Points

  • The sum of the measures of the interior angles in a triangle is 180∘. We can prove this result by drawing a line parallel to the base of the triangle through the other vertex and using the fact that alternate interior angles are congruent.
  • An exterior angle of a triangle is the angle formed outside the triangle between any side and the extension of another side. Both exterior angles at a vertex of a triangle are congruent.
  • An interior angle in a triangle and an adjacent exterior angle are supplementary.
  • 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: π‘₯=𝑦+𝑧.
  • The sum of the measures of the exterior angles in a triangle is 360∘.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.