How to find the Exterior Angle of a Triangle in 3 Easy Steps

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Key Points

The Exterior Angle of a Triangle Theorem states that the measure of an exterior angle at a vertex of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle, applicable to all types of triangles.
An exterior angle of a triangle is formed by extending one side of the triangle, and its measure is equal to the sum of its two non-adjacent interior angles, playing a significant role in geometry for proving theorems, solving problems, and determining triangle congruence.
The text provides examples of calculating exterior angles using given interior angles, presents the formula that the exterior angle is equal to the sum of its opposite interior angles, and notes that the sum of exterior angles in a triangle is always 360 degrees.

Here’s how to Solve Exterior Angles of a Triangle

The Exterior Angle of a Triangle is created by the extension of one side of the triangle and the adjacent side. There are two rules for solving for an Exterior Angle of a Triangle. The first rule states that the Exterior Angle of a Triangle is equal to the sum of the two non adjacent angles. The second rule states that the Exterior Angle of a Triangle is supplementary to the adjacent angle. That means the sum of the two angles will be equal to 180 degrees. You can use either rule for finding the Exterior Angle of a Triangle by determining what information is given to you in the problem and then using the correct rule to solve.

Common Core Standard: 8.G.5
Related Topics: Pythagorean Theorem, Parallel Lines Cut by a Transversal, Triangle Angle Sum, Volume of a Cylinder, Volume of a Cone, Volume of a Sphere
Return To: Home, 8th Grade

Exterior Angle of a Triangle Theorem

The Exterior Angle of a Triangle Theorem, also known as the high school exterior angle theorem (HSEAT), is a fundamental concept in geometry. It states that the measure of an exterior angle at a vertex of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle.

In other words, if you extend one side of a triangle, the angle formed outside the triangle at the vertex is equal to the sum of the two angles inside the triangle that are not adjacent to it. This theorem applies to all triangles, whether they are acute, right, or obtuse.

The Exterior Angle of a Triangle Theorem can be expressed mathematically using the following formula:

∠1 = ∠2 + ∠3

Where:

∠1 is the exterior angle at the vertex
∠2 and ∠3 are the two remote interior angles of the triangle

This theorem is important because it helps to solve various problems related to triangles. For example, if you know the measures of two interior angles of a triangle, you can use the Exterior Angle of a Triangle Theorem to find the measure of the third angle.

The proof of the Exterior Angle of a Triangle Theorem is relatively simple. It involves drawing a line parallel to one side of the triangle, creating a transversal, and using the properties of parallel lines and alternate angles to show that the exterior angle is equal to the sum of the two remote interior angles.

In conclusion, the Exterior Angle of a Triangle Theorem is a fundamental concept in geometry that helps to solve various problems related to triangles. It states that the measure of an exterior angle at a vertex of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle.

What is an Exterior Angle of a Triangle?

An exterior angle of a triangle is the angle formed by any side of a triangle and the extension of its adjacent side. In other words, it is an angle that is formed outside a triangle by extending one of its sides.

Interior Angles

Before discussing exterior angles, it is important to understand interior angles. An interior angle is formed by two adjacent sides of a triangle and is located inside the triangle. The sum of all three interior angles of a triangle is always equal to 180 degrees. This is known as the Triangle Angle Sum Theorem.

Exterior Angles

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent interior angles. This is known as the Exterior Angle Theorem.

For example, if angle A is an exterior angle of triangle ABC, then its measure is equal to the sum of angles B and C. Mathematically, we can write this as:

Angle A = Angle B + Angle C

It is important to note that the measure of an exterior angle is always greater than the measure of either of its corresponding interior angles.

Exterior angles play an important role in geometry, particularly in the study of triangles. They are used to prove theorems, solve problems, and determine the congruence of triangles.

In summary, an exterior angle of a triangle is formed by extending one of its sides, and its measure is equal to the sum of the measures of its two non-adjacent interior angles.

Exterior Angles of a Triangle Solution

3 Simple Exterior Angle of a Triangle Examples

The exterior angle of a triangle is an angle formed by extending one of the sides of a triangle to meet a line that is parallel to the opposite side. In other words, it is the angle between a side of a triangle and the adjacent side of a triangle that is not between the two angles.

The three angles in a triangle add up to 180 degrees.
The degree measure of a straight line add up to 180 degrees.
If you are given the adjacent angle you subtract the angle from 180 degrees to solve for the exterior angle of a triangle.
If you are given the two non adjacent angles you add them together to solve for the exterior angle of a triangle.

Here are a few examples of how to calculate the exterior angle of a triangle:

Example 1

Suppose you have a triangle with angles measuring 50°, 70°, and 60°. To find the exterior angle of the triangle at the vertex that corresponds to the 50° angle, follow these steps:

Identify the side of the triangle that is opposite to the 50° angle, which is the side that is not between the two angles.
Extend this side beyond the vertex.
Draw a line parallel to the opposite side that intersects the extended side.
The exterior angle is the angle between the extended side and the parallel line.

In this case, the exterior angle is equal to the sum of the two opposite interior angles, which are 70° and 60°. Therefore, the exterior angle is equal to 130°.

Example 2

Suppose you have a triangle with angles measuring 40°, 60°, and 80°. To find the exterior angle of the triangle at the vertex that corresponds to the 40° angle, follow these steps:

Identify the side of the triangle that is opposite to the 40° angle, which is the side that is not between the two angles.
Extend this side beyond the vertex.
Draw a line parallel to the opposite side that intersects the extended side.
The exterior angle is the angle between the extended side and the parallel line.

In this case, the exterior angle is equal to the sum of the two opposite interior angles, which are 60° and 80°. Therefore, the exterior angle is equal to 140°.

Example 3

Suppose you have a triangle with angles measuring 80°, 50°, and 50°. To find the exterior angle of the triangle at the vertex that corresponds to the 80° angle, follow these steps:

Identify the side of the triangle that is opposite to the 80° angle, which is the side that is not between the two angles.
Extend this side beyond the vertex.
Draw a line parallel to the opposite side that intersects the extended side.
The exterior angle is the angle between the extended side and the parallel line.

In this case, the exterior angle is equal to the sum of the two opposite interior angles, which are 50° and 50°. Therefore, the exterior angle is equal to 100°.

5 Quick Exterior Angle of a Triangle Practice Problems

Exterior Angle of a Triangle Formula

The exterior angle of a triangle is formed when a side of a triangle is extended. The angle is formed outside the triangle by extending one of its sides. The exterior angle of a triangle is equal to the sum of its two opposite interior angles.

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. The remote interior angles are the two angles that are not adjacent to the exterior angle. The formula for the exterior angle of a triangle is:

Exterior Angle = Sum of Interior Opposite Angles

The exterior angle of a triangle can be found using the above formula. If we know the measure of one of the interior angles of a triangle and the measure of the exterior angle, we can find the measure of the other interior angle using the following formula:

Interior Angle = Exterior Angle - Opposite Interior Angle

In a triangle, the sum of all interior angles is always equal to 180 degrees. Therefore, the sum of the two remote interior angles of a triangle is equal to the measure of the third interior angle.

The exterior angle of a triangle can also be used to find a missing angle. If we know the measures of the two remote interior angles, we can use the following formula to find the measure of the missing angle:

Missing Angle = 180 - Exterior Angle

In summary, the exterior angle of a triangle is formed by extending one of its sides. The exterior angle is equal to the sum of its two opposite interior angles. The exterior angle theorem provides us with a formula to find the measure of the exterior angle of a triangle. We can also use the exterior angle to find a missing angle in a triangle.

Exterior Angle of a Triangle is Equal to What?

The exterior angle of a triangle is formed when a side of the triangle is extended. This angle is formed outside the triangle, opposite to the vertex of the extended side. The exterior angle of a triangle is equal to the sum of its remote interior angles. This property is known as the Exterior Angle Theorem.

In other words, if a triangle has sides AB, BC, and AC, and the side BC is extended to form angle ACD, then the exterior angle ACD is equal to the sum of the remote interior angles, which are angles A and B. Mathematically, we can express this as:

∠ACD = ∠A + ∠B

This property holds true for all triangles, regardless of their size or shape. The exterior angle theorem is a fundamental property of triangles and is used in various geometric proofs and calculations.

It is also important to note that the exterior angle and its adjacent interior angle are supplementary angles, which means they add up to 180 degrees. This property can be helpful in solving problems involving exterior angles of triangles.

In summary, the exterior angle of a triangle is equal to the sum of its remote interior angles. This property is known as the Exterior Angle Theorem and is a fundamental concept in geometry. The exterior angle and its adjacent interior angle are supplementary angles, which can be useful in solving problems involving triangles.

FAQ about Exterior Angle of a Triangle

What is the exterior angle formula?

The formula for the exterior angle of a triangle is the sum of its two interior opposite angles. That is, the exterior angle is equal to the sum of the two non-adjacent interior angles of the triangle.

What is the formula for exterior angle theorem of a triangle?

The formula for the exterior angle theorem of a triangle is that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. This theorem is useful in solving problems related to triangles.

Does the exterior of a triangle equal 360?

No, the sum of the exterior angles of a triangle is always equal to 360 degrees. This is known as the Exterior Angle Sum Theorem.

How many exterior angles does a triangle have?

A triangle has three exterior angles, one at each vertex.

What is the exterior angle theorem formula?

The formula for the exterior angle theorem is that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles of the triangle. This theorem is useful in solving problems related to triangles.

What is the exterior angle property?

The exterior angle property states that an exterior angle of a triangle is greater than either of the two non-adjacent interior angles. In other words, the exterior angle is supplementary to the adjacent interior angle.

Does the exterior of a triangle equal 180?

No, the sum of the exterior angles of a triangle is always equal to 360 degrees. The sum of the interior angles of a triangle is always equal to 180 degrees.

Is the exterior of a triangle 80 degrees?

The measure of the exterior angle of a triangle depends on the measures of the two non-adjacent interior angles. It is not always 80 degrees.

Exterior Angle of a Triangle Worksheet: Video Explanation

Watch our free video on how to solve Triangle Exterior Angle Theorem. This video shows how to solve problems that are on our free Triangle Exterior Angle Theorem worksheet that you can get by submitting your email above.

Watch the free Exterior Angle of a Triangle video on YouTube here: Exterior Angle of a Triangle

Video Transcript:
This video is about exterior angles of a triangle. Here we are at our exterior angles of a triangle worksheet that you can get our on website and we’re going to do a couple practice problems from it.

We’re looking at our first problem about exterior angle theorem examples here which gives us our triangle and it also gives us our exterior angle here. You can see that we have our triangle and then we have a line that’s extended from one side of the triangle and then we’re trying to find the angle in between that line and the side of the triangle.

Now in the case of this problem you have to know that all lines are all straight lines add up to 180 degrees. This right here the entire angle from one side of the line to the other is 180 degrees. When they give us a side of the triangle and they tell us that the angle of this side from here to here is a hundred and thirteen degrees that only leave.

Many degrees for this angle here so in order to find that out or in order to figure out the degree measure for this angle, all you have to do is you have to take whatever they give you in the case of this problem it’s a hundred and thirteen. We know this much of the 180 is already taken, in order to figure out what’s left over you just do 180 minus 113.

We know the entire thing from here to here has to be 180 degrees from here to here is 11, you just subtract that from 180. In the case of this problem you will be left with 67 degrees. We know that X has to be equal to 67 degrees because 113 plus 67 is 180 or it’s one full line which is 180 degrees.

Number four on the triangle exterior angles worksheet gives us another triangle and a line and extends from that triangle and then it tells us to find the exterior angles of triangles that is created by that extension. Here is our angle that we are looking for now in the case of this one we don’t have an angle here and the other problem we could have just taken this angle subtracted it from 180 and it would have given us the angle that we need to figure out but in the case of this problem we do not have that.

The easiest way to do this is to take the two opposite or non adjacent angles from the exterior angle and you add them together. In the case of this triangle we have a right angle here which is of course 90 degrees and we also have 33 degrees. We’re going to take ninety degrees and we’re going to add it to 33 degrees to get 123 degrees.

We know that this angle and this angle add up to 120 degrees and we know that all the angles in a triangle have to add up to 180. We could have done 180 minus 123 which gets us 57 degrees, we now know that this question mark angle is 57 degrees if you remember back to the original problem this entire thing is a straight line. It all has to add up to 180 degrees, if this amount is 57 you can do 180 minus 57 to get the remaining amount. In this case 180 minus 57 is a hundred and 23 degrees, so we know that X is 123 degrees. This is an explanation how to find angles of a triangle.

Now you may notice that this amount and this amount are the same they are equal to each other they are exactly the same, and a shortcut for solving the exterior angle of a triangle. When they give you the two non adjacent angles is to just add them together and we could have skipped these entire steps down here had we just known the rule was to add them together and that’s going to be your exterior angle measure.

If you ever get a problem like this again and you have non adjacent angles to the exterior angle you just add them up and whatever they add up to is the angle measure of the exterior angle of triangle. Try all the practice problems by downloading the free triangle exterior angle worksheet above.