Lesson Video: Exterior Angles of a Triangle Mathematics • 8th Grade

Video Transcript

In this video, we will learn how to find the measure of the exterior angle of a triangle and compare between an exterior angle and its corresponding remote interior angles. We will begin by defining what we mean by an exterior and interior angle.

An interior angle is the angle between adjacent sides of a polygon. An exterior angle is the angle between one side of a polygon and the extension of an adjacent side. Whilst these definitions are true for all polygons, in this video, we will only deal with triangles. We can show these angles by considering the triangle 𝐴𝐵𝐶 shown. There are three interior angles at vertex 𝐴, vertex 𝐵, and vertex 𝐶. We can show the exterior angle by extending one of the sides. In this case, we have extended 𝐶𝐵. The exterior angle is the angle made between this extended line and the side length 𝐴𝐵. We will now look at a question that will lead us to some important properties of interior and exterior angles.

The figure shows the exterior angles of a given triangle. Find the sum of the exterior angles.

We recall that an exterior angle of a triangle lies between one side of the triangle and an extended adjacent side. In the figure drawn, we have a triangle 𝐴𝐵𝐶. The line 𝐴𝐵 has been extended to create the exterior angle 108.4 degrees. The side 𝐶𝐴 has been extended to create 116.6 degrees. Finally, our third exterior angle of 135 degrees has been created by extending the side 𝐵𝐶. We want to calculate the sum of these three angles. This means that we need to add 135, 116.6, and 108.4. 135 is the same as 135.0.

Adding the numbers in the tenths column gives us 10, so we carry a one to the ones column. Five plus six plus eight plus one is equal to 20. We put a zero in the ones column and carry the two. Adding our numbers in the tens column gives us six, and adding the numbers in the hundreds column gives us three. The sum of the exterior angles in the triangle is 360 degrees. This is true for any triangle. The exterior angles will always sum to 360 degrees. It is also true for any polygon, although in this video, we will only deal with triangles.

We now have our first rule when dealing with exterior and interior angles of a triangle. The exterior angles of any triangle sum to 360 degrees. If we consider the diagram we saw earlier of triangle 𝐴𝐵𝐶 where line 𝐶𝐵 has been extended, angle 𝑥 is an exterior angle. The interior angle at vertex 𝐵 lies on a straight line with 𝑥. This means that the adjacent interior and exterior angles sum to 180 degrees, as angles on a straight line add up to 180. The interior angle at vertex 𝐵 would therefore be equal to 180 minus 𝑥.

If we let the other two interior angles be 𝑦 and 𝑧, this leads us to another property of interior and exterior angles. We know that the angles in a triangle sum to 180 degrees. This means that 𝑦 plus 𝑧 plus 180 minus 𝑥 is equal to 180. If we add 𝑥 to both sides of this equation, we have 𝑦 plus 𝑧 plus 180 is equal to 180 plus 𝑥. Subtracting 180 from both sides gives us 𝑦 plus 𝑧 is equal to 𝑥. The sum of the other two interior angles are equal to the exterior angle. This gives us our third rule. An exterior angle of a triangle is equal to the sum of the opposite interior angles.

We will now use these three rules to solve some more problems involving exterior and interior angles.

Is the measure of two right angles less than, equal to, or greater than the sum of the interior angles of a triangle?

In order to answer this question, we need to recall two properties of angles: firstly, the properties of a right angle and, secondly, the properties of the interior angles of a triangle. We recall that a right angle is equal to 90 degrees. This means that the measure of two right angles will be equal to two multiplied by 90 degrees. This is equal to 180 degrees. We also recall that the sum of the interior angles of a triangle is equal to 180 degrees. As these two angles are the same, the correct answer is equal to. The measure of two right angles is equal to the sum of the interior angles of a triangle.

In our next question, we will look at a special type of triangle.

What is the measure of an exterior angle of an equilateral triangle?

We recall that the three sides of an equilateral triangle are equal in length. This means that the three interior angles at vertexes 𝐴, 𝐵, and 𝐶 are also equal. An exterior angle is created by extending one of our side lengths, in this case, 𝐵𝐶. The exterior angle is the angle between this extended side length and the adjacent side length 𝐴𝐶. We can now answer this question using our properties of angles.

The sum of the exterior angles of a triangle is equal to 360 degrees. As there are three exterior angles, one way of calculating the measure of one of them would be to divide 360 by three. 36 divided by three is equal to 12. Therefore, 360 divided by three is 120. The exterior angle of an equilateral triangle is therefore equal to 120 degrees. An alternative method here would be to recall that the sum of interior angles of any triangle equals 180 degrees. As our triangle is equilateral, we know the interior angles are equal, and 180 divided by three is equal to 60. This means that all three interior angles of an equilateral triangle are 60 degrees.

We know that the exterior angle is equal to the sum of the two nonadjacent interior angles. As shown in the diagram, we could add 60 degrees and 60 degrees to give us 120 degrees. This would once again prove that the exterior angle of an equilateral triangle is 120 degrees. We also notice in this question that we have a straight line, and the angles on a straight line sum to 180 degrees. The exterior angle plus the interior angle must add up to 180 degrees.

In our next question, we will consider different types of angles.

In a triangle 𝑋𝑌𝑍, if 𝑋𝑌 is equal to 𝑋𝑍, what type of angle is the exterior angle at vertex 𝑍? Is it (A) acute, (B) right, (C) obtuse, or (D) reflex?

We begin by recalling the properties of our four angles. An acute angle is less than 90 degrees. A right angle is equal to 90 degrees. An obtuse angle lies between 90 degrees and 180 degrees. Finally, a reflex angle lies between 180 degrees and 360 degrees. In this question, we are told that in the triangle 𝑋𝑌𝑍, the length of 𝑋𝑌 is equal to the length of 𝑋𝑍. This means that we have an isosceles triangle. In any isosceles triangle, two interior angles are also equal. In this case, angle 𝑋𝑌𝑍 is equal to angle 𝑋𝑍𝑌.

We know that angles in a triangle sum to 180 degrees. This means that the interior angles at vertex 𝑌 and vertex 𝑍 must both be acute. They must both be less than 90 degrees. Otherwise the sum would be greater than 180. The exterior angle at vertex 𝑍 is shown on the diagram. It is the angle between the side length 𝑋𝑍 and the extension of side length 𝑌𝑍. We know that angles on a straight line sum to 180 degrees. This means that the interior and exterior angles at vertex 𝑍 must sum to 180. As the interior angle is acute, it is less than 90, the exterior angle must be greater than 90 and obtuse. In the isosceles triangle drawn, the exterior angle at vertex 𝑍 is obtuse.

In our final question, we will need to calculate interior and exterior angles of a triangle.

Which of the following is correct? Is it option (A) the measure of angle 𝑋𝑍𝑇 is greater than angle 𝑋 which is greater than angle 𝑋𝑍𝑌? Option (B) angle 𝑋𝑌𝑆 is greater than angle 𝑋𝑍𝑌 which is greater than angle 𝑋𝑌𝑍. Option (C) angle 𝑋 is greater than angle 𝑋𝑍𝑌 which is greater than angle 𝑋𝑌𝑍. Option (D) angle 𝑋𝑍𝑌 is greater than angle 𝑋𝑌𝑍 which is greater than angle 𝑋. Or finally, option (E) angle 𝑋𝑌𝑍 is greater than angle 𝑋𝑌𝑆 which is greater than angle 𝑋𝑍𝑌.

As many of the angles in our options are the same, it makes sense to fill in all the missing angles on the diagram first. We can do this using our properties of interior and exterior angles. We know that the angles on a straight line sum to 180 degrees. We could use this to calculate the interior angle at vertex 𝑌 and vertex 𝑍. 180 minus 137 is equal to 43. This means that the interior angle at vertex 𝑍 is 43 degrees. 180 minus 115 is equal to 65. So the interior angle at vertex 𝑌 is 65 degrees.

To calculate the third interior angle, the one at vertex 𝑋, we could use the fact that angles in a triangle sum to 180 degrees. We could add 43 and 65 and then subtract this from 180. Alternatively, we could use the fact that any exterior angle of a triangle is equal to the sum of the nonadjacent interior angles. 137 is therefore equal to 65 plus 𝑥. Subtracting 65 from both sides of this equation gives us 𝑥 is equal to 72. We could also have calculated this by subtracting 43 degrees from 115 degrees. The third interior angle is 72 degrees.

We now have the five angles required. Angle 𝑋 is equal to 72 degrees. Angle 𝑋𝑌𝑍 is equal to 65 degrees. Angle 𝑋𝑌𝑆 is equal to 115 degrees. Angle 𝑋𝑍𝑇 is equal to 137 degrees. And finally, angle 𝑋𝑍𝑌 is equal to 43 degrees. We can now substitute these in to our five options. All five of our options want the numbers to be in descending order. This is because the first angle needs to be greater than the second angle which needs to be greater than the third angle.

In option (E), 65 is not greater than 115. In option (D), 43 is not greater than 65. In option (C), while 72 is greater than 43, this is not greater than 65. The same problem occurs with option (B). 115 is greater than 43, but 43 is not greater than 65. In option (A), our numbers are in descending order. 137 is greater than 72 which is greater than 43. This means that the correct answer is angle 𝑋𝑍𝑇 is greater than angle 𝑋 which is greater than angle 𝑋𝑍𝑌.

We will now summarize the key points from this video. We found out in this video that an exterior angle is the angle between one side of a triangle and the extension of an adjacent side. An interior angle of a triangle, on the other hand, lies between two adjacent sides. The angle properties we identified with regards exterior and interior angles include the following. The exterior angles sum to 360 degrees. The interior angles sum to 180 degrees. We found that any exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles. Finally, an exterior angle and its adjacent interior angle sum to 180 degrees.