Lesson Video: Interior and Exterior Angles of Triangles Mathematics

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

17:45

Video Transcript

In this video, we will see how we can complete geometric proofs using the angle sum of a triangle and we’ll also find the interior and exterior angles of triangles.

Let’s start with the fact that the sum of the measures of the interior angles in any triangle is 180 degrees. Of course, we’ve probably used this lots of times in mathematics. But have we ever thought about why this is true?

In order to see why this is true, we’ll start with one geometric fact. The angle measures on a straight line sum to 180 degrees. For example, in this figure, we could say that π‘₯ degrees plus 𝑦 degrees is equal to 180 degrees. So let’s imagine then that we have this triangle 𝐴𝐡𝐢 and we want to determine what their angle measures sum to. We can say that these angles are called π‘₯, 𝑦, and 𝑧. So we want to say, well, what’s π‘₯ degrees plus 𝑦 degrees plus 𝑧 degrees actually equal to?

In order to determine their sum, we’re going to construct an extra line segment in the figure. We can call this line segment 𝐷𝐸. And the very important thing is, is that 𝐷𝐸 is parallel to 𝐴𝐡. We want these lines to be parallel because then we know that we can apply some geometric properties that we know occur in parallel lines. We know, for example, that alternate interior angles have equal measure. So this angle at 𝐷𝐢𝐴 will be the same measure as the angle at vertex 𝐴. It will also be π‘₯ degrees. In the same way, by using the transversal 𝐢𝐡, we can say that this angle 𝐡𝐢𝐸 must be congruent to the angle at vertex 𝐡. They’re both equal to 𝑦 degrees.

We notice then that, of course, π‘₯ and 𝑦 and 𝑧 all lie on a straight line. Therefore, their angle sum must be 180 degrees. And these three angles π‘₯, 𝑦, and 𝑧 on the straight line are the same as the three angles of π‘₯, 𝑦, and 𝑧 degrees within the triangle. And since we have demonstrated this without using any specific angle measurement, then we can say that this must be true for all triangles. That is, we have proved this theorem that we use every day in mathematics that the sum of all the measures of the interior angles of a triangle is 180 degrees.

We will now see a more specific example of how we can apply this method to find missing angles in a triangle.

In the given figure, if 𝐴𝐢 is parallel to 𝐷𝐸, the measure of angle 𝐴𝐡𝐸 equals 55 degrees, and the measure of angle 𝐢 equals 75 degrees, find the measure of angle 𝐴𝐡𝐢.

The first piece of information that we are given here is that 𝐴𝐢 is parallel to 𝐷𝐸. We can also fill in the two angle measures that we are given that the measure of angle 𝐴𝐡𝐸 is 55 degrees and the measure of angle 𝐢 is equal to 75 degrees. We can then establish that the angle that we wish to calculate is here, the measure of angle 𝐴𝐡𝐢.

So there are a couple of ways in which we could find the measure of this unknown angle. But both methods will use the fact that we have these parallel line segments. By using the parallel lines 𝐴𝐢 and 𝐸𝐷 and the transversal 𝐡𝐢, we can identify a pair of alternate interior angles. And since alternate interior angles are congruent, we can say that the measure of angle 𝐷𝐡𝐢 is equal to the measure of angle 𝐢. These will both be 75 degrees.

We can notice then that these three angles made at the vertex 𝐡 all lie on a straight line. We recall that the angle measures on a straight line sum to 180 degrees. Therefore, we can write that the measure of angle 𝐴𝐡𝐸 plus the measure of angle 𝐴𝐡𝐢 plus the measure of angle 𝐷𝐡𝐢 must be equal to 180 degrees. We can then simply fill in the angle information that we know. Then, by adding 55 degrees and 75 degrees, we get 130 degrees. We can then simplify this equation by subtracting 130 degrees from both sides, leaving us with the measure of angle 𝐴𝐡𝐢 is equal to 50 degrees. And so we have found the value of this unknown angle.

But let’s have a look at the alternative method that we could have used. Let’s return to the question as we had it. As previously mentioned, we still use the parallel lines to help us with this method. But this time, let’s consider the transversal 𝐴𝐡. Once again, we would use the property that alternate interior angles are congruent to work out that this angle at vertex 𝐴 is equal to the measure of angle 𝐴𝐡𝐸. They are both 55 degrees.

However, the geometric property that we will then apply is that the sum of the measures of the internal angles in a triangle is 180 degrees. So, if we added 55 degrees, 75 degrees, and the measure of angle 𝐴𝐡𝐢, we would get 180 degrees. And as we have previously worked out, if we add 55 degrees and 75 degrees and subtract that from 180 degrees, it leaves us with 50 degrees. Therefore, either method will give us the answer that the measure of angle 𝐴𝐡𝐢 is 50 degrees.

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

In the following triangle 𝐴𝐡𝐢, if the measure of angle 𝐢 equals the measure of angle 𝐢𝐴𝐷 equals 43 degrees and the measure of angle 𝐡 equals the measure of angle 𝐡𝐴𝐷, find the measure of angle 𝐡𝐴𝐢.

We can start this question by identifying the two pairs of congruent angle measures. We have that the measure of angle 𝐢 is equal to the measure of angle 𝐢𝐴𝐷, and those are both 43 degrees. We also have that the measure of angle 𝐡 is equal to the measure of angle 𝐡𝐴𝐷, although we aren’t given an exact measurement for those. We can then identify that the angle that we wish to calculate is that of the measure of angle 𝐡𝐴𝐢, which occurs at the vertex 𝐴 in the larger triangle 𝐴𝐡𝐢.

A property that we can apply in this question is that the sum of the measures of the interior angles in a triangle is 180 degrees. So then, if we consider the large triangle 𝐴𝐡𝐢, we can say that the measure of angle 𝐡𝐴𝐢 plus the measure of angle 𝐡 plus the measure of angle 𝐢 is equal to 180 degrees. From the diagram then, we can observe that the measure of angle 𝐡𝐴𝐢 actually consists of an angle of 43 degrees and the measure of angle 𝐡𝐴𝐷.

Then, we are given in the question that the measure of angle 𝐡 is equal to the measure of angle 𝐡𝐴𝐷. Adding in the measure of angle 𝐢, which is 43 degrees, we can add the left-hand side, and it will be equal to 180 degrees. We can then simplify this by adding the two 43 degrees, which is 86 degrees. And we know that there will be two lots of the measure of angle 𝐡𝐴𝐷. Subtracting 86 degrees from both sides, we have that two times the measure of angle 𝐡𝐴𝐷 is equal to 94 degrees. Finally, dividing through by two, we have that the measure of angle 𝐡𝐴𝐷 is 47 degrees. Now that we know the measure of this angle, we can calculate the measure of angle 𝐡𝐴𝐢. It will be equal to 43 degrees plus 47 degrees, which gives us a final answer of 90 degrees.

We’ll now consider what we mean by the exterior angle of a triangle. An exterior angle is defined as the angle formed outside the triangle between any side and the extension of another side. It’s worth noting that there are two exterior angles of a triangle at each vertex. For example, if we call this triangle 𝐴𝐡𝐢, then we could extend either side 𝐴𝐡 or side 𝐡𝐢. However, since both of the exterior angles at 𝐡 make a straight angle with angle 𝐴𝐡𝐢, then they have equal measure. Because both exterior angles at a vertex are congruent, we refer to either of the exterior angles as the exterior angle. And because the interior and exterior angles add to 180 degrees, we say that the exterior angle at a vertex of a triangle is supplementary to the adjacent interior angle.

We’ll now see how we can derive the next important geometric property about exterior angles. Let’s continue with the triangle 𝐴𝐡𝐢. And we remember that the sum of the measures of the interior angles of a triangle is 180 degrees. We could say that the measure of angle 𝐴 plus the measure of angle 𝐡 plus the measure of angle 𝐢 is equal to 180 degrees. And if we label the point on the line here with 𝐷, we can also say that the measure of angle 𝐢𝐡𝐷 plus the measure of angle 𝐡 is equal to 180 degrees, because these two angles lie on a straight line.

If we then return to the first equation, we can rearrange it to obtain that the measure of angle 𝐡 is equal to 180 degrees subtract the measure of angle 𝐴 plus the measure of angle 𝐢. In the second equation, we can also make the measure of angle 𝐡 the subject such that this is equal to 180 degrees subtract the measure of angle 𝐢𝐡𝐷. Then, setting the right-hand side of both of these equations equal, we have 180 degrees minus the measure of angle 𝐴 plus the measure of angle 𝐢 is equal to 180 degrees minus the measure of angle 𝐢𝐡𝐷. By subtracting 180 degrees from both sides and then dividing through by negative one, we have that the measure of angle 𝐴 plus the measure of angle 𝐢 is equal to the measure of angle 𝐢𝐡𝐷.

Let’s now consider what exactly we have demonstrated with this final equation. Well, we’re really saying that this exterior angle 𝐢𝐡𝐷 is equal to the sum of the measures of the other two interior angles in the triangle. And we can formalize this property as follows. 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. We’ll now see an example of how we can use this property to find the measure of an exterior angle.

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

If we look at the diagram, we can observe that we are given the measures of two of the interior angles of this triangle 𝐴𝐡𝐢. The measure of angle 𝐴 is 50 degrees, and the measure of angle 𝐡 is 55 degrees. We need to calculate the measure of this exterior angle at 𝐢, which is given as π‘₯.

The most efficient way to answer this question is to recall the property that the measure of any exterior angle of a triangle is equal to the sum of the measures of the opposite interior angles. This means that since 𝐴 and 𝐡 are the opposite interior angles to the exterior angle at 𝐢, we can say that π‘₯ is equal to 50 degrees plus 55 degrees. This gives us an answer of 105 degrees.

Alternatively, we could’ve used the fact that the three interior angles in the triangle add up to 180 degrees. We could therefore write that the measure of this interior angle 𝐡𝐢𝐴 is equal to 180 degrees subtract 50 degrees plus 55 degrees. This gives 75 degrees. We would then need to perform another calculation using the fact that the sum of these two angles at the vertex 𝐢 must be 180 degrees, because they lie on a straight line. So π‘₯ would be equal to 180 degrees subtract 75 degrees, and that would give us an answer of 105 degrees. Either method gives us the answer of 105 degrees.

We’ll now see another question, which will demonstrate the final geometric property that we’ll see in this video.

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

In order to consider this question, let’s sketch a triangle. Here, we have the triangle 𝐴𝐡𝐢. And as we are considering the exterior angles of the triangle, we can mark these in and label them with π‘₯, 𝑦, and 𝑧 degrees. And even though we are going to consider the exterior angles of a triangle, knowing this property about the interior angles will be useful, that is, that the sum of the measures of the interior angles of a triangle is 180 degrees.

We could therefore form the equation that the measure of angle 𝐴 plus the measure of angle 𝐡 plus the measure of angle 𝐢 is equal to 180 degrees. Then, before we do anything further with this equation, let’s consider each vertex. Because we know that the exterior and interior angles at a vertex are supplementary, we could write that π‘₯ degrees plus the measure of angle 𝐴 is equal to 180 degrees. Likewise, we can say that 𝑦 degrees plus the measure of angle 𝐡 is equal to 180 degrees. And 𝑧 degrees plus the measure of angle 𝐢 is equal to 180 degrees. We can then rearrange each of these three equations to make the measure of the interior angle the subject.

We now have these three equations, which we can substitute in to our first equation. This gives us that 180 degrees minus π‘₯ degrees plus 180 degrees minus 𝑦 degrees plus 180 degrees minus 𝑧 degrees is equal to 180 degrees. We can then rearrange this equation. By subtracting 180 degrees and adding π‘₯, 𝑦, and 𝑧 to both equations, we have that 360 degrees is equal to π‘₯ plus 𝑦 plus 𝑧. With this equation then, we have demonstrated that the sum of the three exterior angles of the triangle π‘₯, 𝑦, and 𝑧 degrees is 360 degrees. This question gives us a demonstration of the very important geometric property that the sum of the measures of the exterior angles of a triangle is 360 degrees.

We can now finish this video by summarizing the key points. The sum of the measures of the interior angles of a triangle is 180 degrees. We saw how we can prove this result by drawing a line parallel to the base of a 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. We also saw that the measure of any exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. And finally, as we saw demonstrated in the last example, the sum of the measures of the exterior angles in a triangle is 360 degrees.

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