Lesson Video: Similarity of Triangles Mathematics • 8th Grade

Video Transcript

In this video, we will learn how to identify similar triangles using either the AA rule or the SSS rule.

The first thing we should do is recall that similar shapes have all corresponding angles congruent or equal and all corresponding sides in proportion. Note that it’s different to congruent shapes as congruent shapes have corresponding angles equal and corresponding sides equal. Similar shapes will have sides in proportion. So when it comes to identifying similar triangles, there’s a few shortcuts that we can take rather than trying to demonstrate all the angles are congruent and all the corresponding sides are in proportion. Let’s start by focusing on the angles.

On the two triangles drawn here, we can see that there are three pairs of corresponding angles congruent. Let’s have a look at the larger triangle. Let’s say that we take a point on this larger triangle that’s the same length as the corresponding side in the smaller triangle and draw a line in parallel to the base of this triangle. If we take a look at the angles, then as we have a pair of parallel lines and two transversals, then we’d have two pairs of equal corresponding angles. The triangle proportionality theorem says that if a line is parallel to one side of a triangle and it intersects the other two sides, then it divides those two sides proportionally.

So if we label the vertices of the larger triangle with 𝐴𝐵𝐶 and the smaller triangle with vertices 𝐷𝐸𝐹, then we would say that the side 𝐴𝐵 is proportional to the side 𝐷𝐸, and that would be the same proportion as 𝐴𝐶 to 𝐷𝐹, which would be the same as the proportion of 𝐵𝐶 to 𝐸𝐹. Therefore, showing all three angles are congruent, we’ll also demonstrate that the corresponding sides are in proportion.

In fact, there is an even simpler way than showing our three angles are congruent. And that’s because if we only show two angles, because the angles in a triangle sum to 180 degrees, we know that if we’ve got two pairs of corresponding angles congruent in a triangle, then the third angles must also be congruent. And therefore, if we want to prove that two triangles are similar, we only need to use the AA rule to show that two pairs of corresponding angles are congruent.

Next, let’s think about the sides in these triangles. The proportion of 𝑃𝑄 over 𝐾𝐿 can be written as six over eight. The proportion of 𝑃𝑅 to 𝐾𝑀 can be written as 7.5 over 10. Finally, 𝑄𝑅 over 𝐿𝑀 would be the proportion nine over 12. Each of these three ratios all simplify to three-quarters. Therefore, the sides have the same proportion. As we saw above, if the sides are in proportion, then corresponding pairs of angles will be congruent. This is called the SSS rule, which means that if we show that corresponding pairs of sides are in proportion, then we demonstrate that we have a pair of similar triangles.

Remember, be careful and don’t confuse it with the SSS rule that we might use for congruent triangles; in that case, we’d have to show they’re congruent. But for similar triangles, we show the SSS rule meaning the sides are in proportion.

In the following questions, we’ll be using either of these two rules to help us identify similar triangles. Let’s have a look at our first question.

Given the following four shapes, which two are similar?

In this question, we’re given four different shapes, complete with the side measurements. We’re asked which of these shapes are similar. And we recall that similar shapes have corresponding angles congruent or equal and corresponding sides in proportion. We’re not given any information regarding the angles in these four triangles. So, let’s have a look at the sides.

We can notice that within these four triangles, we do have two that look similar. Shape two and four would both be isosceles triangles, and shape one and shape three also look as though they’re a similar shape. We need to check each pair of triangles, however, to see if the corresponding sides are in proportion. Let’s begin with our isosceles triangles, that’s shapes two and four. Let’s take a corresponding pair of sides.

We would say then that the ratio here would be 12 in the larger triangle over seven in the smaller triangle. Is this equal to the ratio of our other sides eight in the larger triangle and five in the smaller triangle? In order to more easily compare these fractions, let’s make the denominator the same. In order to change twelve-sevenths into a fraction over 35, we would need to multiply the numerator and denominator by five, which would give us a value of 60 on the numerator. On the right-hand side, we would need to multiply the numerator and denominator by seven. But as we see 60 over 35 is not equal to 56 over 35. So, shapes two and four are not similar.

Let’s check shapes one and three. Writing our proportions with the value of the smaller triangle’s lengths as the numerator and the larger triangle’s lengths as the denominator would give us the values of four- eighths, three-sixths, and five-tenths. We should notice that all of these fractions simplify to one-half. So, we’ve demonstrated that these three pairs of corresponding sides are in proportion, and that applies the SSS rule of similar triangles. So, our answer is that out of these four shapes, shape one and shape three are similar.

Let’s have a look at another question.

Are the two triangles similar?

Let’s begin by recalling that similar triangles have corresponding angles congruent and corresponding sides in proportion. If we look at the smaller triangle 𝐴𝐷𝐸 and the larger triangle 𝐴𝐵𝐶, it doesn’t seem as though these two triangles would be the same shape. However, let’s see if we could prove it mathematically, just in case the triangles are not drawn correctly. In order to show that two triangles are similar, we can remember that we would either use the AA rule to show that two pairs of angles are congruent or the SSS rule to demonstrate that there are three pairs of corresponding sides in proportion. We’re not given enough information about the angles here, so let’s see if we can use the SSS rule.

We need to check if there’s the same ratio or proportion between corresponding sides. For example, is this side 𝐴𝐸 over the side 𝐴𝐶 in the same proportion as 𝐴𝐷 over 𝐴𝐵 and the same as 𝐸𝐷 over 𝐶𝐵? We can fill in the numerical values for the lengths into our side information. But as we don’t have any dimensions for sides 𝐸𝐷 and 𝐶𝐵, we won’t be able to prove the two triangles are similar. But if the proportions of 𝐴𝐸 and 𝐴𝐶 and 𝐴𝐷 and 𝐴𝐵 are different, then we could prove that they are not similar. Let’s take a look.

The length of 𝐴𝐸 is given as 46 centimeters. But be careful as the length of 𝐴𝐶 isn’t 32.2, but rather it’s the sum of 46 and 32.2, which is 78.2. 𝐴𝐷 is 22, and 𝐴𝐵 is the sum of 22 and 24.2 centimeters, which is 46.2 centimeters. We now need to compare these two fractions to see if they’re equivalent. We can begin by removing this decimal point from the denominator. We can do this by multiplying both of our numerators and denominators by 10. Dividing the numerator and denominator by 46, then 460 over 782 simplifies to 10 over 17. Dividing the numerator and denominator of 220 over 462 by 22 gives us the fraction 10 over 21. We can, therefore, see that these two ratios are not equal. Therefore, the sides are not in proportion. If the sides are not in proportion, then the triangles would not be similar. So, our answer here would be no.

Before we finish with this question, just a point to note. If we had find that 𝐴𝐸 over 𝐴𝐶 was equal to 𝐴𝐷 over 𝐴𝐵, then we would have also needed the values for 𝐸𝐷 and 𝐶𝐵. As it’s not enough just to show that two sides are in proportion, we would need to show that there are three pairs of corresponding sides in proportion. In this question, it didn’t matter that we didn’t have these two other lengths as it was enough to show that these triangles were not similar.

In the next question, we’ll see if we can apply the AA rule to demonstrate a similarity.

The figure shows two triangles. Are the two triangles similar? Why?

In this question, we’re asked if these two triangles are similar. We can recall that similar triangles have corresponding angles congruent and corresponding sides in proportion. So, let’s have a look at the angles in both of these triangles. There’s a right angle of 90 degrees in each triangle. And one triangle has a 60-degree angle, and the other one has a 30-degree angle. However, if we remember that the angles in a triangle add up to 180 degrees, then we could calculate the third angle in each triangle.

In the top triangle, 60 plus the right angle of 90 degrees would give us 150 degrees, and subtracting that from 180 degrees would leave us with 30 degrees. In the lower triangle, 90 degrees plus 30 gives us 120 degrees, leaving us with a third angle of 60 degrees. What we can see then is that we have three pairs of equal angles. We have two 30-degree angles, two 60-degree angles, and two 90-degree angles. We have shown a three pairs of corresponding angles congruent. But in fact, we only need to show two pairs of corresponding angles congruent in order to show that two triangles are similar.

So, to answer the question “are the two triangles similar?” is yes. To answer the second question “why?,” we could say that if you calculate the measure of the third angle in one of the triangles, you can see that the triangles share two angles. Therefore, by the AA criteria, the triangles are similar.

The figure shows two triangles, 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime. Work out the measure of angle 𝐴𝐵𝐶. What does the AA criterion tell us about these two triangles?

In this question, we have two triangles drawn on grid paper. And the first thing we’re asked to do is find the measure of angle 𝐴𝐵𝐶, which is in the smaller triangle. In order to do this, we should recall that the angles in a triangle add up to 180 degrees. We’ll therefore need to calculate 180 degrees subtract the other two angles of 114.3 degrees and 34.1 degrees, which gives 31.6 degrees. And so that’s our answer for the measure of angle 𝐴𝐵𝐶.

In the second part of this question, we’re asked about the AA criterion, which is the criterion we use to show that two triangles are similar. It’s what we have when we demonstrate that there are two pairs of angles congruent. So, let’s have a closer look at these two triangles. The angle 𝐴𝐵𝐶 that we’ve just worked out as 31.6 degrees has a corresponding angle at angle 𝐴 prime 𝐵 prime 𝐶 prime of the same value, 31.6 degrees. We also have another pair of corresponding congruent angles, the angle 𝐶 prime 𝐴 prime 𝐵 prime and angle 𝐶𝐴𝐵, which are both given as 34.1 degrees. Showing that there are two pairs of corresponding congruent angles or the AA rule, we demonstrate that these two triangles are similar.

We could therefore answer the second part of this question with a statement such as this: as both triangles share two angels of equal measures, they must be similar.

We’ll now have a look at one final question.

Triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime in the given figure are similar. Work out the value of 𝑥.

We’re told here that these two triangles are similar. This means that corresponding angles are congruent and corresponding sides are in proportion. Looking at the triangles, this would mean that the angle at 𝐵 would be congruent with the angle at 𝐵 prime as these two angles are corresponding. Angle 𝐶 and 𝐶 prime would be congruent, and angle 𝐴 and 𝐴 prime are congruent.

We’re given measurements for angle 𝐴 and 𝐴 prime. So, we could write that five 𝑥 plus 90 over six must be equal to three 𝑥 plus 320 over six. We could then solve to find the value of 𝑥. Multiplying both of these fractions by six would give us that five 𝑥 plus 90 is equal to three 𝑥 plus 320. Subtracting three 𝑥 from both sides would give us two 𝑥 plus 90 is equal to 320. Subtracting 90 from both sides would give us two 𝑥 equals 230. And finally, dividing by two would give us 𝑥 equals 115. And so, we have our answer that the value of 𝑥 would be equal to 115. We don’t need to include the degree sign as it was part of the angle definition in 𝐴 and 𝐴 prime.

We can check our answer by substituting this value for 𝑥 back in and find that angle 𝐵𝐴𝐶 is 110 and 56 degrees and so is angle 𝐵 prime 𝐴 prime 𝐶 prime.

We can now summarize what we’ve learned in this video. Firstly, we saw that similar triangles have corresponding pairs of angles congruent and corresponding pairs of sides in proportion. We can identify similar triangles by using the AA rule which is when we have two pairs of corresponding angles congruent. We can also identify similar triangles using the SSS criterion, which is where all three pairs of corresponding sides are in the same proportion. Remember that we only need to show that the corresponding sides are in proportion and we don’t need to show that they’re congruent. We should also remember when we’re working through problems like these that triangles can be similar even if they’re in a different orientation.