How to Find Missing Sides and Angles of Similarity Figures

In this lesson, we’ll talk about similar triangles and how do we determine missing sides and angles when we are given two similar triangles. We’re given two triangles PQR which is this one here and triangle XYZ and we are told that they are similar. So, triangle PQR is similar to triangle XYZ. This is the sign for similarity by the way. I’m actually is going to erase that because we don’t want to confuse the sides. This is the sign of similarity. Using the given side and angle measurements that are given on the diagram, find this two.

What do we have to do? We have to find the length of the side PQ that’s not given to us and the angle PQR. This is what we need to determine. We are told that these two triangles are similar. Here’s what we know. For this triangle, what do we know? We know that PQ is not given to us. QR which is this side is 22 inches and PR which is the third side is 23 inches. We’re also given on this triangle which is XYZ; XY is given to us as 38 inches, YZ which is this one is given to us as 44 inches and XZ is given to us just 46 inches.

We’re also given all three angles here and one angle here. So, this is what we’ve been told. We need to find the missing angle and the missing line. Let’s do this. What do we know about triangles that are similar? The two key things to remember, I’m going to use a different pen to highlight that was if the two triangles are similar then the ratio of their corresponding sides is proportional or equivalent. That’s one fact and the second fact is the corresponding angles are equal. We’ll actually use both of these but let’s write the ratio first.

What do we know about the ratio of their corresponding sides? The corresponding sides are PQ and XY which means PQ/XY is the same as QR and the corresponding side to QR is XYZ which is equal to PR/XZ. So, all three corresponding sides of one triangle and of the other triangle are equal. The ratios are equal. Now that we know five out of the six anyway, let’s substitute the values.

Notice that I've got five out of the six; PQ we don’t know. XY we are given is 38 equals QR is 22 and YZ is 44 and all of these are inches by the way, equals PR which is 23 inches and XZ which is 46 inches. I can use any of these two ratios to find the value of PQ. Since PQ/38=22/44, we can cross multiply which gives us 44 inches times PQ, multiplication of these two terms equals 38 inches times 22 inches. If I solve for this, PQ will become 19 inches because 38×22/44 is 38/2 which is 19. So, PQ we know is 19 inches. If we know PQ is 19 inches then let’s look at solving the second part since we now know this is 19 inches.

We have to find this angle. What it tells us is the corresponding angles are equal. Let’s test it. This angle is 50, the corresponding angle is 50. If this is 65, this is also 65 and if this is 65, this is 65 as well.

Quickly what have we learned? If there are two triangles that are similar like these two, the triangles are similar. We express it with a similarity sign. What that means is the ratio of their corresponding sides are proportional which means PQ/XY or the ratio of PQ and XY will be the same as the ratio of QR and YZ and the third side as well. Using that, this is what we get and the second fact is their angles that are corresponding are equal. So, this and this angle is equal. This angle and this angle are equal and so is the third one. Using this and using cross multiplication, we could calculate the value of PQ as 19 inches and the value of the angle PQR was 65 degrees. That was what we calculated as well and we didn’t have to do much for that because we just knew since this angle was 65, the corresponding angle was 65 as well.