Question Video: Similarity of Right-Angled Triangles Mathematics • 8th Grade

Video Transcript

Fill in the blank: Triangle 𝐴𝐵𝐶 is similar to triangle blank is similar to triangle blank.

We should remember that this wiggly line indicates similarity. So, here, we’re looking for three similar triangles. Triangle 𝐴𝐵𝐶 is the largest triangle in the diagram. There are also two smaller triangles, triangle 𝐴𝐶𝐷 and triangle 𝐴𝐵𝐷. It can be really difficult when we’re looking at a diagram like this to think about similarity as we’re always trying to rotate triangles in our heads. So, let’s see if we can draw these three triangles separately in order to better investigate similarity.

Here, we have the largest triangle 𝐴𝐵𝐶, and we’ve also remembered to include that 90-degree angle at angle 𝐴. It’s important that we label the vertices on our diagram to help us keep track of them. Let’s see if we can draw a triangle 𝐴𝐶𝐷 next, and let’s see if we can keep it in the same orientation as the largest triangle 𝐴𝐵𝐶.

For example, the smallest angle in triangle 𝐴𝐵𝐶 is 𝐶. So, when we draw 𝐴𝐶𝐷, we’ll also look for the smallest angle and put it in the same position and it’s going to be angle 𝐶 in triangle 𝐴𝐶𝐷 as well. So, we can draw 𝐴𝐶𝐷 like this. And don’t forget, we’ve got the right angle here at angle 𝐷. Next, let’s try drawing triangle 𝐴𝐵𝐷. So, remember, if we want the smallest angle on the triangle on the left-hand side, then that’s going to be angle 𝐴 in triangle 𝐴𝐵𝐷. Now that we have drawn this triangle, you might already be wondering about this angle at 𝐴𝐷𝐵, and there is something that we can say about it. Since 𝐶𝐵 is a straight line — and we remember that the angles on a straight line sum to 180 degrees — then this means that the angle 𝐴𝐷𝐵 must also be 90 degrees.

Now that we have drawn these three triangles, we can see that there’s one corresponding angle of 90 degrees that’s the same in each triangle. However, showing one angle is the same is not sufficient to show similarity. One way to demonstrate that two triangles are similar is to see if we can prove the AA rule. In this rule, we’re proving that there are two sets of corresponding angles congruent and, therefore, the triangles are similar.

So, let’s see if we can prove if there’s another set of angles that are the same in each triangle. Let’s look at this angle highlighted in pink. In the original diagram, we can see that it’s angle 𝐴𝐶𝐵, and it’s exactly the same angle that we have in triangle 𝐴𝐶𝐷. However, if we look on our third triangle, we can’t say anything for sure about this angle at 𝐷𝐴𝐵. We don’t know that this angle is the same as the angle at 𝐶 on our other triangles.

Okay, so let’s then look at the large triangle and compare it with triangle 𝐴𝐵𝐷. If we highlight the angle 𝐴𝐵𝐶 in the large triangle, then this angle is going to be exactly the same as the angle 𝐴𝐵𝐷 on the smallest triangle. So, let’s go back and look again at the first two triangles that we’ve drawn. We have the same angle of 90 degrees, and we’ve also confirmed that these two angles at 𝐶 are the same. Because the angles in a triangle sum to 180 degrees, this means that the third angle in each triangle must also be congruent. If we then consider our second and third triangles, we’ve got this pair of angles which are 90 degrees. And we have a pair of congruent angles here at angle 𝐷𝐴𝐶 and angle 𝐷𝐵𝐴, which means that the third angle of 𝐷𝐴𝐵 is congruent to angle 𝐷𝐶𝐴 in the second triangle and angle 𝐴𝐶𝐵 in the first triangle.

Now we have shown that in each triangle there are two sets of corresponding angles congruent. In fact, we’ve demonstrated that there are three sets of corresponding angles congruent. So, we have shown that these three triangles are similar. However, we need to make sure that when we write our similarity statement, we get the order of letters correct. We have been given the first part of the similarity statement as triangle 𝐴𝐵𝐶, meaning that we read from 𝐴 to 𝐵 to 𝐶. So, the second triangle must be read in the same way as triangle 𝐷𝐴𝐶. The third triangle must be stated as triangle 𝐷𝐵𝐴.

Therefore, we say that triangle 𝐴𝐵𝐶 is similar to triangle 𝐷𝐴𝐶 is similar to triangle 𝐷𝐵𝐴. Therefore, the two missing blanks would be 𝐷𝐴𝐶 and 𝐷𝐵𝐴.