Video: Congruent Triangles

Video Transcript

In this video, we will learn how to prove that two triangles are congruent by using one of four different criteria. We will see the RHS criterion, which stands for right angle-hypotenuse-side; the SSS criterion, which is side-side-side; the ASA criterion, which stands for angle-side-angle; and finally the SAS rule, which stands for side-angle-side. But first, we can recap what we mean by congruent triangles.

We say that two triangles are congruent if their corresponding sides are congruent and their corresponding angles are congruent. Let’s consider these three triangles. It doesn’t matter if the triangles have different orientations or are flipped; we can still compare their side lengths and their angle measures. The corresponding sides and angles in these three triangles are highlighted below. So, when all three pairs of corresponding sides are congruent and all three angle measures are congruent, then the three triangles are congruent.

We could write that triangle 𝐴𝐵𝐶 is congruent to triangle 𝐷𝐸𝐹 is congruent to triangle 𝑃𝑄𝑅. But the order in which we write this congruent relationship is very important. So we could write it in this way or as triangle 𝐶𝐴𝐵 is congruent to triangle 𝐹𝐷𝐸 is congruent to triangle 𝑅𝑃𝑄. But we could not write that triangle 𝐴𝐵𝐶 is congruent to triangle 𝐸𝐷𝐹 because the vertices do not correspond.

So, when it comes to determining if two triangles are congruent or not, there are some shorter methods we can use rather than establishing that all the corresponding sides and all the corresponding angles are congruent. These are called the congruence criteria. And let’s have a look at the first one of these. This is called the side-angle-side criterion, which is abbreviated to SAS. This rule states that two triangles are congruent if two sides and the included angle in one triangle are congruent to the corresponding parts in the other triangle. In the diagram below, we can see that there are two pairs of corresponding congruent sides. The angles between these sides in each triangle are also congruent.

As an illustration of why this criterion works, let’s consider this triangle, which has sides labeled with five centimeters and six centimeters and the included angle of 40 degrees. Let’s now try and draw another triangle that has these same two lengths and included angle measure, but we won’t constrain the size of the third side or the two other angles in any way. But when we begin sketching a triangle, we can see that there will be only one way in which to create a complete triangle. That complete triangle will be congruent to the original triangle. So by knowing that two triangles have two pairs of congruent sides and the included angle measures in each triangle are equal, then that would show that the triangles themselves are congruent.

Let’s have a look at the second congruency criterion. The angle-side-angle, or ASA, congruence criterion states that two triangles are congruent if two angles and the side drawn between their vertices in one triangle are congruent to the corresponding parts in the other triangle. If we look at the figure, we can see that we have two pairs of congruent angle measures in each triangle, and it is the included side between these two angles which is congruent. Now we have seen two congruence criterion: the side-angle-side rule and the angle-side-angle rule. Let’s see if we can apply these in the following example.

Determine whether the triangles in the given figure are congruent, and, if they are, state which of the congruence criteria proves this.

We recall that two triangles are congruent if their corresponding sides are congruent and corresponding angles are congruent. We can use a number of different criteria to help us establish if two triangles are congruent, so let’s see what we have in the given figure. Let’s start with the two congruent sides, which measure 2.53 length units. They are 𝐴𝐶 and 𝐴 prime 𝐶 prime. We also have two sides measuring 3.68 length units, so we can say that 𝐵𝐶 is equal to 𝐵 prime 𝐶 prime. Both triangles have the same corresponding angle measure of 60.34 degrees. We recall that the SAS, or side-angle-side, congruence criterion states that two triangles are congruent if they have two congruent sides and an included congruent angle. We can therefore give the answer that these two triangles are congruent by the SAS congruence criterion.

We will now see another example.

Can you use SAS to prove the triangles in the given figure are congruent? Please state your reason.

We can recall that two triangles are congruent if corresponding sides are congruent and corresponding angles are congruent. Here, we are asked specifically if the congruency criterion SAS can be used to prove these triangles are congruent. So, let’s have a look at the measures that we are given in the figure. We do have two pairs of corresponding congruent sides. 𝐴𝐵 is equal to 𝐴 prime 𝐵 prime, and 𝐴𝐶 is equal to 𝐴 prime 𝐶 prime. We also have a pair of corresponding angle measures that are congruent. If we consider the SAS congruence criterion, that states that two triangles are congruent if they have two congruent sides and an included congruent angle. However, in this figure, the given angle in each triangle is not the included angle between the sides.

To have the included angle here, we would need to know and compare the measure of angle 𝐵𝐴𝐶 and the measure of angle 𝐵 prime 𝐴 prime 𝐶 prime. When we give our answer, a good statement would reference the fact that we can’t say the triangles are congruent because the angle is not the appropriate angle. Therefore, to answer the question “Can we use SAS to prove the triangles are congruent?” we would say no because the angle must be contained, or included, between the two sides.

We will now see a third criterion for triangle congruence. This is the side-side-side, or SSS, congruence criterion. It states that two triangles are congruent if each side in one triangle is congruent to the corresponding side in the other triangle. This arises simply from the fact that if we have a triangle with three given side lengths, we couldn’t create a noncongruent triangle using the same three side lengths.

The fourth and final congruence criterion, we will see, applies specifically and only to right triangles. The right angle-hypotenuse-side, or RHS, criterion states that two right triangles are congruent if the hypotenuse, that’s the longest side, and a side of one triangle are congruent to the corresponding parts in the other triangle. This criterion is in fact a special application of the SSS criterion. That’s because if we know two sides in a right triangle, we can calculate the third by using the Pythagorean theorem. Notice that if this was not a 90-degree angle, then we would in fact be trying to apply a side-side-angle criterion, and no such criterion exists.

To understand why it doesn’t exist, let’s take this triangle for an example. In this triangle, there are two sides of eight centimeters and four centimeters and a nonincluded angle of 30 degrees. If we then constructed another triangle with the same properties, we would in fact find that there is more than one possible triangle we could create. And so side-side-angle is not sufficient to demonstrate congruency. If we do have two sides and an angle in two triangles, then the angle must be the included angle between the two sides to give us the SAS criterion.

We will now take a look at another example.

Which congruence criteria can be used to prove that the two triangles in the given figure are congruent?

In this question, we need to determine by which criteria the two triangles are congruent, which means that corresponding sides are congruent and corresponding angles are congruent. Let’s take a look at the figure. We can see that we have these two lengths, which are given as 2.57 units, that would mean that 𝐴𝐵 is equal to 𝐴 prime 𝐵 prime. We also have a pair of angle measures of 65.03 degrees and a second pair of angles, which measure 58.55 degrees. We have therefore found that there are a pair of congruent sides and two pairs of congruent angles in these triangles.

Now, there is an angle-side-angle criterion, which relates two angles and the side. It states that two triangles are congruent if they have two angles congruent and the included side congruent. However, in this figure, the side is not the included side because it doesn’t lie between the two angles. This means that we can’t immediately apply the angle-side-angle criterion. But as we do have two angles in each triangle, we can work out the third angle in each triangle by using the fact that the internal angle measures in a triangle sum to 180 degrees.

In triangle 𝐴 prime 𝐵 prime 𝐶 prime, we could calculate the missing angle as 180 degrees subtract 58.55 degrees plus 65.03 degrees. This would give us that the measure of angle 𝐵 prime 𝐴 prime 𝐶 prime is 56.42 degrees. In triangle 𝐴𝐵𝐶, the measure of angle 𝐵𝐴𝐶 is equal to 180 degrees minus 58.55 degrees plus 65.03 degrees. Of course, as these are the same values, then it’s going to give us the same angle measure of 56.42 degrees. And now, if we compare the two angles of 56.42 degrees and 65.03 degrees in both triangles, we have got these two angles plus their included side. Since these corresponding parts are congruent in each triangle, then we can say that these triangles are congruent, and we did that using the angle-side- angle congruence criterion.

We would need to state angle-side-angle, or ASA, in our answer. Notice that if we were answering this question in an exam, it would be very important that we demonstrated the value of the third angle. If we didn’t know that third angle, we couldn’t have applied the angle-side-angle criterion.

We will now take a look at one final example.

In the given figure, points 𝐿 and 𝑁 are on a circle with center 𝑂. Which congruence criteria can be used to prove that triangles 𝑂𝐿𝑀 and 𝑂𝑁𝑀 are congruent?

In this problem, we need to determine how we can prove that triangles 𝑂𝐿𝑀 and 𝑂𝑁𝑀 are congruent, which means that their corresponding sides will be congruent and corresponding angles will be congruent. We might notice that both of these triangles have a right angle, and there is a congruence criteria which is used in right triangles. It’s the RHS criterion, which states that two triangles are congruent if they have a right angle and the hypotenuse and one other side are equal. Let’s see if we can apply this criterion here.

We notice that we aren’t given any length measurements, but we can apply our knowledge of geometry to help. Because we are told that 𝑂 is the center of the circle, then the line segments 𝑂𝐿 and 𝑂𝑁 are radii of the circle. And importantly, that means that they are congruent. We can also see that the line segment 𝑂𝑀 is common to both triangles. And importantly, it’s also the hypotenuse. If we look then at the RHS criterion, both triangles have a right angle, the hypotenuse is congruent because it’s a shared side, and there is a congruent side in both triangles. Therefore, it is by applying the RHS congruency criterion that we can prove that triangles 𝑂𝐿𝑀 and 𝑂𝑁𝑀 are congruent.

We can now summarize the key points of this video. We saw that two triangles are congruent if their corresponding sides are congruent and corresponding angles are congruent. The congruence criteria allow us to more easily prove if triangles are congruent. We saw four different congruence criteria we can apply. They are side-angle-side, angle-side-angle, side-side-side, or the right angle-hypotenuse-side congruence criteria, which applies specifically in right triangles. And finally, we also saw that there is no side-side-angle criterion, since noncongruent triangles can be created with equivalent measurements. If we do have two sides and an angle congruent with another triangle, then this would need to be the included angle.