Lesson Video: Congruent Triangles Mathematics

Video Transcript

In this video, we will learn how to prove that two triangles are congruent by using one of four different criteria. These criteria are the SSS, SAS, ASA, or RHS criteria. But first let’s 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. In other words, the corresponding sides are the same length and the corresponding angles are the same measure. Let’s take 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 angles and sides are marked below. So, when all three sides are congruent and all three angles are congruent, then the triangles themselves are congruent.

We can say that triangle 𝐴𝐵𝐶 is congruent to triangle 𝐷𝐸𝐹 is congruent to triangle 𝑃𝑄𝑅. But the order in which we write this congruency relationship is very important. So we could write it like this 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. But when it comes to determining if two triangles are congruent, there are some shorter methods we can use rather than establishing that all corresponding sides and all corresponding angles are congruent. These are called the congruency criteria, and we can have a look at the first one of these.

The first criterion is 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 corresponding sides which are marked as congruent and the angles between these two sides — that’s the included angles — are also marked as congruent.

As an illustration of why this criterion exists, let’s see this triangle, which has two sides of six centimeters and five centimeters labeled and the included angle of 20 degrees. We could try to draw a different triangle but which has the same side lengths of five centimeters and six centimeters and the same size of included angle. But we wouldn’t constrain the length of the third side or the other two angle measures. But when we start sketching another triangle, we can see that there will be only one way in which to create a triangle. And this complete triangle will in fact be congruent to the original triangle. So, by knowing that two corresponding side measures in a triangle are congruent to those in another triangle and that the included angle measures are equal, then the two triangles must be congruent.

Let’s have a look at the second congruence criterion. This is the ASA or angle-side-angle congruence criterion. It 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. In the figure drawn, we have two pairs of congruent angles. And it is the included side — that’s the side between their vertices — which is marked as congruent.

We have now seen two congruency criteria. So let’s see if we can apply these in the following examples.

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

We can begin by recalling that congruent means that the two triangles will have corresponding sides congruent and corresponding angles congruent. We can use a number of different congruence criteria to help us prove that two triangles are congruent. So let’s see the information that we are given in the figure. Well, we notice that we have two pairs of sides which are congruent. Since 𝐴𝐶 and 𝐴 prime 𝐶 prime are both 2.53 length units, then they are congruent. And since 𝐵𝐶 and 𝐵 prime 𝐶 prime are both 3.68 length units, then they are congruent.

In each of these triangles, the angle between the two given sides is the same measure. Both of these are 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. As this is what we have demonstrated, we can then say that the two triangles are congruent by the SAS congruence criterion.

We’ll now see another example.

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

Two triangles are congruent if their corresponding sides are congruent and corresponding angles are congruent. Here, we are specifically asked if we can use the congruence criterion SAS to prove this. If we look at the measures that we’re given in this diagram, we can say that 𝐴𝐵 is equal to 𝐴 prime 𝐵 prime because they’re both given as 2.36 length units. And 𝐴𝐶 is equal to 𝐴 prime 𝐶 prime because these lengths are both given as 5.52 length units. We also have a corresponding pair of angle measures that are congruent. The measure of angle 𝐴𝐶𝐵 and the measure of angle 𝐴 prime 𝐶 prime 𝐵 prime are both given as 25.22 degrees.

The SAS congruence criterion tells us that two triangles are congruent if they have two congruent sides and an included angle congruent. But in this figure, the angle which we are given in each triangle is not the included angle between the two sides. For the included angle here, we would need to be able to know and compare the measure of angle 𝐶𝐴𝐵 and the measure of angle 𝐶 prime 𝐴 prime 𝐵 prime. When we give our answer for this question, a good answer needs to reference the fact that we can’t say the triangles are congruent because the angle that we are given is not the appropriate angle. Therefore, to answer the question “Can we use SAS to prove the triangles are congruent?,” we can say no, because the angle must be contained or included between the two sides.

We’ll now see the third criterion for triangle congruence. This criterion is called 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 from the fact that if we have a triangle with three known side lengths, we couldn’t create a noncongruent triangle with the same three side lengths.

The fourth congruence criterion that we will see applies specifically and only to right triangles. It is called the right angle-hypotenuse-side congruence criterion, which we abbreviate to RHS. And we recall that the hypotenuse of a right triangle is the longest side. This rule states that two right triangles are congruent if the hypotenuse 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 congruency criterion. That’s because if we know two sides in a right triangle, then by using the Pythagorean theorem, we can calculate the third side in each triangle.

Notice that if this angle wasn’t a 90-degree angle, we’d be trying to apply a side-side-angle criteria, where the angle isn’t the included angle, and there is no such criterion. Let’s consider why there is no side-side-angle criterion. And to understand this, let’s consider this triangle. Here, we have two side lengths of eight centimeters and four centimeters and a nonincluded angle of 30 degrees. If we then constructed another triangle with the same properties, then we would in fact have more than one possible triangle. So, knowing that two sides and a nonincluded angle are congruent is not sufficient to demonstrate that two triangles are congruent.

Now that we have seen all four congruence criterion, let’s have 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 how we can show these triangles are congruent, which means that their corresponding sides are congruent and corresponding angles are congruent. Let’s take a look at the measurements that we’re given on the figure. We can see that we have a pair of lines, which are both given as 2.57 length units, which means that 𝐴𝐵 must be equal to 𝐴 prime 𝐵 prime. We also have two pairs of congruent angle measures. The measure of angle 𝐴𝐵𝐶 is equal to the measure of 𝐴 prime 𝐵 prime 𝐶 prime because they both measure 65.03 degrees. And the measure of angle 𝐴𝐶𝐵 is equal to the measure of angle 𝐴 prime 𝐶 prime 𝐵 prime, as these are both equal to 58.55 degrees.

Therefore, we have found that there are a pair of congruent sides and two pairs of corresponding congruent angles. Now, there is a congruency criterion ASA which relates two angles and a side. It states that two triangles are congruent if they have two angles congruent and the included side congruent. However, in these triangles, 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 ASA criterion.

But as we are given two angles in the triangle, we could calculate the third angle in each triangle by using the fact that the angle measures in a triangle sum to 180 degrees. In triangle 𝐴 prime 𝐵 prime 𝐶 prime, we can calculate the third 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 third angle will also be equal to 180 degrees subtract 58.55 degrees plus 65.03 degrees. And as these are the same values, then we know that the measure of angle 𝐵𝐴𝐶 will also be 56.42 degrees.

So now if we compare the triangles, we have two angles of 56.42 degrees and another pair of angles of 65.03 degrees. The side of 2.57 length units is the included side between these two angles. We could now apply the ASA congruence criterion to show that these two triangles are congruent. For the answer then, we would need to give angle side angle or ASA.

Note, however, that if we were answering a question like this in an exam, we would need to very clearly show that we had calculated the third angle in the triangle. Without knowing this third angle, we could not have applied the ASA congruence criterion.

We’ll 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 criterion can be used to prove that triangles 𝑂𝐿𝑀 and 𝑂𝑁𝑀 are congruent?

In this problem, we need to determine how we can prove that these two triangles 𝑂𝐿𝑀 and 𝑂𝑁𝑀 are congruent, which means that corresponding sides are congruent and corresponding angles are congruent. We might notice that both of these triangles have a right angle. And there is a congruency criterion which applies in right triangles. It is the RHS criterion, which tells us 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. And even though we aren’t given any length measurements, we can apply our knowledge of geometry to help. Because we’re given that 𝐿 and 𝑁 are on the circle and the center is 𝑂, then the line segments 𝑂𝐿 and 𝑂𝑁 are both radii of the circle. And importantly, that means that these are congruent. We also have this line segment 𝑂𝑀, which is a shared side between the two triangles. So this length will be equal in both triangles. Very helpfully, we can also recognize that the line segment 𝑂𝑀 is the hypotenuse in both of these triangles.

Now, if we look at the RHS criterion, we know that both triangles do have a right angle. We also know that the hypotenuse is congruent because this is a common side. And we have another pair of sides which are congruent. Therefore, it is by applying the RHS congruence 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. We saw that congruence criteria allow us to more easily prove that two triangles are congruent. We covered four different congruence criterion. They are the side-angle-side, angle-side-angle, side-side-side, or right angle-hypotenuse-side criteria. Finally, there is not a side-side-angle criterion since noncongruent triangles can be created with equivalent measurements.