Video: Congruence of Triangles: ASA and AAS

In this video, we will learn how to prove that two triangles are congruent using either the angle-side-angle (ASA) or the angle-angle-side (AAS) criterion and determine whether angle-side-side is a valid criterion for triangle congruence or not.

15:34

Video Transcript

In this video, we will prove that two triangles are congruent using either the angle-side-angle rule or the angle-angle-side rule. We can recall that congruent means exactly the same shape and size.

We can start this video by thinking about how we might prove that two shapes are congruent. In any polygon, we’d have to show that all the corresponding pairs of angles are congruent. And we’d have to show that all the corresponding sides are also congruent. So, if we had a pair of triangles, we’d need to show that we have three angles equal. We could write that rule as AAA for angle-angle-angle. And that we have three pairs of corresponding sides congruent, which we could write as SSS.

However, there are some shortcuts that we can take instead of having to show these two rules for every pair of triangles that we want to prove are congruent. You may already know that just the SSS rule is sufficient to show that two triangles are congruent. What we’re going to check first is if the AAA rule by itself is sufficient to show two triangles would be congruent.

Let’s draw a triangle with our three different angles marked on. Could we draw a different triangle that has these same three angles? We’re not constraining the length of the sides in anyway. Here are some sides around the first angle. And if I move this second angle and the third angle, I could create a triangle that looks like this. The three angles in our second triangle are exactly the same size as the angles in the first triangle. In fact, the second triangle is twice as large as the first triangle. The triangles are the same shape, but a different size.

The mathematical word for this is “similar.” So, the AAA rule just proves that two triangles are similar. But importantly, it would not show that they are congruent. So, so far, we have recalled that the SSS rule shows congruence but the AAA rule does not. So, let’s try a different conjecture and see if that works. This time, we’ll also have a side.

When we see ASA for angle-side-angle, the side in the middle represents a side that’s included between the two angles. So, let’s have a look at this triangle. We have two angles and the included side marked. Could we draw a different noncongruent triangle that has these two angles and the same length of side? What I’m going to do here isn’t a proof, but it is a reasoning.

Let’s begin by drawing a side with the same length as the original green side. We would have to have two pairs of congruent angles, but we’re not going to constrain the length of the sides. So, we could draw a very long line or even a shorter line or even shorter still. The same is true for the length of the other side. We’re not going to constrain that length in anyway. So, it could be of any size. However, there is only one way in which we could create a triangle. And that triangle will be congruent. The third angle in the triangles would be congruent.

What we have, in fact, shown here is that we have the AAA rule. But we’ve also shown one other important fact. And that is that the ratio of sides here is one to one. So, our triangles are similar. But we know that this second length on both sides would be congruent, and the same for the third pair of sides. So therefore, if we have a pair of triangles, and we can demonstrate that two angles and the included side are congruent, that would mean that the two triangles themselves are also congruent. And we can update our list on the side here that we have another congruency rule.

For the next conjecture, let’s see if having two angles and any side would show that two triangles are congruent. Let’s take a different triangle. I’ve marked on two angles and a nonincluded side. Let’s see if we can draw a noncongruent triangle to this one. All that we are constraining here is the side length and two angles. So, I could draw this green angle here or even here. I could draw this length anywhere of any size here or here, but let’s start with the line coming from this pink angle. Again, this line could be of any length. It just has to have the same angle as the original pink angle.

Somewhere along this line, we have constrained an angle that’s the same as the green angle denoted by the two arcs. As we have constrained the length of our orange side however, that means there’s only one triangle that we could create. So, we would have two congruent triangles. Once again, this isn’t a proof, just a reasoning. However, we can consider that this third angle in each of the triangles must be the same since we could’ve subtracted the original two angles from 180 degrees.

Therefore, we have demonstrated the AAA rule, which alone wouldn’t be sufficient. But we do also have a corresponding pair of congruent sides. Therefore, we have similar triangles but with a scale factor of one, which means that they would be congruent. And so, our rule would show congruence. Before we go through some questions on these two rules, let’s have a look at one final conjecture.

Let’s see if having two corresponding congruent pairs of sides and a corresponding pair of angles would show that two triangles are congruent. Let’s take our starter triangle with the two sides and the angle marked. On our second triangle, we have our length marked on the base. We’re constraining the angle marked in green. And we’re constraining the length of this second line, the line in pink. It could, however, be drawn at any angle.

Let’s try drawing a line from this green angle. And what do we notice? We have, in fact, created two alternative triangles. Although this diagram isn’t perfectly accurate, simply knowing that two pairs of sides are congruent and a pair of angles are congruent in two triangles would not show that the triangles themselves are congruent. So, the SSA rule is not a rule for showing congruence. We’ll now look at some questions using the angle-side-angle and the angle-angle-side rules.

Which congruence criteria can be used to prove that the two triangles in the given figure are congruent? Option (A) SSS, option (B) SAS, option (C) ASA.

In this question, we’re asked for a congruence criteria. If we look at the options here, we can see that the S refers to side and the A represents angle. So, let’s look at our two triangles, 𝐴𝐵𝐶 on the left and triangle 𝐸𝐷𝐹 on the right. We’ll make a note of any corresponding pairs of angles or sides which are congruent.

In triangle 𝐴𝐵𝐶, we can see that this angle 𝐴𝐵𝐶 is marked as 104 degrees. The same is true of angle 𝐸𝐷𝐹 in triangle 𝐸𝐷𝐹. Therefore, we could say that these two angles are congruent. We can see that angle 𝐴𝐶𝐵 is 22.8 degrees and so is angle 𝐸𝐹𝐷. So, we have another pair of congruent angles. We can see that there are two sides which are marked as 7.1, side 𝐴𝐶 on triangle 𝐴𝐵𝐶 and side 𝐸𝐹 on triangle 𝐸𝐷𝐹. Therefore, these sides are congruent.

What we’ve shown here is that we have angle-angle-side, so we could use the angle-angle-side rule to prove congruence. We could say that triangle 𝐴𝐵𝐶 and triangle 𝐸𝐷𝐹 are congruent using the AAS rule.

A quick reminder that the order of letters is important when describing congruence. For example, we know that angle 𝐶 in triangle 𝐴𝐵𝐶 is congruent with angle 𝐹 in triangle 𝐸𝐷𝐹. We know that angle 𝐵 and angle 𝐷 are congruent, and angle 𝐴 and 𝐸 are congruent. So, when we look at the answer options, we see a problem. The AAS rule is not listed as an option. So, let’s see if we could prove congruence using another rule too.

We don’t know any additional information about the length of the sides. So, let’s have a look at the angles. If we look at angle 𝐵𝐴𝐶 in the first triangle and angle 𝐷𝐸𝐹 in the second triangle, we could actually work out the value of these angles by subtracting 104 and 22.8 from 180 degrees, as we know that there are 180 degrees in total in the triangle. So, both of these angles would be equal to each other; they’re congruent.

We’ve also just proved that these two triangles are congruent. Therefore, we know that these third angles must also be congruent. So therefore, if we take into account these last three pieces of information, we have two angles and the included side. Therefore, we have the ASA rule. So, we’ve shown that these triangles are congruent using the ASA rule as well, which was the answer given in option (C).

Let’s now look at a question where we use congruency to find the lengths of some missing sides.

Find the lengths of 𝐶𝐵 and 𝐴𝐷.

When we first look at this diagram, it might not be immediately obvious how to approach this. It’s not clear if we have any lengths the same. And we can simply see from the diagram that there are two pairs of angles which are marked as congruent. So, let’s see if we can prove if the triangles themselves are congruent.

Let’s look at each of these triangles, triangle 𝐵𝐶𝐷 on the left and triangle 𝐵𝐴𝐷 on the right. We can see from the markings on the angle that angle 𝐶𝐵𝐷 and angle 𝐴𝐵𝐷 are congruent. And we can see that angle 𝐵𝐷𝐶 and angle 𝐵𝐷𝐴 are congruent. However, if we look at the sides, we have a side of 20 centimeters and a side of 12 centimeters, which are obviously not congruent. But if we look at the line 𝐵𝐷, we can see that it’s common to both triangles. Therefore, this would be a congruent side.

So, we’ve demonstrated that we have two angles and an included side which are congruent. We could say that triangle 𝐵𝐶𝐷 and triangle 𝐵𝐴𝐷 are congruent using the angle-side-angle rule. Now we know that these triangles are congruent, let’s find our first missing length, 𝐶𝐵.

This line in triangle 𝐵𝐶𝐷 will be congruent with the line 𝐴𝐵 in triangle 𝐵𝐴𝐷. Sometimes, in particular drawings of diagrams, it can be difficult to establish which lengths are actually congruent. So, if we’ve been careful with the lettering, we can see that line 𝐶𝐵 or 𝐵𝐶 would be congruent with the line 𝐵𝐴 or 𝐴𝐵 in triangle 𝐵𝐴𝐷. And for our missing length, we can see that 𝐴𝐵 is 20 centimeters. So, 𝐶𝐵 will also be 20 centimeters.

To find the next missing length 𝐴𝐷, this will be congruent with the line 𝐶𝐷. And as we’re told that that was 12 centimeters, then that means that 𝐴𝐷 will also be 12 centimeters. And therefore, we’ve found our missing length. 𝐶𝐵 is 20 centimeters, and 𝐴𝐷 is equal to 12 centimeters.

Let’s have a look at one final congruency question.

In the given figure, 𝑆𝑃 equals three 𝑥 minus seven and 𝑆𝑀 equals two 𝑥 minus two. Find 𝑆𝑃.

We can begin by filling in the given information about the lengths onto the diagram. We’re not told any information about this figure other than the fact that there is a pair of right angles and there’s also another pair of congruent angles. So, let’s check if these two triangles are congruent.

Let’s start by noting that we have two right angles, angle 𝐽𝑀𝑆 and angle 𝐽𝑃𝑆. So, we have a pair of congruent angles. We can see that angle 𝑀𝐽𝑆 and angle 𝑃𝐽𝑆 from the diagram are also congruent. We’re given the lengths for 𝑆𝑃 and 𝑆𝑀, but we’re not told that these are congruent. And we’re not given any information about the line 𝐽𝑀 or 𝐽𝑃. We can see, however, that the line 𝐽𝑆 is common to both triangles, which means that we have a pair of congruent sides.

As we found two angles and a nonincluded side, we could say that triangle 𝐽𝑀𝑆 and triangle 𝐽𝑃𝑆 are congruent using the angle-angle-side or AAS rule. Let’s see if this will help us to find the length 𝑆𝑃. Using the fact that these triangles are congruent, we know that the length 𝑆𝑃 in triangle 𝐽𝑃𝑆 is congruent with the length 𝑆𝑀 in triangle 𝐽𝑀𝑆. And so, two 𝑥 minus two must be equal to three 𝑥 minus seven. So, we’ll need to solve to find the value of 𝑥.

If we rewrite this so that we have our higher coefficient of 𝑥 on the left-hand side, we’ll have three 𝑥 minus seven equals two 𝑥 minus two. Subtracting two 𝑥 from both sides of the equation, we’ll have 𝑥 minus seven equals negative two. We can then add seven to both sides of the equation, so 𝑥 equals negative two plus seven. And therefore, 𝑥 is equal to five. It can be easy to stop at this point and think that we’ve found the answer. However, we weren’t asked for 𝑥. We were asked for the length 𝑆𝑃.

We were given that 𝑆𝑃 equals three 𝑥 minus seven. So, we plug in our value of 𝑥 equals five into this equation, which gives us 𝑆𝑃 equals three times five minus seven. And as three times five is 15, we’ll have 15 minus seven, which is eight. And so, we have our answer 𝑆𝑃 equals eight, which we found by proving that the two triangles were congruent.

Let’s take a look at the key things that we learnt in this video. We saw how to use and apply the ASA or angle-side-angle rule, which is when we have two angles and the included side. We also saw the angle-angle-side rule, which is two angles and any side. We also saw that angle-angle-angle would just prove that two triangles are similar but not congruent. In the same way, SSA would not be a congruency rule as it isn’t sufficient to prove that two triangles are congruent. And a reminder that when we’re writing congruency relationships, we must ensure that the letter ordering is correct.

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