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.
