Video Transcript
In this video, we’re going to look
at congruent triangles. We’re going to use the SSS, the
SAS, and the ASA rules to find congruence. We’ll then look at how we can use
this congruency to find missing angles or sides in congruent triangles.
Let’s begin by reminding ourselves
of these congruency rules. The first congruency rule is the
SSS rule, which stands for showing that we have three pairs of corresponding sides
congruent. So, if we take these two triangles,
we would start by establishing if we have a pair of congruent sides, then a second
pair of congruent sides, and then finally a third pair of congruent sides. So, demonstrating that there are
three pairs of congruent sides in two triangles would fulfill the SSS rule and prove
that triangles are congruent.
The important thing in any of these
congruency relationships is that it doesn’t matter if the triangles are rotated or
flipped. They would still be congruent. They’re still the same shape and
the same size no matter what orientation they’re in.
The second rule is the SAS
rule. This time the A stands for the
included angle between two sides. So, in our two triangles, we would
demonstrate that there are two corresponding sides congruent and the angle in
between them is congruent. That would show that these two
triangles are congruent.
The ASA rule is the
angle-side-angle rule. And this time, the side is included
between two angles. So, we’d show a pair of sides
congruent, like this, and two pairs of congruent angles, remembering that the side
is in between or included between these two angles.
The AAS rule is similar in that
it’s two pairs of corresponding angles congruent and a pair of sides congruent. However, when we’re using the AAS
rule, the side does not have to be included between the two angles.
The final congruency rule is the
RHS rule, which applies in right triangles. The R stands for right angle, and
the H stands for hypotenuse, which is the longest side in a right triangle. To show congruence using this rule,
we’d need to show that both triangles have a right angle, the hypotenuse on both
triangles is congruent. And that out of the other two sides
on the triangle, there’s a pair of congruent sides.
Now that we have the five rules
listed, we’re going to use the first three rules in particular to solve
problems. Knowing that we have congruent
triangles can help us to find any missing sides or angles. Let’s have a look at our first
question.
Are two triangles congruent if both
triangles have the same side lengths?
Let’s start by reminding ourselves
that congruent means the same shape and size. All the corresponding pairs of
sides in these triangles would be the same length, and all the corresponding pairs
of angles would be the same size. Sometimes, when we’re answering a
question like this, it can be helpful to draw out a few triangles to
investigate.
Let’s draw out this triangle which
has lengths of four, five, and six units. We could draw another triangle that
also has the same lengths of four, five, and six. We could even draw another one that
looks like this. So, even though these triangles are
all in different orientations, are they still congruent? And the answer is yes. We couldn’t draw a differently
shaped triangle that has these side lengths of four, five, and six.
We can also apply the congruency
rule SSS. This stands for three pairs of
corresponding sides congruent. So, we can see that we have got a
set of sides here in these three triangles which are all of length four. We have another corresponding set
of lengths five units and a third set of corresponding length of six units.
In this example, we use the lengths
of four, five, and six units. But this works for any size of
triangles. If we can show that there are three
pairs of corresponding side lengths or that the triangles in other words here have
the same side lengths, then we would say that these are congruent. And so, our answer to the question
would be yes.
In the next question, we’re told
that there’s a congruency. And we’ll need to work out a
missing angle.
Given that triangle 𝐴𝐵𝐶 is
congruent to triangle 𝑋𝑌𝑍, find the measure of angle 𝐵.
Because we’re told that these two
triangles are congruent, that means that they will have pairs of corresponding
angles congruent. What we need to do here is to work
out exactly which angles are congruent. Sometimes, it can be very easy to
tell from a diagram and other times not quite so easy. But we can use the order of the
letters to help us.
As we’re told that triangle 𝐴𝐵𝐶
is congruent to triangle 𝑋𝑌𝑍, that means that the angle at 𝐴 is corresponding
with the angle at 𝑋. So, both the angle at 𝐴 and the
angle at 𝑋 will be 54 degrees. In the same way, this angle 𝐵 in
triangle 𝐴𝐵𝐶 is corresponding to the angle 𝑌 in triangle 𝑋𝑌𝑍. And both of these will be 52
degrees. As we were asked to find the
measure of angle 𝐵, then our answer would be that this is 52 degrees.
For completeness, we can note that
the angle at 𝐶 would be corresponding to the angle at 𝑍. And these would both be 74
degrees. But as we’re asked for 𝐵, that’s
52 degrees.
In the next question, before
finding a missing length, we’ll need to prove that the triangles are congruent.
In the figure, 𝐵𝐷 meets 𝐴𝐸 at
𝐶, which is also the midpoint of 𝐵𝐷. Find the length of 𝐶𝐸.
The missing length that we need to
find out is the length 𝐶𝐸. It may not be immediately obvious
how to find this length of 𝐶𝐸. However, these triangles look very
close to being the same shape and the same size, in other words, congruent. Let’s see if we have enough
information about any sides or angles to prove congruence.
In the question, we’re told that
𝐵𝐷 meets 𝐴𝐸 at this point 𝐶. 𝐶 is the midpoint of 𝐵𝐷. As it’s the midpoint, then the
length 𝐵𝐶 of 27 will also be the same for the length 𝐶𝐷. Then, if we consider our triangles
𝐸𝐷𝐶 and 𝐴𝐵𝐶, we could write that the length 𝐷𝐶 is equal or congruent to 𝐵𝐶
as they’re both of the length 27.
Let’s have a look at some angles
next. We have this right angle at angle
𝐴𝐵𝐶, but will there be a right angle in triangle 𝐸𝐷𝐶? Well, yes, there will be. But let’s think about why. We have this line 𝐷𝐵, which is
perpendicular to the line 𝐴𝐵, and so creating a right angle. But there’s also a parallel line to
𝐴𝐵. And that’s the line 𝐸𝐷. And that’s why we’ll also have a
right angle here at 𝐸𝐷𝐶.
Now that we’ve shown that we have a
pair of sides congruent and a pair of angles congruent, what else can we see from
the diagram? We have this length 𝐴𝐵, which is
36. But we can’t say for sure that
there’s any other length that’s also the same length on triangle 𝐸𝐷𝐶.
But let’s have a think about this
angle at 𝐵𝐶𝐴. There would, in fact, be a
congruent angle. Angle 𝐷𝐶𝐸 would be equal to the
angle 𝐵𝐶𝐴 because these are vertically opposite angles.
If we look at what we’ve shown on
these two triangles, we’ve got a pair of corresponding sides, a pair of congruent
angles, and another pair of congruent angles. The side here is included between
the two angles, so we could use the ASA rule to say that triangle 𝐸𝐷𝐶 is
congruent to triangle 𝐴𝐵𝐶.
It’s important to remember that
even though we have a right triangle, the RHS congruency rule would not be
applicable here. To use the RHS rule, we need to
show that there’s a right angle, hypotenuse, and side congruent. As we don’t know the hypotenuse
length here, then we can’t use this rule. But let’s go ahead and see if we
can work out the length of 𝐶𝐸.
The length that corresponds to 𝐶𝐸
in triangle 𝐴𝐵𝐶 will be this length of 𝐴𝐶. We’re not told what this length of
𝐴𝐶 is, but there’s a way that we can work it out. The Pythagorean theorem tells us
that the square on the hypotenuse is equal to the sum of the squares on the other
two sides. If we take a look at the triangle
𝐴𝐵𝐶, we don’t know the hypotenuse. We want to find that out. So, let’s define this length as
𝑥.
With the other two sides of 27 and
36, we can fill these into the Pythagorean theorem to give us 𝑥 squared equals 27
squared plus 36 squared. Evaluating our squares, we’ll have
𝑥 squared equals 729 plus 1296. And adding these, we have 𝑥
squared equals 2025. Taking the square root of both
sides of our equation will give us that 𝑥 equals 45. Now that we have found this length
of 𝐶𝐴 is 45, we can say that the corresponding length of 𝐶𝐴 on triangle 𝐸𝐷𝐶
will also be 45. And that’s our answer for the
question to find the length of 𝐶𝐸.
In the final question, we’ll see
how we can use congruency to help us find the area of a triangle.
The two triangles in the given
figure are congruent. Work out the area of triangle
𝐴𝐵𝐶.
We’re told here that the two
triangles are congruent. That means that pairs of
corresponding angles will be equal and pairs of corresponding sides will be
equal. We’ll need to use this fact to help
us work out the area of triangle 𝐴𝐵𝐶.
We can recall that to find the area
of a triangle, we multiply half times the base times the perpendicular height. When we look at triangle 𝐴𝐵𝐶, we
can see that we don’t know the base length of this triangle, which is why we’ll need
to use the fact that this is congruent with triangle 𝐷𝐸𝐹 to help us work out the
length of 𝐵𝐶.
In this question, we weren’t given
a congruency relationship, so we’ll need to establish which sides correspond to
which sides. Let’s start with the hypotenuse,
the longest side on triangle 𝐴𝐵𝐶. This will correspond with the
longest side or hypotenuse on our other triangle. So, 𝐴𝐶 and 𝐷𝐹 will be
congruent.
On triangle 𝐴𝐵𝐶, if we go from
the hypotenuse down to the right angle along the line 𝐴𝐵, this corresponds to the
same journey or path from the hypotenuse down to the right angle on triangle
𝐷𝐸𝐹. So, 𝐴𝐵 and 𝐷𝐸 will be the same
length of 5.1. The final pair of sides 𝐵𝐶 and
𝐸𝐹 will also be congruent, and they’ll be of length 4.1.
We now have enough information to
work out the area of triangle 𝐴𝐵𝐶. Filling in the values for our base
length of 4.1 and the perpendicular height of 5.1, we’ll have a half times 4.1 times
5.1. We can work out 4.1 times 5.1 by
calculating 41 times 51. As our values had a total of two
decimal digits, then our answer will also have two decimal digits. Half of 20.91 will give us
10.455. The units here would be square
units. This is our answer for the area of
triangle 𝐴𝐵𝐶. Note that if we worked out the area
of triangle 𝐷𝐸𝐹 instead, we would’ve got the same answer as both of these
triangles are congruent.
We can now summarize what we’ve
learned in this video. We saw that triangles can still be
congruent even if they’re reflected or rotated; they don’t have to be in the same
orientation. We reminded ourselves of the
congruency rules: SSS, SAS, ASA, AAS, and RHS. And finally, when we’re proving
triangles are congruent, we may need to use other angle rules, for example,
remembering that vertically opposite angles are equal.
