Video Transcript
In this video, we will learn how to
identify similar triangles using either the AA rule or the SSS rule.
The first thing we should do is
recall that similar shapes have all corresponding angles congruent or equal and all
corresponding sides in proportion. Note that it’s different to
congruent shapes as congruent shapes have corresponding angles equal and
corresponding sides equal. Similar shapes will have sides in
proportion. So when it comes to identifying
similar triangles, there’s a few shortcuts that we can take rather than trying to
demonstrate all the angles are congruent and all the corresponding sides are in
proportion. Let’s start by focusing on the
angles.
On the two triangles drawn here, we
can see that there are three pairs of corresponding angles congruent. Let’s have a look at the larger
triangle. Let’s say that we take a point on
this larger triangle that’s the same length as the corresponding side in the smaller
triangle and draw a line in parallel to the base of this triangle. If we take a look at the angles,
then as we have a pair of parallel lines and two transversals, then we’d have two
pairs of equal corresponding angles. The triangle proportionality
theorem says that if a line is parallel to one side of a triangle and it intersects
the other two sides, then it divides those two sides proportionally.
So if we label the vertices of the
larger triangle with 𝐴𝐵𝐶 and the smaller triangle with vertices 𝐷𝐸𝐹, then we
would say that the side 𝐴𝐵 is proportional to the side 𝐷𝐸, and that would be the
same proportion as 𝐴𝐶 to 𝐷𝐹, which would be the same as the proportion of 𝐵𝐶
to 𝐸𝐹. Therefore, showing all three angles
are congruent, we’ll also demonstrate that the corresponding sides are in
proportion.
In fact, there is an even simpler
way than showing our three angles are congruent. And that’s because if we only show
two angles, because the angles in a triangle sum to 180 degrees, we know that if
we’ve got two pairs of corresponding angles congruent in a triangle, then the third
angles must also be congruent. And therefore, if we want to prove
that two triangles are similar, we only need to use the AA rule to show that two
pairs of corresponding angles are congruent.
Next, let’s think about the sides
in these triangles. The proportion of 𝑃𝑄 over 𝐾𝐿
can be written as six over eight. The proportion of 𝑃𝑅 to 𝐾𝑀 can
be written as 7.5 over 10. Finally, 𝑄𝑅 over 𝐿𝑀 would be
the proportion nine over 12. Each of these three ratios all
simplify to three-quarters. Therefore, the sides have the same
proportion. As we saw above, if the sides are
in proportion, then corresponding pairs of angles will be congruent. This is called the SSS rule, which
means that if we show that corresponding pairs of sides are in proportion, then we
demonstrate that we have a pair of similar triangles.
Remember, be careful and don’t
confuse it with the SSS rule that we might use for congruent triangles; in that
case, we’d have to show they’re congruent. But for similar triangles, we show
the SSS rule meaning the sides are in proportion.
In the following questions, we’ll
be using either of these two rules to help us identify similar triangles. Let’s have a look at our first
question.
Given the following four
shapes, which two are similar?
In this question, we’re given
four different shapes, complete with the side measurements. We’re asked which of these
shapes are similar. And we recall that similar
shapes have corresponding angles congruent or equal and corresponding sides in
proportion. We’re not given any information
regarding the angles in these four triangles. So, let’s have a look at the
sides.
We can notice that within these
four triangles, we do have two that look similar. Shape two and four would both
be isosceles triangles, and shape one and shape three also look as though
they’re a similar shape. We need to check each pair of
triangles, however, to see if the corresponding sides are in proportion. Let’s begin with our isosceles
triangles, that’s shapes two and four. Let’s take a corresponding pair
of sides.
We would say then that the
ratio here would be 12 in the larger triangle over seven in the smaller
triangle. Is this equal to the ratio of
our other sides eight in the larger triangle and five in the smaller
triangle? In order to more easily compare
these fractions, let’s make the denominator the same. In order to change
twelve-sevenths into a fraction over 35, we would need to multiply the numerator
and denominator by five, which would give us a value of 60 on the numerator. On the right-hand side, we
would need to multiply the numerator and denominator by seven. But as we see 60 over 35 is not
equal to 56 over 35. So, shapes two and four are not
similar.
Let’s check shapes one and
three. Writing our proportions with
the value of the smaller triangle’s lengths as the numerator and the larger
triangle’s lengths as the denominator would give us the values of four- eighths,
three-sixths, and five-tenths. We should notice that all of
these fractions simplify to one-half. So, we’ve demonstrated that
these three pairs of corresponding sides are in proportion, and that applies the
SSS rule of similar triangles. So, our answer is that out of
these four shapes, shape one and shape three are similar.
Let’s have a look at another
question.
Are the two triangles
similar?
Let’s begin by recalling that
similar triangles have corresponding angles congruent and corresponding sides in
proportion. If we look at the smaller
triangle 𝐴𝐷𝐸 and the larger triangle 𝐴𝐵𝐶, it doesn’t seem as though these
two triangles would be the same shape. However, let’s see if we could
prove it mathematically, just in case the triangles are not drawn correctly. In order to show that two
triangles are similar, we can remember that we would either use the AA rule to
show that two pairs of angles are congruent or the SSS rule to demonstrate that
there are three pairs of corresponding sides in proportion. We’re not given enough
information about the angles here, so let’s see if we can use the SSS rule.
We need to check if there’s the
same ratio or proportion between corresponding sides. For example, is this side 𝐴𝐸
over the side 𝐴𝐶 in the same proportion as 𝐴𝐷 over 𝐴𝐵 and the same as 𝐸𝐷
over 𝐶𝐵? We can fill in the numerical
values for the lengths into our side information. But as we don’t have any
dimensions for sides 𝐸𝐷 and 𝐶𝐵, we won’t be able to prove the two triangles
are similar. But if the proportions of 𝐴𝐸
and 𝐴𝐶 and 𝐴𝐷 and 𝐴𝐵 are different, then we could prove that they are not
similar. Let’s take a look.
The length of 𝐴𝐸 is given as
46 centimeters. But be careful as the length of
𝐴𝐶 isn’t 32.2, but rather it’s the sum of 46 and 32.2, which is 78.2. 𝐴𝐷 is 22, and 𝐴𝐵 is the sum
of 22 and 24.2 centimeters, which is 46.2 centimeters. We now need to compare these
two fractions to see if they’re equivalent. We can begin by removing this
decimal point from the denominator. We can do this by multiplying
both of our numerators and denominators by 10. Dividing the numerator and
denominator by 46, then 460 over 782 simplifies to 10 over 17. Dividing the numerator and
denominator of 220 over 462 by 22 gives us the fraction 10 over 21. We can, therefore, see that
these two ratios are not equal. Therefore, the sides are not in
proportion. If the sides are not in
proportion, then the triangles would not be similar. So, our answer here would be
no.
Before we finish with this
question, just a point to note. If we had find that 𝐴𝐸 over
𝐴𝐶 was equal to 𝐴𝐷 over 𝐴𝐵, then we would have also needed the values for
𝐸𝐷 and 𝐶𝐵. As it’s not enough just to show
that two sides are in proportion, we would need to show that there are three
pairs of corresponding sides in proportion. In this question, it didn’t
matter that we didn’t have these two other lengths as it was enough to show that
these triangles were not similar.
In the next question, we’ll see if
we can apply the AA rule to demonstrate a similarity.
The figure shows two
triangles. Are the two triangles
similar? Why?
In this question, we’re asked
if these two triangles are similar. We can recall that similar
triangles have corresponding angles congruent and corresponding sides in
proportion. So, let’s have a look at the
angles in both of these triangles. There’s a right angle of 90
degrees in each triangle. And one triangle has a
60-degree angle, and the other one has a 30-degree angle. However, if we remember that
the angles in a triangle add up to 180 degrees, then we could calculate the
third angle in each triangle.
In the top triangle, 60 plus
the right angle of 90 degrees would give us 150 degrees, and subtracting that
from 180 degrees would leave us with 30 degrees. In the lower triangle, 90
degrees plus 30 gives us 120 degrees, leaving us with a third angle of 60
degrees. What we can see then is that we
have three pairs of equal angles. We have two 30-degree angles,
two 60-degree angles, and two 90-degree angles. We have shown a three pairs of
corresponding angles congruent. But in fact, we only need to
show two pairs of corresponding angles congruent in order to show that two
triangles are similar.
So, to answer the question “are
the two triangles similar?” is yes. To answer the second question
“why?,” we could say that if you calculate the measure of the third angle in one
of the triangles, you can see that the triangles share two angles. Therefore, by the AA criteria,
the triangles are similar.
The figure shows two triangles,
𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime. Work out the measure of angle
𝐴𝐵𝐶. What does the AA criterion tell
us about these two triangles?
In this question, we have two
triangles drawn on grid paper. And the first thing we’re asked
to do is find the measure of angle 𝐴𝐵𝐶, which is in the smaller triangle. In order to do this, we should
recall that the angles in a triangle add up to 180 degrees. We’ll therefore need to
calculate 180 degrees subtract the other two angles of 114.3 degrees and 34.1
degrees, which gives 31.6 degrees. And so that’s our answer for
the measure of angle 𝐴𝐵𝐶.
In the second part of this
question, we’re asked about the AA criterion, which is the criterion we use to
show that two triangles are similar. It’s what we have when we
demonstrate that there are two pairs of angles congruent. So, let’s have a closer look at
these two triangles. The angle 𝐴𝐵𝐶 that we’ve
just worked out as 31.6 degrees has a corresponding angle at angle 𝐴 prime 𝐵
prime 𝐶 prime of the same value, 31.6 degrees. We also have another pair of
corresponding congruent angles, the angle 𝐶 prime 𝐴 prime 𝐵 prime and angle
𝐶𝐴𝐵, which are both given as 34.1 degrees. Showing that there are two
pairs of corresponding congruent angles or the AA rule, we demonstrate that
these two triangles are similar.
We could therefore answer the
second part of this question with a statement such as this: as both triangles
share two angels of equal measures, they must be similar.
We’ll now have a look at one final
question.
Triangles 𝐴𝐵𝐶 and 𝐴 prime
𝐵 prime 𝐶 prime in the given figure are similar. Work out the value of 𝑥.
We’re told here that these two
triangles are similar. This means that corresponding
angles are congruent and corresponding sides are in proportion. Looking at the triangles, this
would mean that the angle at 𝐵 would be congruent with the angle at 𝐵 prime as
these two angles are corresponding. Angle 𝐶 and 𝐶 prime would be
congruent, and angle 𝐴 and 𝐴 prime are congruent.
We’re given measurements for
angle 𝐴 and 𝐴 prime. So, we could write that five 𝑥
plus 90 over six must be equal to three 𝑥 plus 320 over six. We could then solve to find the
value of 𝑥. Multiplying both of these
fractions by six would give us that five 𝑥 plus 90 is equal to three 𝑥 plus
320. Subtracting three 𝑥 from both
sides would give us two 𝑥 plus 90 is equal to 320. Subtracting 90 from both sides
would give us two 𝑥 equals 230. And finally, dividing by two
would give us 𝑥 equals 115. And so, we have our answer that
the value of 𝑥 would be equal to 115. We don’t need to include the
degree sign as it was part of the angle definition in 𝐴 and 𝐴 prime.
We can check our answer by
substituting this value for 𝑥 back in and find that angle 𝐵𝐴𝐶 is 110 and 56
degrees and so is angle 𝐵 prime 𝐴 prime 𝐶 prime.
We can now summarize what we’ve
learned in this video. Firstly, we saw that similar
triangles have corresponding pairs of angles congruent and corresponding pairs of
sides in proportion. We can identify similar triangles
by using the AA rule which is when we have two pairs of corresponding angles
congruent. We can also identify similar
triangles using the SSS criterion, which is where all three pairs of corresponding
sides are in the same proportion. Remember that we only need to show
that the corresponding sides are in proportion and we don’t need to show that
they’re congruent. We should also remember when we’re
working through problems like these that triangles can be similar even if they’re in
a different orientation.
