### Video Transcript

Which two of these triangles are
similar? And we’re given four different
triangles numbered (1) through (4).

Let’s begin by recalling the
meaning of similar. Similar triangles, like any similar
shapes, will have corresponding angles equal and corresponding sides in
proportion.

If we take a quick look at the
triangles that we’re given, we can see that triangles (1) and (2) both have an angle
of 40 degrees and triangle (4) has an angle of 100 degrees. There’s not enough information
given in any of these triangles to prove that they’re similar. So let’s take a look and see if we
can work out any of the other angles in each triangle.

To help us do this, we’ll need to
remember that the angles in a triangle add up to 180 degrees. So let’s take a look at triangle
number (1). Triangle (1) has this markation on
the lines, indicating that there are two sides the same length. We remember that a triangle with
two sides the same length is an isosceles triangle. An isosceles triangle also has a
pair of equal angles. The two equal angles will be these
ones marked in orange. And we’re given that the measure of
one of these angles is 40 degrees. Therefore, they’re both 40
degrees. We can then find the third unknown
angle in this triangle using the fact that these three angles must add up to 180
degrees. 40 plus 40 gives us 80. And subtracting that from 180
degrees leaves us with 100 degrees.

Now that we know the three angles
in this triangle, we can see if any of the other triangles will have similar angle
measurements. Let’s have a look then at triangle
number (2). And we notice that once again this
is also an isosceles triangle. It’s got three line markings this
time to indicate that these signs are not necessarily the same length as those in
triangle (1). When we consider which two angles
will be the same size in this isosceles triangle number (2), it will be these two
angles at the base of the triangle. Because we know that these angles
are the same size and we’re given that the other angle is 40 degrees, then we can
work out their value.

Let’s say that we call this unknown
angle 𝑥 degrees. We know that the three angles must
add up to give 180 degrees. So we could write the equation two
𝑥 degrees plus 40 degrees is equal to 180 degrees. We could then subtract 40 degrees
from both sides. And dividing through by two gives
us that 𝑥 is equal to 70 degrees. Therefore, in triangle (2), we have
two angles of 70 degrees and an angle of 40 degrees. We now know that triangles (1) and
(2) are not similar because we don’t have corresponding pairs of angles equal.

Let’s take a look then at triangle
number (3). Once again, we can see that this is
an isosceles triangle. And although we’re not given any
angle measurement, we are given the indication that we have a right angle of 90
degrees. In this isosceles triangle, we’ll
have two pairs of equal angles, which will be here in orange. So let’s see if we can work out the
value of these. Using the same approach as before,
we can take the value of the unknown angle as 𝑦. Therefore, 𝑦 plus 𝑦 plus 90
degrees is 180 degrees. Simplifying to two 𝑦 degrees plus
90 degrees is 180 degrees. Subtracting 90 degrees from both
sides and then dividing through by two gives us that 𝑦 must be equal to 45
degrees.

Now that we have the three angle
measurements in triangle number (3), we can also see that none of these three
triangles would be similar to each other. So let’s have a look at the fourth
triangle.

Triangle (4) is also an isosceles
triangle with a pair of equal angles here in pink. As these two angles are the same
and all three angles add up to 180 degrees, then the two unknown angles must be 40
degrees. The other triangle in this set,
which has angles of 100 degrees, 40 degrees, and 40 degrees, is triangle number
(1).

Therefore, we can give the answer
for which two of these triangles are similar as triangles (1) and (4).

As we were working through this
problem, we found all three angles in each triangle. However, in order to prove that two
triangles are similar, we only need to prove that there are two pairs of
corresponding angles equal. Proving that means that the AA rule
is fulfilled and the triangles are similar. However, showing that all three
corresponding angles are equal in triangles (1) and (4) as we did here is also
valid.