Which two of these triangles are
In order for shapes to be similar,
they must have corresponding sides that are proportional and corresponding angles
that are equal. So for sides to be proportional,
that means maybe one is smaller than the other. So say one that’s smaller has the
side length of one and then the other corresponding side on a larger triangle has a
side length of three. That means its side of the larger
triangle is three times larger than the smaller triangle.
So if the smaller triangle had
another side length of two, that corresponding side of the larger triangle must be
two times three. It must be three times bigger. So it must be six. And the corresponding angles must
be equal. So they must be exactly the
So let’s look at the sides being
proportional. These two triangles have two side
lengths that are congruent. So they would be considered
isosceles. And these two triangles are both
equilateral because all of their sides are equal. So one and three have corresponding
side lengths that are proportional.
Let’s say that the smaller triangle
— number three — has side length of one, then the larger triangle — triangle number
one — say it has a side length of three, then they all will be three. So each of those sides would be
three times larger than the smaller one, just like we said in our example. And the same could be true for two
and four. Say these are lengths of four and
these are lengths of eight. Those sides will be twice as
large. They will be proportional.
Would this make the last two sides
proportional? Well, before we give that too much
thought, what about the fact that the corresponding angles must be equal. How are we going to know what these
angles are equal to? We won’t.
However, with equilateral
triangles, all of their angles are equal. And actually a triangle has a total
of 180 degrees. So if we split that equally between
three angles, we get 60 degrees. Therefore, we would know that for
triangles one and three, their corresponding angles will be equal. And we already know that the
corresponding sides are proportional. Therefore, triangles one and three
will be similar.