# The relationship between similar triangles

### Ratios and Proportions - Similar figures - First Glance Two geometrical objects are called similar if they both have the same shape, or one has the Due to this theorem, several authors simplify the definition of similar triangles to . The relationship holds for figures that are not rectifiable as well. Two figures that have the same shape are said to be similar. When two figures are similar, the ratios of the lengths of their corresponding sides are equal. Two geometrical figures, not only triangles, are similar if one can become congruent to the other by resizing (uniformly scaling), in addition to.

We can see it in just the way that we've written down the similarity. If this is true, then BC is the corresponding side to DC. So we know that the length of BC over DC right over here is going to be equal to the length of-- well, we want to figure out what CE is. That's what we care about. The corresponding side over here is CA. It's going to be equal to CA over CE. This is last and the first. Last and the first. And we know what BC is. BC right over here is 5.

We know what DC is. We know what CA or AC is right over here. And now, we can just solve for CE. Well, there's multiple ways that you could think about this.

### Similar Triangles: Perimeters and Areas

You could cross-multiply, which is really just multiplying both sides by both denominators. So you get 5 times the length of CE. We were able to use similarity to figure out this side just knowing that the ratio between the corresponding sides are going to be the same. Now, let's do this problem right over here.

## Solving similar triangles

Let's do this one. Let me draw a little line here to show that this is a different problem now. This is a different problem. So in this problem, we need to figure out what DE is.

And we, once again, have these two parallel lines like this. And so we know corresponding angles are congruent. So we know that angle is going to be congruent to that angle because you could view this as a transversal. We also know that this angle right over here is going to be congruent to that angle right over there.

Once again, corresponding angles for transversal. Once again, we could have stopped at two angles, but we've actually shown that all three angles of these two triangles, all three of the corresponding angles, are congruent to each other. And once again, this is an important thing to do, is to make sure that you write it in the right order when you write your similarity.

## Similar Triangles: Perimeters and Areas

We now know that triangle CBD is similar-- not congruent-- it is similar to triangle CAE, which means that the ratio of corresponding sides are going to be constant.

So we know, for example, that the ratio between CB to CA-- so let's write this down. And we know what CB is. CB over here is 5. We know what CA is. And we have to be careful here. The triangles have two congruent angles,  which in Euclidean geometry implies that all their angles are congruent. All the corresponding sides have lengths in the same ratio: This is equivalent to saying that one triangle or its mirror image is an enlargement of the other.

Two sides have lengths in the same ratio, and the angles included between these sides have the same measure. This is known as the SAS similarity criterion. Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other transitivity of similarity of triangles. Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio. The statement that the point F satisfying this condition exists is Wallis's postulate  and is logically equivalent to Euclid's parallel postulate. In the axiomatic treatment of Euclidean geometry given by G.