Theory:

In this section, we will learn about the criteria for similarities of triangles.

Let us first recall what the similarity of triangles is.

The similarity of triangles:
A triangle is said to be similar if:

1. their corresponding angles are equal.

2. their corresponding sides are in the same ratio (or proportion).
Example:
Let us look at the two similar triangles, $$PQR$$ and $$XYZ$$, to understand better.
In this case:

$$\angle P$$ $$=$$ $$\angle X$$, $$\angle R$$ $$=$$ $$\angle Z$$, and $$\angle Q$$ $$=$$ $$\angle Y$$.

Similarly, $$\frac{PQ}{XY}$$ $$=$$ $$\frac{QR}{YZ}$$ $$=$$ $$\frac{RP}{ZX}$$
But, should we consider equality relations of all corresponding angles and equality relations of the ratios of their corresponding sides?

In the previous year, we were introduced to the congruence of triangles.

We proved the congruence of triangles using a few criteria.

Similarly, we can prove the similarity of triangles with the help of a few criteria, and we can look at it in detail in the sections that follow.

Important!
Let us look at some important symbolical conditions.
1. The similar triangles $$PQR$$ and $$XYZ$$ can be represented as $$\triangle PQR \sim \triangle XYZ$$.

2. In expressing the similarity between two triangles, appropriate expressions has to be followed.

That is, the similar triangles $$PQR$$ and $$XYZ$$ can be represented as $$\triangle PQR \sim \triangle XYZ$$.

It cannot be represented as $$\triangle PQR \sim \triangle YZX$$. But, similar triangles can be represented as $$\triangle QRP \sim \triangle YZX$$.