Side F O is congruent to side H E side O X is congruent to side E N, and ∠ O and ∠ E are the included, congruent angles. Both ∠ O and ∠ E are included angles between sides F O and O X on △ F O X, and sides H E and E N on △ H E N. Notice that ∠ O on △ F O X corresponds to ∠ E on △ H E N. To make your life easy, we made them both equilateral triangles. You cannot compare two sides of two triangles and then leap over to an angle that is not between those two sides. The included angle refers to the angle between two pairs of corresponding sides. If the ratios are congruent, the corresponding sides are similar to each other. You can establish ratios to compare the lengths of the two triangles' sides. Their comparative sides are proportional to one another their corresponding angles are identical.

Triangles are easy to evaluate for proportional changes that keep them similar. You look at one angle of one triangle and compare it to the same-position angle of the other triangle. The three theorems for similarity in triangles depend upon corresponding parts. Even if two triangles are oriented differently from each other, if you can rotate them to orient in the same way and see that their angles are alike, you can say those angles correspond. In geometry, correspondence means that a particular part on one polygon relates exactly to a similarly positioned part on another. These three theorems, known as Angle - Angle (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS), are foolproof methods for determining similarity in triangles. Similar triangles are easy to identify because you can apply three theorems specific to triangles. An equilateral triangle with sides 21 cm and a square with sides 14 cm would not be similar because they are different shapes. You could have a square with sides 21 cm and a square with sides 14 cm they would be similar. In geometry, two shapes are similar if they are the same shape but different sizes. This is an everyday use of the word 'similar,' but it not the way we use it in mathematics. Mint chocolate chip ice cream and chocolate chip ice cream are similar, but not the same. Similar triangles are triangles with the same shape but different side measurements. Similarity in mathematics does not mean the same thing that similarity in everyday life does. Similar Triangles (Definition, Proving, & Theorems)

Aa similarity postulate