What defines similar triangles?

Similar triangles are defined by the property that their corresponding angles are equal and their corresponding sides are proportional. This means that while the triangles may vary in size, the shape remains consistent—every angle in one triangle matches with an angle in the other triangle, and the lengths of the sides of both triangles can be related by a common ratio. This proportionality allows for the application of various geometric principles and theorems, such as those related to area, similarity, and scale factors.

For instance, if one triangle has sides of lengths 3, 4, and 5, and a second triangle has sides of lengths 6, 8, and 10, these triangles are similar. The angles are all the same, and the sides of the larger triangle are precisely double those of the smaller triangle, demonstrating the proportional relationship that defines similarity in triangles. This property is a fundamental concept in geometry and is widely used in solving problems involving geometric figures.