Triangles - Similarity

Two figures are similar if they have the same shape but not necessarily the same size.
Congruent figures are always similar, but similar figures are not necessarily congruent.
All circles are similar, all squares are similar and all equilateral triangles are similar.
Polygons with the same number of sides are similar if:
- Their corresponding angles are equal.
- Their corresponding sides are in the same ratio (or proportion).
The ratio of corresponding sides is known as the scale factor.

Two triangles are similar if:
- Their corresponding angles are equal.
- Their corresponding sides are in the same ratio (or proportion).
If corresponding angles of two triangles are equal, they are called equiangular triangles.

Theorem 6.1: If a line is drawn parallel to one side of a triangle intersecting the other two sides at distinct points, then the other two sides are divided in the same ratio.
- If $DE || BC$ , then $\frac{AD}{DB} = \frac{AE}{EC}$ .

Theorem 6.2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
- If $\frac{AD}{DB} = \frac{AE}{EC}$ , then $DE || BC$ .

AAA (Angle-Angle-Angle) Similarity Criterion (Theorem 6.3): If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio, and the triangles are similar.
AA Similarity Criterion: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
SSS (Side-Side-Side) Similarity Criterion (Theorem 6.4): If in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal, and the triangles are similar.
SAS (Side-Angle-Side) Similarity Criterion (Theorem 6.5): If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.