Basic Proportionality Theorem (Thales' Theorem)
Theorem Statement:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
In $\Delta ABC$, if line $DE \parallel BC$:
$$\frac{AD}{DB} = \frac{AE}{EC}$$
Converse of BPT:
If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.
Criteria for Similarity of Triangles
Two triangles are similar ($\sim$) if their corresponding angles are equal and their corresponding sides are in the same ratio.
- AAA Similarity: If in two triangles, corresponding angles are equal, then their corresponding sides are proportional and hence the triangles are similar (AA similarity is sufficient).
- SSS Similarity: If corresponding sides of two triangles are proportional, then their corresponding angles are equal.
- SAS Similarity: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional.
