If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar

A Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

A Theorem

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

A Corollary

If the sides of two triangles are in proportion, then the trianlges are similar.

A Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.