Transformations and SymmetryStep

Similarity on Rays

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

We can extend this theorem to a situation outside of triangles where we have multiple parallel lines cut by transverals.

Theorem: If three or more parallel lines are cut by two transversals, then they divide the transversals proportionally.

Think about a midsegment of a triangle. A midsegment is parallel to one side of a triangle and divides the other two sides into congruent halves. The midsegment divides those two sides proportionally.

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

Triangle Proportionality Theorem Converse: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.