Proportionality are important in many relationships. The fundamental properties of proportions are useful in geometric proofs.
Midsegment: The segment that joins the midpoints of two sides of a triangle.
Transversal: A line that intersects a set of lines (may or may not be parallel).
Angle Bisector: A ray that divides an angle into two congruent angles, each with a measure equal to exactly half of the original angle.
Recall that every triangle has three midsegments.
The midsegment divides the other two sides of the triangle proportionally.
The Midsegment Theorem is a special case of the Triangle Proportionality Theorem.
Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides into proportional segments.
The converse is also true.
Converse of the Triangle Proportionality Theorem: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Theorem: If two transversals intersect the same set of parallel lines, then the parallel lines divide the transversals into proportional segments.
This theorem can be expanded to any number of parallel lines and any number of transverals.
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.