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.**

*Midsegment Theorem:*The midsegment of a triangle is parallel to one side of a triangle and divides the other two sides in half.

The midsegment divides the other two sides of the triangle proportionally.

- The ratio for the triangle below is a : a or b : b, which both simplify to a ratio of 1:1.
- Creates two similar triangles

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.

- If \(\overline{XY} || \overline{DF}\), then \(\frac{EX}{XD} = \frac{EY}{YF}\)

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.

- \(\frac{AB}{BC}\) = \(\frac{DE}{EF}\)

This theorem can be expanded to any number of parallel lines and any number of transverals.

- All the corresponding segments of the transversals are proportional.

*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.

- \(\overrightarrow{AC}\) bisects \(\angle BAD\) so that \(\angle BAC \cong CAD\), so \(\frac{BC}{CD}\) = \(\frac{AB}{AD}\).