Geometry

Proportionality Relationships

Big Picture

Proportionality are important in many relationships. The fundamental properties of proportions are useful in geometric proofs.

Key Terms

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.

Triangle Proportionality

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.

Midsegment TheoremConverse of the Triangle Proportionality

Parallel Lines and Transversals

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.
Parallel Lines and Transversals

Angle Bisector

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 \(\frac{BC}{CD}\) =  \(\frac{AB}{AD}\).
Angle Bisector