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