Corollary of Proportionality Theorem

Definition, similar triangles: Two triangles are similar if corresponding angles are congruent and if the ratio of corresponding sides is constant. If the triangles ΔABC and ΔDEF are similar we can write this relation as ΔABC∼ΔDEF.

(The difference between similar and congruent triangles is that similar triangles do not have to be the same size.)

There is one useful corollary of the triangle proportionality theorem:

Corollary: A transversal that is parallel to a side in a triangle defines a new smaller triangle that is similar to the original triangle.

Proof:

Show that corresponding angles are congruent (equal).

Then show that \[\frac{a+b}{a}=\frac{c+d}{c}\]

Draw another transversal parallel to another side and show that \[\frac{a+b}{a}=\frac{c+d}{c}=\frac{f}{e}\]