03.06 Applications of Congruence and Similarity

Triangle Proportionality

If DE is parallel to side BC and intersects the other two sides, it divides those sides in the same ratio:

Keep pieces with pieces or wholes with wholes. Do not mix them.

DE ∥ BC creates congruent corresponding angles, so ΔADE ~ ΔABC by AA. Corresponding sides then give the proportions.

The converse also works: if a line divides two sides proportionally, it is parallel to the third side.

Altitude on the hypotenuse

In right triangle ABC, draw the altitude from right-angle vertex B to hypotenuse AC at D. Then ΔABC ~ ΔADB ~ ΔBDC. The course calls this the Pieces of Right Triangles Similarity Theorem.

Each smaller triangle shares a 90° angle and one acute angle with the original, so AA proves all three similar.

Those similarities can prove a² + b² = c². The converse says that if a² + b² = c² for longest side c, then the angle opposite c is 90°.

Two proof shapes that keep showing up

In a parallelogram with a diagonal, opposite sides are parallel, so alternate interior angles match. The diagonal is shared by Reflexive Property. That gives ASA congruence.

For two triangles sharing a vertex with a perpendicular through it, both have a 90° angle. One more matching angle pair gives AA similarity.

For crossed segments between parallel lines, use vertical angles at the crossing and alternate interior or corresponding angles from the parallels. That also gives AA.