Similar Triangles

Given a correspondence between two triangles. If corresponding angles are congruent and corresponding sides are proportional, then the correspondence is called a similarity, and the triangles are said to be similar

Definition of Similar Triangles

If a line intersects any two sides of any triangle and is parallel to the third side, then that line divides the two intersected sides of the triangle proportionally.

The Basic Proportionality Theorem

If two triangles have two pairs of congruent corresponding angles, then the third pair of corresponding angles are also congruent.

Corollary to Triangle Sum Theorem

Given a correspondence between two triangles. If corresponding angles are congruent, then the correspondence is a similarity

AAA Similarity Theorem

Given a correspondence between two triangles. If two pairs of corresponding angles are congruent, then the correspondence is a similarity.

AA Corollary

If a line intersects two sides of a triangle, and cuts off segments proportional to these two sides, then it is parallel to the third side.

Converse of Basic Proportionality Theorem

Given a correspondence between two triangles. If two pairs of corresponding sides are proportional, and their included angles are congruent, then the correspondence is a similarity

SAS Similarity Theorem

Given a correspondence between two triangles. If corresponding sides are proportional, then the correspondence is a similarity

SSS Similarity Theorem

The segment between the midpoints of two sides of a triangle is parallel to the third side and half as long.

Midline Theorem

An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the two other sides

The Angle Bisector Proportionality Theorem

In any right triangle, the altitude to the hypotenuse separates the triangle into two triangles which are similar to each other and to the original triangle.

Right Triangle Similarity Theorem