Similar triangles

Ratio

A comparison between 2 or more quantities.

Proportion

An equation that states 2 ratios are equal.

Scale Factor / Ratio of Similitude

The ratio of the lengths of corresponding sides of similar polygons.

Geometric Mean / Mean Proportional

If a, b, and x are positive numbers and a/x = x/b, then x is the geometric mean of a and b.

Divided Proportionally

Two segments are divided proportionally iff the ratio of the lengths of the pieces of one segment equals the ratio of the lengths of the pieces of the other.

Cross Multiplication Theorem

In a proportion, the product of the extremes equals the product of the means.

Triangle Angle Bisector Theorem

If a ray bisects an angle of a triangle, it divides the opposite sides into segments proportional to the other two sides.

Right Triangle Altitude Theorem

If an altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to the leg.

SAS Similarity Theorem

If one angle of a triangle is congruent to one angle of another triangle and the sides including those angles are in proportion, the triangles are similar.

SSS Similarity Theorem

If the corresponding sides of two triangles are in proportion, then the triangles are similar.

AA Similarity Postulate

If two triangles have two congruent angles, then the triangles are similar.

Triangle Proportionality Theorem

If a segment is parallel to one side of a triangle, it divides the other sides proportionally.

Converse of Triangle Proportionality Theorem

If a segment divides two sides of a triangle proportionally, then it is parallel to the third side.

Corollary of Triangle Proportionality Theorem

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

Similar Triangles Area Ratio

The ratio of the areas of similar triangles is equal to the square of their scale factor.