Geometry - Similarity

Similar Polygons via Measurement

Two polygons are similar if and only if:
1) Each pair of corresponding angles are congruent
2) Each pair of corresponding sides have the same scale factor

Definition of Similar Polygons via Transformation

Two polygons are similar if and only if one polygon can be mapped to the other using a series of rigid motions that also includes one dilation

Angle-Angle Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then those triangles are similar

Side-Angle-Side Similarity Theorem

If an angle of one triangle is congruent to the angle of another triangle and the sides included in those angles are in proportion (have the same scale factor) then those triangles are similar

Side-Side-Side Similarity Theorem

If all three sides of a triangle have the same scale factor (in proportion) then those triangles are similar

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally

Midsegment

A segment is a midsegment of a triangle if and only if its endpoints are the midpoints of two sides of the triangle

Midsegment Theorem

If a segment of a triangle is a midsegment, then it is parallel to the base and half the distance of the base it is parallel to

Triangle 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 segment sides of the angle it bisects

Three Parallel Lines Theorem

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

Geometric Mean

The geometric mean of two positive numbers a and b is the positive number x that satisfies a/x = x/b. So, x² = ab and x = √ab

Right Triangle Similiarity Theorem

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and each other

Geometric Mean (Leg) Theorem

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each of the right triangles is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg

Geometric Mean (Altitude) Theorem

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse