Geometry, Unit 5-Proportions and Similarity

Similar polygons

Polygons that have the same shape, but not necessarily the same size. Corresponding sides of similar polygons are proportional and corresponding angles are congruent.

Triangle Proportionality Theorem: Easy Error Tip

The Triangle Proportionality Theorem only works for the non-parallel parts of the triangle. It does not work for the parallel parts or whole sides of the triangle. For those you must think of the triangles as whole sides.

Geometric Mean

The geometric mean, x, between two numbers, a and b, can be found by setting up a proportion, a/x = x/b, and then solved using cross multiplication

Right Triangle Leg Theorem

The leg of a right triangle is equal to the geometric mean of the hypotenuse and the adjacent part of the hypotenuse when it is intersected by the altitude drawn from the right angle.

Right Triangle Altitude Theorem

The length of the altitude of a right triangle is equal to the geometric mean of the divided parts of the hypotenuse.

Cross Multiplication

The product of the numerator of one ratio and the denominator of the other ratio; used with proportions to solve for a missing variable

Scale factor

the ratio of the lengths of two corresponding sides

Extended ratio

A comparison of three or more quantities by division

Ratio

A comparison of two quantities by division

Triangle Midsegment Theorem

A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of the parallel side

Midsegment of a triangle

A segment that joins the midpoints of two sides of the triangle

Proportion

An equation stating that two ratios are equal

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides into segments of proportional lengths

Side-Side-Side (SSS) Similarity Theorem

If the corresponding side lengths of two triangles are proportional, then the triangles are similar

Side-Angle-Side (SAS) Similarity Theorem

If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar

Proportional Parts of Parallel Lines

If three or more parallel lines intersect two transversals, then they divide the transversals into proportional parts

Angle-Angle (AA) Similarity Postulate

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

Perimeters of Similar Polygons

If two polygons are similar, then their perimeters are proportional to the ratios of the scale factor between them