Classical Geometry: Euclidean, Transformational, Inversive, and Projective

CHAPTER 3

SIMILARITY

3.1 Similar Triangles

The word similar is used in geometry to describe two figures that have identical shapes but are not necessarily the same size. A working definition of similarity can be obtained in terms of angles and ratios of distances.

Two polygons are similar if corresponding angles are congruent and the ratios of corresponding sides are equal.

Notation. We use the symbol ~ to denote similarity. Thus, we write

to denote that the polygons ABCDE and QRSTU are similar. As with congruency, the order of the letters is important.

To say that ΔABC ~ ΔDEF means that

and

where k is a positive real number.

The constant k is called the proportionality constant or the magnification factor. If k > 1, triangle ABC is larger than triangle DEF; if 0 < k < 1, triangle ABC is smaller than triangle DEF; and if k = 1, the triangles are congruent.

Note that congruent figures are necessarily similar, but similar figures do not need to be congruent.

3.2 Parallel Lines and ...