The Art and Craft of Problem Solving

274 CHAPTER 8 GEOMETRY FOR AMERICANS

Similar Triangles

Contemplating area gives our geometric investigations a powerful "multiplicative" in­

sight. The concept of similar triangles lets us include division as well.

Triangles ABC and DEF are similar (denoted ABC rv DEF) if their respective

angles are equal and respective sides are proportional. In other words,

LA = LD, LB = LE, LC = LF

and

AB/DE = AC/DF = BC/EF.

In other words, the two triangles "have the same shape."

Fact 8.3.8 We can relax the "equal angles" and "proportional sides" conditions in the

definition of similarity:

(a) If the angles of two triangles are respectively equal, then the triangles are similar.

(b) If the sides of two triangles are respectively proportional, then the triangles are

similar.

(c) "Proportional SAS." If two corresponding sides of two triangles are in pro­

portion, and the angles between these corresponding sides are equal, then the

two triangles are similar. For example, in the picture below, LC = LF and

CB/FE = CA/FD, and this is enough to guarantee that MBC rv 6.DEF.

C F

L:1

L2

E D

B A

Many geometric investigations depend on finding pairs of similar triangles. Often,

auxiliary constructions such as parallel or perpendicular lines are employed, because

of the following (which you should have no trouble proving).