Page 163 - ArithBook5thEd ~ BCC

1. If two sides of a triangle are joined by a line segment parallel to the third side, the resulting
(smaller) triangle is similar to the original triangle.

B B

E

A C A C
D

Starting with triangle ABC on the left, we drew the segment DE, parallel to AB, creating (smaller)
triangle DEC,similar to ABC. Both triangles are visible on the right (the smaller overlapping
the larger), so there was really no need to draw the first picture. By similarity, the corresponding
sides are proportional:

smaller triangle DE CD CE
= = .
larger triangle AB CA CB

2. If a pair of parallel line segments is “cross-connected” as shown, the two triangles formed are
similar.

C D C D

E

A B A B

Starting with the parallel segments AB and CD,we connected D to A and C to B by means of
straight lines (called transversals) which cross at point E.The upper and lower angles at E are
equal. In addition, ∠D = ∠A and ∠C = ∠B. (These facts are visually plausible; they would be
proved in a geometry class.) It follows that triangles ABE and CDE are similar and hence

upper triangle CD EC ED
= = .
lower triangle AB EB EA

Page 163