Geometry: An Interactive Journey to Mastery

Example 2
Each of the following pictures shows a pair of triangles. Which pairs of triangles are certain to be similar?
Which are certain to be congruent? Of those that are similar/congruent, explain which principle gives you this.
D௘ͽ

Figure 10.2

 E௘ͽ

Figure 10.3

F௘ͽ

Figure 10.4

 G௘ͽ a

b^3 b^3 a

Figure 10.5

Solution
D௘ͽ 7KHWZRWULDQJOHVDUHVLPLODUEHFDXVHRI6667KHVFDOHIDFWRULV
clearly 1, so they are, in fact, congruent.
E௘ͽ /DEHOWKHVDPHVLGHLQWHULRUDQJOHVIRUWKHSDUDOOHOOLQHVxDQGíx,
as shown in Figure 10.6. We have a second angle of measure x. The two
WULDQJOHVLQWKHGLDJUDPDUHVLPLODUE\$$ͼ௘7KH\VKDUHDFRPPRQDQJOHy,
and each possess an angle of measure x௘ͽ
F௘ͽ $FWXDOO\WKHUHDUHWKUHHWULDQJOHVLQWKLVGLDJUDP7KHUHLVQRUHDVRQWREHOLHYHWKDWDQ\WZRRIWKHP
are similar.
G௘ͽ 7KHVHWZRWULDQJOHVDUHVLPLODUE\6$6

x

x

y

íx

Figure 10.6