Similar triangles have the same angles between the sides, however it is the lengths of the sides that make a triangle similar, and not congruent (recall that if you had congruent triangles, both the angles and the side lengths would be the same.)
Another way to write that triangle ABC is congruent to triangle DEF is like this: 
Notice that, between the triangles, the angles are equal, and the side lengths are different, but proportional -
First, look at the lengths:
AB = 2(DE)
BC = 2(EF)
AC = 2(DF)
All of the corresponding sides are proportional.
Now, look at the angles:
< ABC = < DEF
< BCA = < EFD
< CAB = < FDE
All of the corresponding angles are equal.
Conditions for Similarity
1. Side Side Side
Three pairs of corresponding sides are proportional (they have the same enlargement or reduction ratio). You need to know the lengths of all of the sides. For example,
2. Side Angle Side
Two pairs of corresponding sides are proportional, and their contained angle (the angle between the sides) is equal. For example,
3. Angle Angle
Two pairs of corresponding angles are equal. For example,
Example 1: Prove that the following triangles are congruent, and find the missing side.
Using the F-pattern,
< ABE = < CDE
Thus, by Angle-Angle, triangle ABE is similar to triangle CDE.
Therefore, AB/CD = 15/10
AB/CD = 3/2
Thus, the ratio, or scale factor, from triangle ABE to triangle CDE is 3/2.
Whenever we find other sides, we must write triangle ABE first, and triangle CDE second to find the correct length.
AE/CE = AB/CD
AE/12 = 3/2
multiply both sides by 12
AE = 12 (3/2)
AE = 18
Therefore, AE is 18 units long.
Practice Questions
Page 462 # 1a, 4c, 6cd, 7cef,
