236 Part 3: The Shape of the World
Similar Triangles
Triangles are similar if they are the same shape, but not necessarily the same size. Because they
have to be the same shape, and angles control shape, corresponding angles are congruent. They
don’t have to be the same size, so you don’t have to have corresponding sides congruent, but to
change the size without distorting the triangle, corresponding sides will be in proportion. If the
longest side of one triangle is twice the size of the longest side of the other triangle, then the
shortest side of the first triangle will be twice the size of the shortest side of the second triangle.
All the sides will be multiplied by the same number, called the scale factor.DEFINITION
Two triangles are similar if each pair of corresponding angles are congruent and cor-
responding sides are in proportion.
The scale factor of two similar triangles is the ratio of a pair of corresponding sides.To conclude that triangles are similar, you must know that all three pairs of corresponding
angles are congruent, and you must check that sides are in proportion. Just like with congruent
triangles, however, there are shortcuts. The most common one for similar triangles is called AA.
You must know that two angles of one triangle are congruent to the corresponding angles of the
other. Of course, if two pairs of angles are congruent, the third pair will be congruent, too. The
second shortcut is to check that two pairs of sides are in proportion and the angles between them
are congruent to each other. The third is to check that all the sides are in proportion. In fact, AA
is the one used most often.CHECK POINT
Decide if the information given is enough to guarantee that the triangles are congru-
ent. If so, state the rule that guarantees the congruence. If not, write “cannot be
determined.”- A # X, B # Y, ABXYù
- CT INù , A # W, C # I
- BI MAù , IG ANù , B # M
- CA DOù , ATOù G, CT DGù
- BO CAù , O # A, OX ARù