MTH 232
Section 12.3
Similar Triangles
Definition
• Triangle ABC is similar to triangle DEF if, and
only if:
1. Corresponding angles are congruent;
2. The ratio of lengths of corresponding sides
are all equal (this common ration is called the
scale factor).
Notation
ABC ~ DEF
A  D
B  E
C  F
DE EF DF


AB BC AC
Pictures
An Activity
1. Give your students triangles of various sizes
(or have them create their own). Have them
measure the lengths of the sides and the
angles.
2. Put the triangles on an overhead projector,
Elmo, or document camera so that the image
projects onto a screen.
3. Perform the same measurements to verify
the angles and to find the scale factor.
Proving Triangle Similarity
• The AA Similarity Property: if two angles of one
triangle are respectively congruent to two angles
of a second triangle, then the triangles are
similar.
• The SSS Similarity Property: if the three sides of
one triangle are proportional to the three sides of
a second triangle, then the triangles are similar.
• The SAS Similarity Property: if, in two triangles,
the ratio of any two pairs of corresponding sides
are equal and the included angles are congruent,
then the two triangles are similar.
Examples
Important Considerations
1. Make sure you establish the proper
correspondence between angles and sides of
the two triangles—even when the two
triangles are not oriented the same.
2. Emphasize labeling of sides and angles to
verify the correspondences established in #1.
Notation is critical here.
Problem Solving with Similar Triangles
1. Determine the similar triangles in your
picture, if one is given. Again, notation is
important.
2. Set up the ratios for your scale factor. You will
likely use them to solve the problem.
3. Solve using cross-multiplication. Emphasize
proper units.
4. Make sure your answer is reasonable.
Examples
• 3(d); 4(a); 9; 29