Section 12.3 Similar Triangles MTH 232 Section 12.3 Similar Triangles

Definition Triangle ABC is similar to triangle DEF if, and only if: Corresponding angles are congruent; The ratio of lengths of corresponding sides are all equal (this common ration is called the scale factor).

Notation

Pictures

An Activity Give your students triangles of various sizes (or have them create their own). Have them measure the lengths of the sides and the angles. Put the triangles on an overhead projector, Elmo, or document camera so that the image projects onto a screen. 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 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. Emphasize labeling of sides and angles to verify the correspondences established in #1. Notation is critical here.

Problem Solving with Similar Triangles Determine the similar triangles in your picture, if one is given. Again, notation is important. Set up the ratios for your scale factor. You will likely use them to solve the problem. Solve using cross-multiplication. Emphasize proper units. Make sure your answer is reasonable.

Examples 3(d); 4(a); 9; 29