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When a triangle has two sides that are in the same proportion to another triangle and their included angle is equal, these triangles are similar. Be careful not to confuse this theorem with the Side-Angle-Side theorem for congruence. For congruence, the two sides with their included angle must be identical; for similarity, the proportions of the sides must be same and the angle must be identical. For example: Triangle ABC and DEF are similar is angle A = angle D and AB/DE = AC/DF. Using a ruler, measure two sides of triangle ABC and label them with that measure. Make sure triangle DEF is oriented in the same direction and measure the same two sides. Label these sides as well. Example: Measures of triangle ABC; side AB = 4 cm and side AC = 8 cm. Measures of triangle DEF; side DE = 2 cm and side DF = 4 cm. Using a protractor, measure the included angle, or, the angle between the two sides that you already measured. For this theorem, the measure of the angle should be identical in both triangles. Example: Angle A in triangle ABC is 26. Angle D in triangle DEF is also 26. To use the SAS theorem, the sides of the triangles must be proportional to each other. To calculate this, simply use the formula AB/DE = AC/DF. Example: AB/DE = AC/DF; 4/2 = 8/4; 2 = 2. The proportions of the two triangles are equal. Once you have determined that the proportions of two sides of a triangle and their included angle are equal, you can use the SAS theorem in your proof. Example: Because AB/DE = AC/DF and angle A = angle D, triangle ABC is similar to triangle DEF. Note: If angle A did not equal angle D, the triangles would not be similar. Also, if the proportions were not equal, the triangles would not be similar.
Define the Side-Angle-Side (SAS) Theorem for similarity. Measure the same two sides of each triangles. Identify the measure of the angle between those two sides. Calculate the proportion of the side lengths between the two triangles. Apply the Side-Angle-Side Theorem to prove similarity.