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Proving Similarity and Congruence of Triangles Proving Triangles Are Congruent Many times triangles may look like they are congruent. However, you cannot assume two triangles are congruent just because they look congruent. Congruent triangles have the following characteristics: Corresponding sides have the same length. Corresponding angles have the same measures. But, you do not have to show that ALL sides and ALL angles are congruent to prove that triangles are congruent. The following are the four congruency theorems that prove two triangles are congruent. 5 Congruency Theorems Side– Side–Side (SSS) Side–Angle–Side (SAS) Angle–Side–Angle (ASA) Angle–Angle–Side (AAS) Hypotenuse–Leg (H-L) Side-Side-Side If each side of one triangle has the same length as the corresponding side of another triangle, then those triangles are congruent. Example B 5 A 4 5 C 6 E D ΔABC ΔDEF 4 F 6 Side–Angle–Side If two sides and the angle created by those two sides have the same measures as two sides and the angle in between them of another triangle, then those triangles are congruent. Example B 5 A 50o 4 5 C E 50o D © LaurusSoft, Inc. 4 F Angle–Side–Angle If two angles and the side shared by those two angles have the same measures as two angles and the side shared by those two angles of another triangle, then those triangles are congruent. B 5 A 70o 40o E Example 5 C D ΔABC ΔDEF 70o 40o F Angle–Angle–Side If two angles and the side not shared by those two angles have the same measures as two angles and the side not shared by those two angles of another triangle, then those triangles are congruent. B 70o A 40o 6 E Example 70o C D 40o 6 ΔABC ΔDEF F Hypotenuse-Leg If the hypotenuse and leg of one right triangle have the same measure of a leg and the hypotenuse of another right triangle, then the right triangles are congruent. Example E ΔABC ΔDEF B 11 6 6 A C D 11 F Proving Triangles Are Similar Many times triangles may look like they are similar. However, you cannot assume two triangles are similar just because they look similar. Similar triangles have the following characteristics: Corresponding sides are proportional Corresponding angles have the same measures. But, you do not have to show that ALL corresponding sides are proportional and ALL corresponding angles are congruent to prove that triangles are similar. The following are the three similarity theorems that prove two triangles are similar. © LaurusSoft, Inc. 3 Similarity Theorems Side–Side–Side (SSS) Side–Angle–Side (SAS) Angle–Angle (AA) Side-Side-Side If each side of one triangle has a length that is proportional to the corresponding side of another triangle, then those triangles are similar. Example B 3 A 2 E ΔABC ~ ΔDEF 4 6 C 3 F 6 D Side–Angle–Side If two pairs of corresponding sides have lengths that are proportional and the angles created by those two pairs of sides have the same measure, then those triangles are similar. E B Example 4 70o 2 8 70o ΔABC ~ ΔDEF 4 C A F D Angle–Angle If two angles of one triangle have the same measure as two angles of another triangle, then those triangles are similar. By the way, the third angle of both triangles is congruent as well. Example E ΔABC ~ ΔDEF B 70o o 70 A 40o C D 40o © LaurusSoft, Inc. F