Triangle Congruence Theorems Unit 5 Lessons 2-4

Side-Side-Side (SSS)  If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

Side-Angle-Side (SAS)  If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (The included angle is the angle formed by the sides being used.)

Angle-Side-Angle (ASA)  If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (The included side is the side between the angles being used. It is the side where the rays of the angles would overlap.)

Angle-Angle-Side (AAS)  If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (The non- included side can be either of the two sides that are not between the two angles being used.)

Hypotenuse-Leg (HL)  If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. (Either leg of the right triangle may be used as long as the corresponding legs are used.)

Angle-Angle-Angle (AAA)  AAA works fine to show that triangles are the same SHAPE (similar), but does NOT work to also show they are the same size, thus congruent!

Side-Side-Angle (SSA)  SSA (or ASS) is humorously referred to as the "Donkey Theorem".  This is NOT a universal method to prove triangles congruent because it cannot guarantee that one unique triangle will be drawn!!

Example #1 GIVEN:  ABC and  EDC;  1   2;  A   E; and AC  EC PROVE:  ABC   EDC  Which method, if any, should be used to prove these triangles are congruent?  SAS  ASA  SSS  AAS  HL  not possible

Example #2 GIVEN: AB = CB; AD = CD PROVE:  ABD   CBD  Which method, if any, should be used to prove these triangles are congruent?  SAS  ASA  SSS  AAS  HL  not possible

Example #3 GIVEN: quadrilateral PQRS; PR = ST;  PRT   STR PROVE:  PRT   STR  Which method, if any, should be used to prove these triangles are congruent?  SAS  ASA  SSS  AAS  HL  not possible

Example #4 GIVEN: MO = QP;  M   Q PROVE:  MOR   QPR  Which method, if any, should be used to prove these triangles are congruent?  SAS  ASA  SSS  AAS  HL  not possible

Example #5 GIVEN: quadrilateral PQRS; PQ  QR; PS  SR; QR = SR PROVE:  PQR   PSR  Which method, if any, should be used to prove these triangles are congruent?  SAS  ASA  SSS  AAS  HL  not possible

Example #6 GIVEN: segments LS and MT intersect at P,  M   T;  L   S PROVE:  MPL   TPS  Which method, if any, should be used to prove these triangles are congruent?  SAS  ASA  SSS  AAS  HL  not possible

Example #7 GIVEN: segment KT bisects  IKE and  ITE PROVE:  KIT   KET  Which method, if any, should be used to prove these triangles are congruent?  SAS  ASA  SSS  AAS  HL  not possible