Triangle Congruence Theorems Unit 5 Lessons 11-2

Proving Triangles Congruent There are 5 theorems that can prove triangles congruent There are 2 theorems that cannot prove triangles are congruent. Make sure you draw the pictures with the notes!! This is important!!!

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 – Two sides and a non-included angle 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 C is the midpoint of AE. PROVE: ABC  EDC Which method, if any, should be used to prove these triangles are congruent? SAS ASA SSS AAS HL not possible ASA – all necessary information is given

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 SSS – the triangles share side BD

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 SAS – the triangles share side RT

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 Don’t assume O and P are right angles!! NOT POSSIBLE – don’t assume the triangles are right triangles! Not enough information is given.

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 HL – the perpendiculars form right angles, giving us right triangles, and the triangles share side PR (the hypotenuse)

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 Vertical Angles are congruent!! AAA cannot prove triangles are congruent!! NOT POSSIBLE – no information is given about the sides

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 ASA – the bisector forms two sets of congruent angles and the triangles share side KT