Triangle Congruence Theorems Unit 6 Lessons 1 & 2

Congruent triangles . . . have the same shape and and the same size. they may be flipped or turned. Of the six triangles pictured below, some are congruent. Identify the congruent triangles. Triangles a, c & d are congruent. Triangles b and f are congruent.

Congruent  which means the same shape (similar) and Objects that are exactly the same size and shape are said to be congruent. The mathematical symbol used to denote congruent is . The symbol is made up of two parts:  which means the same shape (similar) and  which means the same size (equal).

Congruent When you are looking at congruent figures, be sure to find the sides and the angles that "match up" (are in the same places) in each figure.  Sides and angles that "match up" are called corresponding sides and corresponding angles. In congruent figures, these corresponding parts are also congruent.  The corresponding sides will be equal in measure (length) and the corresponding angles will be equal in degrees.

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 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 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 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