1. Triangle Congruence Theorems
Unit 5
Lessons 11-2

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!!!

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

4. 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.)

5. 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.)

6. 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.)

7. 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.)

8. 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!

9. 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!!

10. 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

11. 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

12. 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

13. 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.

14. 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)

15. 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

16. 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