1. Essential Question: What do I need to know about two triangles before I can say they are congruent?

10. The Idea of Congruence
Two geometric figures with exactly the same size and shape. A C B D E F

11. How much do you
need to know. . .
. . . about two triangles
to prove that they
are congruent?

12. Corresponding Parts
Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. B A C
AB  DE
BC  EF
AC  DF
 A   D
 B   E
 C   F
ABC   DEF E D F

13. Do you need all six ?
NO !
SSS
SAS
ASA
AAS HL

14. The triangles are congruent by SSS.
Side-Side-Side (SSS)
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. B A C
Side E
Side F D
Side
AB  DE
BC  EF
AC  DF
ABC   DEF
The triangles are congruent by SSS.

15. side-angle-side, or just SAS.
Included Angle
The angle between two sides
GIH I
GHI H
HGI G
This combo is called
side-angle-side, or just SAS.

16. The other two angles are the NON-INCLUDED angles.
Name the included angle:
YE and ES
ES and YS
YS and YE S Y E
 YES or E
 YSE or S
 EYS or Y
The other two angles are the NON-INCLUDED angles.

17. Side-Angle-Side (SAS) The triangles are congruent by SAS.
If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.
included
angle B E
Side F A C
Side D
AB  DE
A   D
AC  DF
Angle
ABC   DEF
The triangles are congruent by SAS.

18. angle-side-angle, or just ASA.
Included Side
The side between two angles GI GH HI
This combo is called
angle-side-angle, or just ASA.

19. The other two sides are the NON-INCLUDED sides.
Name the included side:
Y and E
E and S
S and Y S Y E YE ES SY
The other two sides are the NON-INCLUDED sides.

20. Angle-Side-Angle (ASA) The triangles are congruent by ASA.
If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.
included
side B E
Angle
Side F A C D
Angle
A   D
AB  DE
 B   E
ABC   DEF
The triangles are congruent by ASA.

21. Angle-Angle-Side (AAS) The triangles are congruent by AAS.
If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent.
Non-included
side B A C
Angle E D F
Side
Angle
A   D
 B   E
BC  EF
ABC   DEF
The triangles are congruent by AAS.

22. Warning: No SSA Postulate
There is no such thing as an SSA postulate!
Side
Angle
Side
The triangles are NOTcongruent!

23. There is no such thing as an SSA postulate!
Warning: No SSA Postulate
There is no such thing as an SSA postulate!
NOT CONGRUENT!

24. If we know that the two triangles are right triangles!
BUT: SSA DOES work in one
situation!
If we know that the two triangles are right triangles!
Side
Side
Side
Angle

25. These triangles ARE CONGRUENT by HL!
We call this
HL,
for “Hypotenuse – Leg”
Remember! The triangles must be RIGHT!
Hypotenuse
Hypotenuse
Leg
RIGHT Triangles!
These triangles ARE CONGRUENT by HL!

26. The triangles are congruent by HL.
Hypotenuse-Leg (HL)
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Right Triangle
Leg
Hypotenuse
AB  HL
CB  GL
C and G are rt.  ‘s
ABC   DEF
The triangles are congruent by HL.

27. There is no such thing as an AAA postulate!
Warning: No AAA Postulate
There is no such thing as an AAA postulate!
Different Sizes!
Same Shapes! E B A C F D
NOT CONGRUENT!

28. Congruence Postulates
and Theorems
SSS
SAS
ASA
AAS
AAA?
SSA? HL

29. Name That Postulate
(when possible)
SAS
ASA
SSA
AAS
Not enough info!