1. Lessons 4-4 and 4-5
Proving Triangles Congruent

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

3. C.P.C.T.C.
In Lesson 4-3, you 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

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

5. 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.

6. Included Angle
The angle between two sides
GIH I
GHI H
HGI G

7. 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.

8. 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. B E
Side F A C
Side D
AB  DE
A   D
AC  DF
Angle
ABC   DEF
The triangles are congruent by SAS.

9. Included Side
The side between two angles GI GH HI

10. 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.

11. 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. B E
Angle
Side F A C D
Angle
A   D
AB  DE
 B   E
ABC   DEF
The triangles are congruent by ASA.

12. 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. B A C
Angle E D F
Side
Angle
A   D
 B   E
BC  EF
ABC   DEF
The triangles are congruent by AAS.

13. Warning: No ASS or SSA Postulate
The triangles are NOT congruent!
Side
Angle
Side

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

15. DISCLAIMER: It DOES work in one situation!
If we know that the two triangles are right triangles!
Side
Side
Side
Angle

16. Hypotenuse Leg Theorem (HL) These triangles ARE CONGRUENT by HL!
for “Hypotenuse – Leg”
Remember! The triangles must be RIGHT!
Hypotenuse
Hypotenuse
Leg
RIGHT Triangles!
These triangles ARE CONGRUENT by HL!

17. Hypotenuse-Leg (HL) AB  HL CB  GL C and G are rt.  ‘s
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

18. 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!

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

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

21. Name That Postulate AAA SSS SSA SSA HL Not enough info!

22. Name That Postulate SSA SSA AAA HL Not enough info! Not enough info!

23. Reflexive Sides and Angles
Vertical Angles,
Reflexive Sides and Angles
When two triangles touch, there may be additional congruent parts.
Vertical Angles
Reflexive Side
side shared by two
triangles

24. Name That Postulate SAS SAS SSA AAS Vertical Angles Reflexive Property
Not enough info!

25. Reflexive Sides and Angles
When two triangles overlap, there may be additional congruent parts.
Reflexive Side
side shared by two
triangles
Reflexive Angle
angle shared by two

26. Indicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
A  F
For AAS:

27. What’s Next
Try Some Proofs
End Slide Show

28. Choose a Problem. Problem #1 SSS Problem #2 SAS Problem #3 ASA
End Slide Show
Problem #1
SSS
Problem #2
SAS
Problem #3
ASA

29. Problem #4
AAS

30. Problem #5 HL
Given ABC, ADC right s,
Prove: