1. CONGRUENT
TRIANGLES
Sections 4-3, 4-4, 4-5

2. This is called a common side. It is a side for both triangles.
You need to mark it!!
It is called the Reflexive property
when a side or angle is  to itself.

3. Things you can assume :
Anything marked in the picture is True.
Anything written in words is True.
Vertical angles are congruent.
Reflexive Property

4. When we talk about congruent triangles,
we mean everything about them Is congruent.
All 3 pairs of corresponding angles are equal….
And all 3 pairs of corresponding sides are equal

5. When you prove triangles congruent, you only need a specific combination of sides and angles.

6. Let’s look at all the possible combinations
SSS
ASA
SAS
AAS
SSA
AAA
Which ones do you think work and why???

7. If we can show all 3 pairs of corr. sides are congruent, the triangles
SSS
If we can show all 3 pairs of corr.
sides are congruent, the triangles
have to be congruent. J A C B K L

8. Included angle- An angle between two sides
Ex.
Name the included angle C B A
AC&BC_____
AB&BC_____
AC&AB_____ C
 B
 A

9. SAS Show 2 pairs of sides and the included angles are congruent and
the triangles have to be congruent.
Non-included
angles
Included
angle

10. A B J L K M N D C B Z A X C Y Which method can be used to
prove the triangles are congruent?
Write an congruence statement
Vertical angles A B J L
Reflexive
SSS K M
∆ABC≅ ∆CDA N
SAS D C
 JKN   MKL B Z A
SSS X
SAS C Y
 ABC   XYZ
 IJH   KML

11. What about SSA?

12. Will AAA work???
AAA doesn’t preserve the size of the triangles

13. Included side- A side between 2 angles.
Name the included side.
A&B_____
C&B_____
A&C_____

14. ASA, AAS A ASA – 2 angles and the included side S A AAS – 2 angles and
The non-included side A A A Or S S S A A A

16. C D B E J A L M N K
∆JKL≅ ∆NLM:ASA
∆ABE≅ ∆CDB: AAS
∆ABC≅ ∆ADC : ASA

17. How do we find the missing side of a right triangle?
So, if we had a leg and hypotenuse, then we could find the 3rd side.. Proving triangles congruent 5 ? 4 3

18. HL ( hypotenuse leg ) is used
only with right triangles, BUT,
not all right triangles.

19. Decide if the following triangles are congruent by SSS,SAS,ASA,AAS, or HL. If they are not congruent, then write not congruent.

20. When Starting A Proof, Make The Marks On The Diagram Indicating
The Congruent Parts. Use The Given
Info, Properties, Definitions, Etc.
We’ll Call Any Given Info That Does Not Specifically State Congruency
Or Equality A PREREQUISITE

21. SOME REASONS WE’LL BE USING
DEF OF MIDPOINT
DEF OF A BISECTOR
VERT ANGLES ARE CONGRUENT
DEF OF PERPENDICULAR BISECTOR
REFLEXIVE PROPERTY (COMMON SIDE)
PARALLEL LINES ….. ALT INT ANGLES

22. SAS Our Outline P rerequisites S ides A ngles Triangles ˜ A C = B 1 2
Given: AB = BD
EB = BC
Prove: ∆ABE ˜ ∆DBC A C = B 1 2
Our Outline
P rerequisites
S ides
A ngles
Triangles ˜
SAS E D =

23. A C B = SAS E D P S A ∆’s none AB = BD Given 1 = 2 Vertical angles
Given: AB = BD
EB = BC
Prove: ∆ABE ˜ ∆DBC B 1 2 =
SAS E D
STATEMENTS REASONS P S A
∆’s
none
AB = BD Given
1 = Vertical angles
EB = BC Given
∆ABE ˜ ∆DBC SAS =

24. AAS C Given: CX bisects ACB A ˜ B Prove: ∆ACX ˜ ∆BCX = 1 2 = A X B P A
CX bisects ACB Given
1 = Def of angle bisc
A = B Given
CX = CX Reflexive Prop
∆ACX ˜ ∆BCX AAS =

25. Proofs with perpendicular lines ⟘A B 1
Statement Reason 2 C D