The document discusses different methods for proving triangles are congruent: SSS, SAS, ASA, AAS, and HL. It states that if the specified corresponding parts of two triangles are congruent using one of these methods, then the triangles are congruent. The document provides examples of congruence proofs using these methods and their reasoning.

1. Sections 4-3 - 4-5

2. 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

3. For us to prove that 2 people are
identical twins, we don’t need to show
that all “2000” body parts are equal. We
can take a short cut and show 3 or 4
things are equal such as their face, age
and height. If these are the same I think
we can agree they are twins. The same
is true for triangles. We don’t need to
prove all 6 corresponding parts are
congruent. We have 5 short
cuts or methods.

4. SSS
If we can show all 3 pairs of corr.
sides are congruent, the triangles
have to be congruent.

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

6. This is called a common side.
It is a side for both triangles.
We’ll use the reflexive property.

7. Which method can be used to
prove the triangles are congruent

8. Common side
SSS
Vertical angles
SAS
Parallel lines
alt int angles
Common side SAS

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

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

12. 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

13. 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

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

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

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

17. Can you prove these triangles
are congruent?
A B Given: AB ll DC
X is the midpoint of AC
X Prove: AXB = CXD
˜
D C

18. A B Given: AB ll DC
X is the midpoint of AC
X
Prove: AXB = CXD
˜
D C
ASA

19. Triangle Congruence Triangle Congruence
SSS – If three sides of one triangle are congruent to three sides of SSS – If three sides of one triangle are congruent to three sides of
another triangle, then the two triangles are congruent another triangle, then the two triangles are congruent
SAS – If two sides and the included angle of one triangle are congruent SAS – If two sides and the included angle of one triangle are congruent
to those of another triangle, then the two triangles are congruent to those of another triangle, then the two triangles are congruent
ASA – If two angles and the included side of one triangle are congruent ASA – If two angles and the included side of one triangle are congruent
to those of another triangle, then the two triangles are congruent to those of another triangle, then the two triangles are congruent
AAS – If two angles and a non-included side of one triangle are AAS – If two angles and a non-included side of one triangle are
congruent to those of another triangle, then the two triangles are congruent to those of another triangle, then the two triangles are
congruent congruent
HL – If the hypotenuse and one leg of a right triangle are congruent to HL – If the hypotenuse and one leg of a right triangle are congruent to
those of another right triangle, then the two triangles are those of another right triangle, then the two triangles are
congruent congruent
. .