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Objectives
Use the SSS Postulate
 Use the SAS Postulate
 Use the HL Theorem
 Use ASA Postulate
 Use AAS Theorem
 CPCTC Theorem

Vocabulary

Bisect: to cut into two equal parts
Postulate (SSS)
Side-Side-Side  Postulate
3 sides of one Δ are  to 3
sides of another Δ, then the
Δs are .
If
More on the SSS Postulate
If seg AB  seg ED, seg AC  seg EF, & seg
BC  seg DF, then ΔABC  ΔEDF.
E
A
F
C
B
D
EXAMPLE 1
Use the SSS Congruence Postulate
Write a proof.
GIVEN
PROVE
Proof
KL
NL, KM
KLM
NM
NLM
It is given that KL
NL and KM
By the Reflexive Property, LM
So, by the SSS Congruence
Postulate,
KLM
NLM
NM
LN.
GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true.
Explain your reasoning.
1.
DFG
HJK
SOLUTION
Three sides of one triangle are congruent to three
sides of second triangle then the two triangle are
congruent.
Side DG
HK, Side DF
JH,and Side FG JK.
So by the SSS Congruence postulate,
Yes. The statement is true.
DFG
HJK.
GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true.
Explain your reasoning.
2.
ACB
CAD
SOLUTION
GIVEN : BC
PROVE :
PROOF:
AD
ACB
CAD
It is given that BC AD By Reflexive property
AC AC, But AB is not congruent CD.
GUIDED PRACTICE
for Example 1
Therefore the given statement is false and
ABC is not
Congruent to CAD because corresponding sides
are not congruent
GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true.
Explain your reasoning.
3.
QPT
RST
SOLUTION
GIVEN : QT TR , PQ
SR, PT TS
PROVE :
RST
PROOF:
QPT
It is given that QT TR, PQ SR, PT TS. So by
SSS congruence postulate, QPT
RST.
Yes the statement is true.
Postulate (SAS)
Side-Angle-Side  Postulate
2 sides and the included  of
one Δ are  to 2 sides and the
included  of another Δ, then
the 2 Δs are .
 If
More on the SAS Postulate
seg BC  seg YX, seg AC  seg
ZX, & C  X, then ΔABC 
ΔZXY.
B
Y
 If
(
A
C
X
Z
EXAMPLE 2
Use the SAS Congruence Postulate
Write a proof.
GIVEN
BC
DA, BC AD
ABC
PROVE
CDA
STATEMENTS
S
REASONS
1.
BC
DA
1. Given
2.
BC
AD
2. Given
A 3.
S 4.
BCA
AC
DAC
CA
3. Alternate Interior
Angles Theorem
4. Reflexive Property of
Congruence
EXAMPLE 2
Use the SAS Congruence Postulate
STATEMENTS
5.
ABC
CDA
REASONS
5. SAS Congruence
Postulate
Example 3:
Given: RS  RQ and ST  QT
Prove: Δ QRT  Δ SRT.
S
Q
R
T
Q
Example 3:
R
T
Statements
Reasons________
1. RS  RQ; ST  QT
1. Given
2. RT  RT
3. Δ QRT  Δ SRT
Postulate
2. Reflexive
3. SSS
R
Example 4:
Given: DR  AG and AR  GR
Prove: Δ DRA  Δ DRG.
D
A
R
G
Example 4:
Statements_______
1. DR  AG; AR  GR
2. DR  DR
3.DRG & DRA are rt.
s
4.DRG   DRA
5. Δ DRG  Δ DRA
Reasons____________
1. Given
2. Reflexive Property
3.  lines form 4 rt. s
4. Right s Theorem
5. SAS Postulate
D
A
R
G
Theroem (HL)
Hypotenuse - Leg  Theorem
 If
the hypotenuse and a leg of a
right Δ are  to the hypotenuse
and a leg of a second Δ, then
the 2 Δs are .
Note: Right Triangles Only
Postulate (ASA):
Angle-Side-Angle Congruence
Postulate

If two angles and the
included side of one
triangle are congruent
to two angles and the
included side of a
second triangle, then
the triangles are
congruent.
Theorem (AAS):
Angle-Angle-Side Congruence
Theorem

If two angles and a
non-included side of
one triangle are
congruent to two
angles and the
corresponding nonincluded side of a
second triangle, then
the triangles are
congruent.
Proof of the Angle-Angle-Side (AAS)
Congruence Theorem
Given: A  D, C  F, BC  EF
Prove: ∆ABC  ∆DEF
A
D
B
F
Paragraph Proof
C
E
You are given that two angles of ∆ABC are congruent to two angles of ∆DEF.
By the Third Angles Theorem, the third angles are also congruent. That is, B
 E. Notice that BC is the side included between B and C, and EF is the
side included between E and F. You can apply the ASA Congruence
Postulate to conclude that ∆ABC  ∆DEF.
Theroem (CPCTC)
Corresponding Parts of Congruent
Triangles are Congruent
When two triangles are congruent, there are
6 facts that are true about the triangles:
 the triangles have 3 sets of congruent (of
equal length) sides and
 the triangles have 3 sets of congruent (of
equal measure) angles.
Use this after you have shown that two figures are
congruent. Then you could say that Corresponding parts of
the two congruent figures are also congruent to each other.
Example 5:
Is it possible to prove these triangles are
congruent? If so, state the postulate or theorem
you would use. Explain your reasoning.
Example 5:
In addition to the
angles and segments
that are marked,
EGF JGH by the
Vertical Angles
Theorem. Two pairs of
corresponding angles
and one pair of
corresponding sides
are congruent. Thus,
you can use the AAS
Congruence Theorem
to prove that ∆EFG 
∆JHG.
Example 6:
Is it possible to prove
these triangles are
congruent? If so, state
the postulate or
theorem you would
use. Explain your
reasoning.
Example 6:
In addition to the
congruent segments
that are marked, NP 
NP. Two pairs of
corresponding sides
are congruent. This is
not enough
information to prove
the triangles are
congruent.
Example 7:
Given: AD║EC, BD  BC
Prove: ∆ABD  ∆EBC
Plan for proof: Notice that
ABD and EBC are
congruent. You are given
that BD  BC. Use the fact
that AD ║EC to identify a
pair of congruent angles.
Writing Proofs
 Proofs
are used to prove what you are
finding.
 Geometric proofs can be written in one of
two ways: two columns, or a paragraph. A
paragraph proof is only a two-column proof
written in sentences
 List the given statements and then list the
conclusion to be proved
 Draw a figure and mark the figure
accordingly along with your proofs
Proof:
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4.
ABD  EBC
5.
∆ABD  ∆EBC
Reasons:
1. Given
2. Given
3. If || lines, then alt. int.
s are 
4. Vertical Angles
Theorem
5. ASA Congruence
Postulate