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Warm-Up #38
1. Line M goes through the points (7, -1) and (-2,
3). Write an equation for a line perpendicular to
M and through the origin.
2. What are the new points of A(-3, 3) and B(2, 5) if
you want to use the rule (x, y)  (x -3, y + 4)?
What type of transformation is this?
Homework
• SSS-SAS-ASA-AAS Congruence page 1&2
Triangle
Congruence
congruent polygons:
are polygons with congruent
corresponding parts - their
matching sides and angles
B
Y
A
X
C
D
Z
Polygon ABCD  Polygon XYZW
W
The Right Match
Congruent
Not congruent
A
D
F
C
B

E
pg. 203
CPCTC:
corresponding parts of
congruent triangles are
congruent
Corresponding sides and angles
C
A
T
B
Corresponding Angles
S
R
Corresponding sides
A  R
C  T
AB
RS
B  S
BC
ST
AC
RT
Like congruence of segments and angles, congruence of
triangles is reflexive, symmetric, and transitive.
Proving Vertical Angle Theorem
THEOREM
Vertical Angles Theorem
Vertical angles are congruent
1
3,
2
4
Proving Vertical Angle Theorem
5 and 6 are a linear pair,
GIVEN
6 and 7 are a linear pair
PROVE
5
7
Statements
Reasons
1
5 and
6 and
6 are a linear pair,
7 are a linear pair
Given
2
5 and
6 and
6 are supplementary,
7 are supplementary
Linear Pair Postulate
3
5
7
Congruent Supplements Theorem
THEOREM
Third Angles Theorem
If two angles of one triangle are
congruent to two angles of another
triangle, then the third angles are also
congruent.
If  A   D and  B   E, then  C   F.
Goal 1
We will use:
• Use the SSS Postulate
• Use the SAS Postulate
• Use the HL Theorem
• Use ASA Postulate
• Use AAS Theorem
to prove that two triangles are congruent.
SSS: Side-Side-Side Postulate
If three sides of one triangle
are congruent to three sides
of another triangle, then the
two triangles are congruent.
H
G
Q
F
P
 GHF   PQR
R
Write a proof statement:
SAS: Side-Angle-Side Postulate
If two sides and included angle of
one triangle are congruent to two
sides and included angle of
another triangle, then the two
triangles are congruent.
B
C
D
A
F
 BCA   FDE
E
Write a proof statement:
ASA: Angle-Side-Angle Postulate
If two angles and included side of
one triangle are congruent to two
angles and included side of
another triangle, then the two
triangles are congruent.
B
G
P
H
K
 HGB   NKP
N
Write a proof statement:
AAS: Angle-Angle-Side Postulate
If two angles and nonincluded
side of one triangle are congruent
to two angles and nonincluded
side of another triangle, then the
two triangles are congruent.
C
D
T
M
G
 CDM   TGX
Chris Giovanello, LBUSD Math Curriculum Office, 2004
X
Write a proof statement:
Chris Giovanello, LBUSD Math Curriculum Office, 2004
HL Theorem
Hypotenuse - Leg Congruent 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 .
Postulate 19 (SSS)
Side-Side-Side  Postulate
• If 3 sides of one Δ are  to 3
sides of another Δ, then the Δs
are .
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.
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
2. Reflexive
3. Δ QRT  Δ SRT
3. SSS Postulate
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 D
5. Δ DRG  Δ DRA
A
R
Reasons____________
1. Given
2. Reflexive Property
3.  lines form 4 rt. s
4. Right s Theorem
5. SAS Postulate
G
Theroem 4.5 (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 .
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.
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.
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
Proof:
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