Download Slide 1

Congruent Triangles
Triangle Congruence 4.2
Holt Geometry
Congruent Triangles
• Congruent Polygons All corresponding sides and all
corresponding angles must be congruent to prove
congruency.
• Triangles require less proof to test for congruence.
• Use the SSS Postulate to test for triangle congruence.
• Use the SAS Postulate to test for triangle congruence.
• Use the ASA Postulate to test for triangle congruence.
• included angle
• included side
Holt Geometry
homework
Congruent Triangles
Congruence of Polygons
If all six pairs of corresponding parts (sides and angles) are
congruent, then the triangles are congruent.
If
Sides are
congruent
and
Angles are
congruent
1. AB
DE
4.
A
D
2. BC
EF
5.
B
E
3. AC
DF
6.
C
F
then
 ABC
Triangles are
congruent
 DEF
homework
Holt Geometry
Congruent Triangles
Vocabulary - SAS
An included angle is an angle formed
by two adjacent sides of a polygon.
B is the included angle between sides
AB and BC.
homework
Holt Geometry
Congruent Triangles
Vocabulary - ASA
B
An included side is a side formed by
two consecutive angles of a polygon.
BC is the included side between angles
homework
B and C.
Holt Geometry
Congruent Triangles
SSS CONGRUENCE POSTULATE
POSTULATE
Side - Side - Side (SSS) Congruence Postulate
If three sides of one triangle are congruent to three sides
of a second triangle, then the two triangles are congruent.
If Side
S MN
QR
Side
S NP
RS
Side
S PM
SQ
then  MNP
 QRS
homework
Holt Geometry
Congruent Triangles
Using the SSS Congruence Postulate
Prove that
 PQW
 TSW.
SOLUTION
Paragraph Proof
The marks on the diagram show that PQ  TS,
PW  TW, and QW  SW.
So by the SSS Congruence Postulate, you know that
 PQW   TSW.
Holt Geometry
homework
Congruent Triangles
SAS CONGRUENCE POSTULATE
POSTULATE
Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of a
second triangle, then the two triangles are congruent.
If
Side
S
Angle
A
Side
S
PQ
WX
Q
X
QS
XY
then  PQS
WXY
homework
Holt Geometry
Congruent Triangles
ASA CONGRUENCE POSTULATE
POSTULATE
Angle-Side-Angle (ASA) 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 two triangles are congruent.
A
If Angle
Q
X
S
Side
Angle
A
PQ
WX
P
W
then  PQS
WXY
homework
Holt Geometry
Congruent Triangles
CONGRUENCE POSTULATES
The SSS Congruence Postulate is a shortcut for proving
two triangles are congruent without using all six pairs
of corresponding parts.
The SAS Congruence Postulate is a shortcut for proving
two triangles are congruent without using all six pairs
of corresponding parts.
The ASA Congruence Postulate is a shortcut for proving
two triangles are congruent without using all six pairs
of corresponding parts.
homework
Holt Geometry
Congruent Triangles
Decide if the triangles are congruent by
SSS, SAS, ASA
a. SAS
e. None
i. ASA
j.
b. SSS
c. None
d. SAS
f.
g. SAS
h. SSS
SSS
None
k. ASA
l.
None
homework
Holt Geometry
Congruent Triangles
Find t so that ∆ADB  ∆CDB
If t = 4 ∆ADB  ∆CDB by SAS
homework
Holt Geometry
Congruent Triangles
Find x so that ∆MNO  ∆PQR
If x = 5 ∆MNO  ∆PQR by SSS
homework
Holt Geometry
Congruent Triangles
∆ABC  ∆DBC and AB = 3x + 10.
DB = 4x + 4. Find x and AB.
26°
x = 6, AB = 28
homework
Holt Geometry
Congruent Triangles
∆ABC  ∆JKL
AB = 2x + 12. JK = 4x – 50.
Find x and AB.
x = 31,
AB = 74
homework
Holt Geometry
Congruent Triangles
Given: AE  DE, BE  CE
Prove:  AEB
1
2
DEC.
Statements
Reasons
1. AE  DE, BE 
CE
2. 1  2
1. Given
3.  AEB   DEC
3. SAS Congruence Postulate
2. Vertical Angles Theorem
homework
Holt Geometry
Congruent Triangles
B
C
Given:
Prove:
A
Statements
1.
2.
3. ΔABG ΔCGB
G
Reasons
1. Given
2. Reflexive
3. SSS
homework
Holt Geometry
Congruent Triangles
DR
GIVEN:
D
AG
ADR GRD
 DRA
PROVE:
 DRG
Statements
1. DR
2.
3.
AG
DRA and DRG
are right angles.
DRA  DRG
4. ADR
Holt Geometry
Reasons
GRD
A
R
G
1. Given
2. Definition of Perpendicular
3. Right Angle Congruence Theorem
4. Given
5. DR  DR
5. Reflexive Property
6.  DRA   DRG
6. ASA Congruence
Postulate
homework
Congruent Triangles
Assignment
Geometry:
4.2A and 4.2B
Section 9 - 19
Holt Geometry