Math 2 Geometry
Based on Elementary Geometry, 3rd ed, by Alexander & Koeberlein
3.1
Congruent Triangles
Definition
Two triangles are congruent when the six
parts of the first triangle are congruent to
the six parts of the second triangle.
Definition
Two triangles are congruent when the six
parts of the first triangle are congruent to
the six parts of the second triangle.
What are the six parts?
Definition
Two triangles are congruent when the six
parts of the first triangle are congruent to
the six parts of the second triangle.
What are the six parts?
How does this relate to our understanding
of congruence as same shape and size?
Converse of Definition
If two triangles are congruent, then the six
parts of one triangle are congruent to the
corresponding parts of the other triangle.
The converse of a definition is also true:
If the six parts of a triangle are congruent
to the corresponding parts of another
triangle, then the triangles are congruent.
Consequence of the definition
Converse of the definition:
“If the six parts of a triangle are congruent
to the corresponding parts of another
triangle, then the triangles are congruent.”
To prove two triangles are congruent, we
can show that the corresponding six parts
of the two triangles are congruent.
Properties of Congruent s
• ABC  ABC (Reflexive Property of )
Properties of Congruent s
• ABC  ABC (Reflexive Property of )
• If ABC  DEF then DEF  ABC
(Symmetric Property of )
Properties of Congruent s
• ABC  ABC (Reflexive Property of )
• If ABC  DEF then DEF  ABC
(Symmetric Property of )
• If ABC  DEF and DEF  GHI, then
ABC  GHI (Transitive Property of )
Construction
Construct a triangle whose sides have the
lengths of the segments provided in the
figure.
A
A
B
B
C
C
SSS (Side-Side-Side)
Postulate 12
• Method for Proving Triangles Congruent
• If the three sides of one triangle are
congruent to the three sides of a second
triangle, then the triangles are congruent.
E
B
A
D
C
F
SAS (Side-Angle-Side)
Postulate 13
• Method for Proving Triangles Congruent
• If two sides and the included angle of one
triangle are congruent to the two sides and
the included angle of a second triangle,
then the triangles are congruent.
B
A  F
D
Seg AB  Seg FE
F
A
C
E
Seg AC  Seg FD
ASA (Angle-Side-Angle)
Postulate 14
• Method for Proving Triangles Congruent
• If two angles and the included side of one
triangle are congruent to two angles and
the included side of a second triangle,
B
A  F
D
 C  D
F
A
C
E
Seg AC  Seg FD
AAS (Angle-Angle-Side)
Theorem 3.1.1
• Method for Proving Triangles Congruent
• If two angles and a non-included side of
one triangle are congruent to two angles
and a non-included side of a second
triangle, then the triangles are congruent.
B
A  F
D
 C  D
F
A
C
E
Seg BC  Seg ED