Title: 2.2a: Exploring Congruent Triangles
12.2a Exploring Congruent Triangles
CCSS
GSEs
M(GM)104 Applies the concepts of congruency by
solving problems on or off a coordinate plane or
solves problems using congruency involving
problems within mathematics or across disciplines
or contexts.
2Definitions
Congruent triangles Triangles that are the same
size and the same shape.
Congruence Statement tells us the order in
which the
sides and
angles are congruent
3If 2 triangles are congruent
- The congruence statement tells us which parts of
the 2 triangles are corresponding match up.
Means
3 Angles
3 Sides
ORDER IS VERY IMPORTANT
4Example
Meaning
?A ? ?T, ?R ? ?E, ?C ? ?F
AR ? TE, RC ? EF, AC ? TF
5Congruent Triangles
Example 2
Write the Congruence Statement
A
Z
B
C
Y
X
Example 3
6Example 3 Congruence Statement
Finish the following congruence statement ?JKL
? ?_ _ _
N
M
L
M
J
L
K
N
7Definition of Congruent Triangles (CPCTC)
Two triangles are congruent if and only if their
corresponding parts are congruent. (tells us when
Triangles are congruent)
Example
Are the 2 Triangles Congruent. If so write The
congruence statement.
8Ex. 2
Are these 2 triangles congruent? If so, write a
congruence statement.
9Reflexive Property
Does the Triangle on the left have any of the
same sides or angles as the triangle on the
right?
10SSS - Postulate
If all the sides of one triangle are congruent to
all of the sides of a second triangle, then the
triangles are congruent. (SSS)
11Example 1 SSS Postulate
Use the SSS Postulate to show the two triangles
are congruent. Find the length of each side.
AC
5
BC
7
AB
MO
5
NO
7
MN
By SSS
12Definition Included Angle
K is the angle between JK and KL. It is
called the included angle of sides JK and KL.
What is the included angle for sides KL and JL?
L
13SAS - 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
triangles are congruent. (SAS)
S
A
S
S
A
S
by SAS
14Definition Included Side
JK is the side between J and K. It is
called the included side of angles J and K.
What is the included side for angles K and L?
KL
15ASA - 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. (ASA)
by ASA
16Identify the Congruent Triangles.
Identify the congruent triangles (if any). State
the postulate by which the triangles are
congruent.
Note is not SSS, SAS, or ASA.
by SSS
by SAS
17AAS (Angle, Angle, Side)
- If two angles and a non-included side of one
triangle are congruent to two angles and the
corresponding non-included side of another
triangle, . . .
then the 2 triangles are CONGRUENT!
18HL (Hypotenuse, Leg)
only used with right triangles
- If both hypotenuses and a pair of legs of two
RIGHT triangles are congruent, . . .
then the 2 triangles are CONGRUENT!
19The Triangle Congruence Postulates Theorems
Only this one is new
20Summary
- Any Triangle may be proved congruent by (SSS)
(SAS) - (ASA)
- (AAS)
- Right Triangles may also be proven congruent by
HL ( Hypotenuse Leg) - Parts of triangles may be shown to be congruent
by Congruent Parts of Congruent Triangles are
Congruent (CPCTC).
21Example 1
D
E
F
22Example 2
- Given the markings on the diagram, is the pair of
triangles congruent by one of the congruency
theorems in this lesson?
No ! SSA doesnt work
23Example 3
- Given the markings on the diagram, is the pair of
triangles congruent by one of the congruency
theorems in this lesson?
YES ! Use the reflexive side CB, and you have
SSS
24Name That Postulate
(when possible)
SAS
ASA
SSA
SSS
25Name That Postulate
(when possible)
AAA
ASA
SSA
SAS
26Name That Postulate
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Reflexive Property
Vertical Angles
SSA
SAS
27Lets Practice
Indicate the additional information needed to
enable us to apply the specified congruence
postulate.
For ASA
?B ? ?D
For SAS
?A ? ?F
For AAS
28Homework Assignment