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2.2a: Exploring Congruent Triangles

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Title: 2.2a: Exploring Congruent Triangles


1
2.2a 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.
2
Definitions
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
3
If 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
4
Example
Meaning
?A ? ?T, ?R ? ?E, ?C ? ?F
AR ? TE, RC ? EF, AC ? TF
5
Congruent Triangles
Example 2
Write the Congruence Statement
A
Z
B
C
Y
X
Example 3
6
Example 3 Congruence Statement
Finish the following congruence statement ?JKL
? ?_ _ _
N
M
L
M
J
L
K
N
7
Definition 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.
8
Ex. 2
Are these 2 triangles congruent? If so, write a
congruence statement.
9
Reflexive Property
Does the Triangle on the left have any of the
same sides or angles as the triangle on the
right?
10
SSS - 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)
11
Example 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
12
Definition 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
13
SAS - 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
14
Definition 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
15
ASA - 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
16
Identify 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
17
AAS (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!
18
HL (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!
19
The Triangle Congruence Postulates Theorems
Only this one is new
20
Summary
  • 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).

21
Example 1

D
E
F
22
Example 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
23
Example 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
24
Name That Postulate
(when possible)
SAS
ASA
SSA
SSS
25
Name That Postulate
(when possible)
AAA
ASA
SSA
SAS
26
Name That Postulate
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Reflexive Property
Vertical Angles
SSA
SAS
27
Lets 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
28
Homework Assignment
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