2.2a: Exploring Congruent Triangles

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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.
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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
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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
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Example
Meaning
?A ? ?T, ?R ? ?E, ?C ? ?F
AR ? TE, RC ? EF, AC ? TF
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Congruent Triangles
Example 2
Write the Congruence Statement
A
Z
B
C
Y
X
Example 3
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Example 3 Congruence Statement
Finish the following congruence statement ?JKL
? ?_ _ _
N
M
L
M
J
L
K
N
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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.
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Ex. 2
Are these 2 triangles congruent? If so, write a
congruence statement.
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Reflexive Property
Does the Triangle on the left have any of the
same sides or angles as the triangle on the
right?
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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)
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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
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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
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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
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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
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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
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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
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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!
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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!
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The Triangle Congruence Postulates Theorems
Only this one is new
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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).

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Example 1

D
E
F
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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
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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
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Name That Postulate
(when possible)
SAS
ASA
SSA
SSS
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Name That Postulate
(when possible)
AAA
ASA
SSA
SAS
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Name That Postulate
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Reflexive Property
Vertical Angles
SSA
SAS
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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
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Homework Assignment