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2.2a: Exploring Congruent Triangles
M(G&M)–10–4 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.
GSE’s
G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent
G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
CCSS
Congruent triangles: Triangles that are the same size and the same shape.
A B
C
D E
F
In the figure DEF ABC
Congruence Statement: tells us the order in which the sides and angles are congruent
If 2 triangles are congruent:
The congruence statement tells us which parts of the 2 triangles are corresponding “match up”.
ABC DEF Means
A D, B E, C F
and
AB DE, BC EF,andAC DF
ORDER IS VERY IMPORTANT
3 Angles:
3 Sides:
A R
C
T E
F
In the figure TEF ARC
A T, R E, C F
AR TE, RC EF, AC TF
Example
Congruent Triangles
A
B C Y
Z
X
ZXYABC
RSTJKL
______J______S
______KL
RK
ST
Example 2
Write the Congruence Statement
Example 3
Example 3 : Congruence Statement
Finish the following congruence statement:
ΔJKL Δ_ _ _
K
J
L
M
N
N M L
Definition of Congruent: Triangles(CPCTC)
Two triangles are congruent if and only if their corresponding parts are congruent. (tells us when Triangles are congruent)
Are the 2 Triangles Congruent. If so writeThe congruence statement.
Ex. 2 Are these 2 triangles congruent? If so, write a congruence statement.
Reflexive Property
Does the Triangle on the left have anyof the same sides or angles as the triangle on the right?
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)
Example #1 – SSS – Postulate
Use the SSS Postulate to show the two triangles are congruent. Find the length of each side.
AC =
BC =
AB =
MO =
NO =
MN =
5
7 2 25 7 74
5
7 2 25 7 74
MONACB By SSS
Definition – Included Angle
K
J
L
K is the angle between JK and KL. It is called the included angle of sides JK and KL.
K
J
L
What is the included angle for sides KL and JL?
L
SAS - Postulate
QP
R
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)
J
L
KS
AS
S
A
S
by SASPQRJKL
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
K
J
L
K
J
L
Z
XY
ASA - Postulate
K
J
L
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 ASAZXYJKL
Note: is not Note: is not SSS, SAS, or ASA.SSS, SAS, or ASA.
Identify the Congruent Triangles.
Identify the congruent triangles (if any). State the postulate by which the triangles are congruent.
ABC STRV V by SSSby SSS
PNO VUWV V
TSC
B
A
R
H I
J
K
M L P N
O
V W
U
by SASby SAS
JHIV
AAS (Angle, Angle, Side)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!
F
E
D
A
C
B
HL (Hypotenuse, Leg)HL (Hypotenuse, Leg)
• If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . .
A
C
B
F
E
D
then the 2 triangles are
CONGRUENT!
***** only used with right triangles****
The Triangle Congruence The Triangle Congruence Postulates &TheoremsPostulates &Theorems
LAHALLHL
FOR RIGHT TRIANGLES ONLY
AASASASASSSS
FOR ALL TRIANGLES
Only this one is new
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).
Example 1Example 1
F
E
D
A
C
B
? DF CB
if determine any way to thereis
diagram, in then informatio Given the
CPCTCby CB so
SASby CAB !YES!
DF
DEF
Example 2Example 2
• Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?
A
C
B
F
E
D
No ! SSA doesn’t work
Example 3Example 3
• Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?
D
A
C
B
YES ! Use the reflexive side CB, and you have SSS
Name That PostulateName That Postulate
SASSASASAASA
SSSSSSSSASSA
(when possible)
Name That PostulateName That Postulate(when possible)
ASAASA
SASASS
AAAAAA
SSASSA
Name That PostulateName That Postulate(when possible)
SASASS
SASSAS
SASASS
Reflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SSSS
AA
Let’s PracticeLet’s PracticeIndicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
For SAS:
B D
For AAS: A F
AC FE
Homework Assignment