4.4-4.5 & 5.2: Proving Triangles Congruent
p. 206-221, 245-251
http://jwelker.lps.org/lessons/ppt/geod_4_4_congruent_triangles.ppt
Adapted from:
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
Example #2 – SAS – Postulate
S
N
L
W
K
Given: N is the midpoint of LW N is the midpoint of SK
Prove:
N is the midpoint of LWN is the midpoint of SK
Given
,LN NW SN NK Definition of Midpoint
LNS WNK Vertical Angles are congruent
SAS
WNKLNS
Statement Reason
1 1
2 2
3 3
4 4WNKLNS
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
W
HA
KS
Example #3 – ASA – Postulate
AW WK
Given: HA || KS
Prove:
HA || KS, Given
HAW SKW Alt. Int. Angles are congruent
HWA SWK Vertical Angles are congruent
ASA Postulate
AW WKSKWHAW
SKWHAW
1 1
2 2
3 3
4 4
Statement Reasons
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
Example
Given:
Prove: MH HTStatement Reason
AHby bisected is MAT
MATex with vertisosceles is AMT
1) AHby bisected is MAT
MATex with vertisosceles is AMT
1) Given
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
Example
Given:
Prove:
Statement Reason
WBofmidpoint theis
||
E
TBAW
TBAW
1) 1) WBofmidpoint theis
||
E
TBAW Given
2)2)
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****
Example
Given:
Prove:Statement Reason
nglesright tria are CBD and ABD
right are BDC andBDA
CBAB
CA
1) 1) Given
2)2)
nglesright tria are CBD and ABD
right are BDC andBDA
CBAB
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
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