Download - Proving Triangles Congruent
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Proving Triangles Congruent
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Two geometric figures with exactly the same size and shape.
The Idea of a CongruenceThe Idea of a Congruence
A C
B
DE
F
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How much do you How much do you need to know. . .need to know. . .
. . . about two triangles to prove that they are congruent?
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You learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.
Corresponding PartsCorresponding Parts
ABC DEF
B
A C
E
D
F
1. AB DE
2. BC EF
3. AC DF
4. A D
5. B E
6. C F
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Do you need Do you need all six ?all six ?
NO !
SSSSASHL
ASAAAS
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Side-Side-Side (SSS)Side-Side-Side (SSS)
1. AB DE
2. BC EF
3. AC DF
ABC DEF
C
B
A
E
D
F
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Side-Angle-Side (SAS)Side-Angle-Side (SAS)
1. AB DE
2. A D
3. AC DF
ABC DEF
B
A
C
E
D
F
included angle
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The angle between two sides
Included AngleIncluded Angle
G I H
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Name the included angle:
YE and ES
ES and YS
YS and YE
Included AngleIncluded Angle
SY
E
E
S
Y
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Hypotenuse Leg (HL)Hypotenuse Leg (HL)
1. AB DE
2. B E
3. AC DF
ABC DEF
B
A
C
E
D
F
right angle
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Angle-Side-Angle-Side-AngleAngle (ASA) (ASA)
1. A D
2. AB DE
3. B E
ABC DEF
B
A
C
E
D
F
included
side
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The side between two angles
Included SideIncluded Side
GI HI GH
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Name the included side:
Y and E
E and S
S and Y
Included SideIncluded Side
SY
E
YE
ES
SY
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Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)
1. A D
2. B E
3. BC EF
ABC DEF
B
A
C
E
D
F
Non-included
side
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Warning:Warning: No SSA or ASS Postulate No SSA or ASS Postulate
A C
B
D
E
F
NOT CONGRUENT
There is no such thing as an SSA
or ASS postulate!
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Warning:Warning: No AAA Postulate No AAA Postulate
A C
B
D
E
F
There is no such thing as an AAA
postulate!
NOT CONGRUENT
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The Congruence PostulatesThe Congruence Postulates SSS
SAS
HL
ASA
AAS
SSA or ASS
AAA
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Name That PostulateName That Postulate
SASSASASAASA
SSSSSSSSASSA
(when possible)
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Name That PostulateName That Postulate(when possible)
ASAASA
SASASS
AAAAAA
SSASSA
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Name That PostulateName That Postulate(when possible)
SASASS
SASSAS
SASASS
Reflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SSSS
AA
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Name That PostulateName That Postulate(when possible)
Reflexive Property SSSSSS
none
SSSSAA
AAAAAA
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(when possible)Name That PostulateName That Postulate
Vertical Angles
ASAASAReflexive Property SSSS
AA
SASASS
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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
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Try it!Try it!Indicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
For SAS:
For AAS:
N V
MK TU
M U