Post on 22-Feb-2016
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Proving Triangles Congruent
Two geometric figures with exactly the same size and shape.
The Idea of a Congruence
A C
B
DE
F
How much do you need to know. . .
. . . about two triangles to prove that they are congruent?
If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.
Corresponding Parts
ABC DEF
B
A C
E
D
F
1. AB DE2. BC EF3. AC DF4. A D5. B E6. C F
Do you need all six ?
NO !
SSSSASASARHS
S means side &
A means angle
1) Side-Side-Side (SSS)
1. AB DE2. BC EF3. AC DF
ABC DEF
B
A
C
E
D
F
2) Side-Angle-Side (SAS)
1. AB DE2. A D3. AC DF
ABC DEF
B
A
C
E
D
F
included angle
The angle between two sidesIncluded Angle
G I H
Name the included angle:
YE and ES ES and YS YS and YE
Included Angle
SY
E
E S Y
3) Angle-Side-Angle (ASA)
1. A D2. AB DE3. B E
ABC DEF
B
A
C
E
D
F
included
side
The side between two anglesIncluded Side
GI HI GH
Name the included side :
Y and E E and S S and Y
Included Side
SY
E
YEESSY
Angle-Angle-Side (AAS)
1. A D2. B E3. AC EF
NOT
B
A
C
E
D
F
Non-included
side
Angle-Angle-Side (AAS)Sometimes will be ( ASA )
1. A D2. B E3. BC EF ABC
DEF
B
A
C
E
D
F
C F ( why ?)
* *
4) Right angle - hypotenuse - side (RHS)
1. A = D = 90
2. AB DE3. BC EF
ABC DEF
B
A C
E D
F
∘
Warning: No SSA Postulate
A C
B
D
E
F
NOT CONGRUENT
There is no such thing as an SSA
postulate!
Warning: No AAA Postulate
A C
B
D
E
F
There is no such thing as an AAA
postulate!
NOT CONGRUENT
The Congruence Postulates SSS
correspondence ASA
correspondence SAS
correspondence RHS
correspondence SSA
correspondence AAA
correspondence
Example In the given figure prove that : AB ≅ AD
••B
CD
A
In ∆∆ ADC , ABC we have :
∆∆ ADC , ABC : فيهما
1) DC = BC ( given )
1) ( معطى )
DC = BC
2) ∡ DCA = ∡BCA ( given )
2) ( معطى )
∡ DCA = ∡BCA
3) AC = AC ( common side )
ضلع ) (3( مشترك
AC = AC
∴ ∆ ACD ≅ ∆ ACB ( SAS postulate)
∴ ∆ ACD ≅ ∆ ACB ( SAS مسلمة )
∴ AB ≅ AD ( CPCTC ) Corresponding parts of ≅ ∆ s are ≅
∴ AB ≅ AD ( ( المثلثات في المتناظرة األجزاءمتطابقة تكون المتطابقة
Name That Postulate
SAS ASA
SSSSSA
(when possible)
Name That Postulate(when possible)
ASA
SAS
AAA
SSA
Name That Postulate(when possible)
SAS
SAS
SAS
Reflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SS
A
ACTIVITY : Name That Postulate(when possible)
(when possible)ACTIVITY : Name That Postulate
Let’s PracticeIndicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
For SAS:
B D
For AAS ASA : A F AC FE
ACTIVITYIndicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
For SAS:
For AAS ASA :