lessons 4-3 and 4-4 visit for 100’s of free powerpoints. this powerpoint has been modified by...
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Lessons 4-3 and 4-4Lessons 4-3 and 4-4
Visit www.worldofteaching.comFor 100’s of free powerpoints.
This Powerpoint has been modified by Lisa Palen from a Powerpoint found at www.worldofteaching.com and a Powerpoint found at IGO Geometry Online
Proving Triangles Congruent
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Two geometric figures with exactly the same size and shape.
The Idea of CongruenceThe Idea of 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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In Lesson 4-2, 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 !
SSSSASASAAASHL
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Side-Side-Side (SSS)Side-Side-Side (SSS)
1. AB DE
2. BC EF
3. AC DF
ABC DEF
B
A
C
E
D
F
Side
Side
Side
The triangles are congruent
by SSS.
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
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The angle between two sides
Included AngleIncluded Angle
HGI G
GIH I
GHI H
This combo is called side-angle-side, or just SAS.
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Name the included angle:
YE and ES
ES and YS
YS and YE
Included AngleIncluded Angle
SY
E
YES or E
YSE or S
EYS or Y
The other two angles are the NON-INCLUDED
angles.
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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 Side
Angle
Side
The triangles are congruent
by SAS.
If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.
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The side between two angles
Included SideIncluded Side
GI HI GH
This combo is called angle-side-angle, or just ASA.
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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
The other two sides are the
NON-INCLUDED sides.
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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
Angle
Side
Angle
The triangles are congruent
by ASA.
If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.
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E
D
F
Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)
1. A D
2. B E
3. BC EF
ABC DEF
Non-included
side
B
AC
SideAngle
Angle
The triangles are congruent
by AAS.
If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent.
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Warning:Warning: No SSA Postulate No SSA Postulate
There is no such thing as an SSA
postulate!
The triangles are NOTcongruent!
Side
Side
Angle
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Warning:Warning: No SSA Postulate No SSA Postulate
NOT CONGRUENT!
There is no such thing as an SSA
postulate!
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BUT:BUT: SSA DOES work in one SSA DOES work in onesituation!situation!
If we know that the two triangles
are right triangles!
Side
Side
Side
Angle
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We call thisWe call this
These triangles ARE CONGRUENT by HL!
HL,for “Hypotenuse – Leg”
Hypotenuse
Leg
Hypotenuse
RIGHT Triangles!
Remember! The
triangles must be RIGHT!
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Hypotenuse-Leg (HL)Hypotenuse-Leg (HL)
1.AB HL
2.CB GL
3. C and G are rt. ‘s
ABC DEF
The triangles
are congruent
by HL.
Right Triangle
LegHypotenuse
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
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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!
Same Shapes
!
Different Sizes!
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Congruence Postulates Congruence Postulates and Theoremsand Theorems
• SSS• SAS• ASA• AAS• AAA?• SSA?• HL
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Name That PostulateName That Postulate
SASSASASAASA
AASAASSSASSA
(when possible)
Not enough info!
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Name That PostulateName That Postulate(when possible)
SSSSSSAAAAAA
SSASSA
Not enough info!
Not enough info!
SSASSAHL
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Name That PostulateName That Postulate(when possible)
SSASSA
AAAAAA
Not enough info!
Not enough info!
HL
SSASSA
Not enough info!
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Vertical Angles, Vertical Angles, Reflexive Sides and AnglesReflexive Sides and Angles
When two triangles touch, there may be additional congruent parts.
Vertical Angles
Reflexive Side
side shared by two
triangles
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Name That PostulateName That Postulate(when possible)
SASASS
AASAAS
SASASS
Reflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SSSS
AANot enough
info!
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When two triangles overlap, there may be additional congruent parts.
Reflexive Sideside shared by two
triangles
Reflexive Angleangle shared by two
triangles
Reflexive Sides and AnglesReflexive Sides and Angles
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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 Some Proofs
End Slide Show
What’s Next
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Choose a Problem.
Problem #1
Problem #2
Problem #3
End Slide Show
D
A B
C
E
C
D
AB
Z
W Y
X
SSS
SAS
ASA
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Problem #4
55
Statements Reasons
AAS
Given
Given
Vertical Angles Thm
AAS Postulate
Given: A C BE BDProve: ABE CBD
E
C
D
AB
4. ABE CBD
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Statements Reasons
CB D
A
57
3. AC AC
HL
Given
Given
Reflexive Property
HL Postulate4. ABC ADC
Problem #5
1. ABC, ADC right s
AB AD
Given ABC, ADC right s,
Prove:
AB AD
ABC ADC
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Congruence Proofs1. Mark the Given.2. Mark …
Reflexive Sides or Angles / Vertical AnglesAlso: mark info implied by given info.3. Choose a Method. (SSS , SAS, ASA)4. List the Parts …
in the order of the method.5. Fill in the Reasons …
why you marked the parts.6. Is there more?
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Given implies Congruent Parts
midpoint
parallel
segment bisector
angle bisector
perpendicular
segments
angles
segments
angles
angles
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Example Problem
CB D
AGiven: AC bisects BAD AB ADProve: ABC ADC
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Step 1: Mark the Given
… and what it implies
CB D
AGiven: AC bisects BAD AB ADProve: ABC ADC
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•Reflexive Sides•Vertical AnglesStep 2: Mark . . .
… if they exist.
CB D
AGiven: AC bisects BAD AB ADProve: ABC ADC
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Step 3: Choose a Method
SSSSASASAAASHL
CB D
AGiven: AC bisects BAD AB ADProve: ABC ADC
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Step 4: List the Parts
STATEMENTS REASONS
… in the order of the Method
CB D
AGiven: AC bisects BAD AB ADProve: ABC ADC
BAC DAC
AB AD
AC AC
S
AS
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Step 5: Fill in the Reasons
(Why did you mark those parts?)
STATEMENTS REASONS
CB D
AGiven: AC bisects BAD AB ADProve: ABC ADC
BAC DAC
AB AD
AC AC
Given
Def. of Bisector
Reflexive (prop.)
S
AS
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S
AS
Step 6: Is there more?
STATEMENTS REASONS
CB D
AGiven: AC bisects BAD AB ADProve: ABC ADC
BAC DAC
AB AD
AC AC
Given
AC bisects BAD Given
Def. of Bisector
Reflexive (prop.)ABC ADC SAS (pos.)
1.2.3.4.5.
1.2.3.4.5.
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Congruent Triangles Proofs
1. Mark the Given and what it implies.2. Mark … Reflexive Sides / Vertical Angles3. Choose a Method. (SSS , SAS, ASA)4. List the Parts …
in the order of the method.5. Fill in the Reasons …
why you marked the parts.6. Is there more?
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Using CPCTC in Proofs
• According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent.
• This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles.
• This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.
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Corresponding Parts of Congruent Triangles
• For example, can you prove that sides AD and BC are congruent in the figure at right?
• The sides will be congruent if triangle ADM is congruent to triangle BCM.– Angles A and B are congruent because they are marked.– Sides MA and MB are congruent because they are marked.– Angles 1 and 2 are congruent because they are vertical
angles.– So triangle ADM is congruent to triangle BCM by ASA.
• This means sides AD and BC are congruent by CPCTC.
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Statement Reason
MA @ MB Given
ÐA @ ÐB Given
Ð1 @ Ð2 Vertical angles
DADM @ DBCM ASA
AD @ BC CPCTC
Corresponding Parts of Congruent Triangles
• A two column proof that sides AD and BC are congruent in the figure at right is shown below:
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Statement Reason
MA @ MB Given
ÐA @ ÐB Given
Ð1 @ Ð2 Vertical angles
DADM @ DBCM ASA
AD @ BC CPCTC
Corresponding Parts of Congruent Triangles
• A two column proof that sides AD and BC are congruent in the figure at right is shown below:
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Statement Reason
FR @ FO Given
RU @ OU Given
UF @ UF reflexive prop.
DFRU @ DFOU SSS
ÐR @ ÐO CPCTC
Corresponding Parts of Congruent Triangles
• Sometimes it is necessary to add an auxiliary line in order to complete a proof
• For example, to prove ÐR @ ÐO in this picture
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Statement Reason
FR @ FO Given
RU @ OU Given
UF @ UF Same segment
DFRU @ DFOU SSS
ÐR @ ÐO CPCTC
Corresponding Parts of Congruent Triangles
• Sometimes it is necessary to add an auxiliary line in order to complete a proof
• For example, to prove ÐR @ ÐO in this picture