g.7 proving triangles similar
DESCRIPTION
G.7 Proving Triangles Similar. (AA~, SSS ~ , SAS ~ ). Similar Triangles. Two triangles are similar if they are the same shape . That means the vertices can be paired up so the angles are congruent. Size does not matter. AA Similarity (Angle-Angle or AA ~ ). - PowerPoint PPT PresentationTRANSCRIPT
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G.7ProvingTrianglesSimilar
(AA~, SSS~, SAS~)
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Similar Triangles
Two triangles are similar if they are the same shape. That means the vertices can be paired up so the angles are congruent. Size does not matter.
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AA Similarity (Angle-Angle or AA~)
A D B E
If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar.
E
DA
B
CF
ABC ~ DEFConclusion:
andGiven:
by AA~
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SSS Similarity (Side-Side-Side or SSS~)
ABC ~ DEF
If the lengths of the corresponding sides of 2 triangles are proportional, then the triangles are similar.
E
DA
B
CF
Given:
Conclusion:
BC
EF
AB
DE
AC
DF
by SSS~
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E
DA
B
CF
Example: SSS Similarity (Side-Side-Side)
Given: Conclusion:
ABC ~ DEFBC
EF
AB
DE
AC
DF
5
11 22
8 1610
8
16
5
10
11
22 By SSS ~
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E
DA
B
CF
SAS Similarity (Side-Angle-Side or SAS~)
ABC ~ DEF
AB ACA D and
DE DF
If the lengths of 2 sides of a triangle are proportional to the lengths of 2 corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.
Given:
Conclusion: by SAS~
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E
DA
B
CF
Example: SAS Similarity (Side-Angle-Side)
Given: Conclusion:
ABC ~ DEF
A DAB
DE
AC
DF
5
11 22
10
By SAS ~
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A
B C
D E80
80
ABC ~ ADE by AA ~ Postulate
Slide from MVHS
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A B
C
D E
CDE~ CAB by SAS ~ Theorem
6
3
10
5
Slide from MVHS
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O
N
L
KM
KLM~ KON by SSS ~ Theorem
63
10
56
6
Slide from MVHS
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CB
A
D
ACB~ DCA by SSS ~ Theorem
24
36
20
3016
Slide from MVHS
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N
L
AP
LNP~ ANL by SAS ~ Theorem
25 9
15
Slide from MVHS
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Similarity is reflexive, symmetric, and transitive.
1. Mark the Given.2. Mark …
Reflexive (shared) Angles or Vertical Angles3. Choose a Method. (AA~, SSS~, SAS~)Think about what you need for the chosen method and be sure to include those parts in the proof.
Steps for proving triangles similar:
Proving Triangles Similar
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Problem #1:
Pr :
Given DE FG
ove DEC FGC
CD
E
G
F
Step 1: Mark the given … and what it implies
Step 2: Mark the vertical angles
Step 3: Choose a method: (AA,SSS,SAS)Step 4: List the Parts in the order of the method with reasons
Step 5: Is there more? Statements Reasons
Given
Alternate Interior <s
AA Similarity
Alternate Interior <s
1. DE FG2. D F 3. E G
4. DEC FGC
AA
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Problem #2
Step 1: Mark the given … and what it implies
Step 2: Choose a method: (AA,SSS,SAS)Step 4: List the Parts in the order of the method with reasons
Step 5: Is there more? Statements Reasons
Given
Division Property
SSS Similarity
Substitution
SSS
: 3 3 3
Pr :
Given IJ LN JK NP IK LP
ove IJK LNP
N
L
P
I
J K
1. IJ = 3LN ; JK = 3NP ; IK = 3LP
2. IJ
LN=3,
JK
NP=3,
IK
LP=3
3. IJ
LN=
JK
NP=
IK
LP
4. IJK~ LNP
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Problem #3
Step 1: Mark the given … and what it implies
Step 3: Choose a method: (AA,SSS,SAS)
Step 4: List the Parts in the order of the method with reasons
Next Slide………….
Step 5: Is there more?
SAS
: midpoint
midpoint
Prove :
Given G is the of ED
H is the of EF
EGH EDF
E
DF
G H
Step 2: Mark the reflexive angles
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Statements Reasons
1. G is the Midpoint of
H is the Midpoint of
Given
2. EG = DG and EH = HF Def. of Midpoint
3. ED = EG + GD and EF = EH + HF Segment Addition Post.
4. ED = 2 EG and EF = 2 EH Substitution
Division Property
Substitution
Reflexive Property
SAS Postulate
ED
EF
7. GEHDEF
8. EGH~ EDF
6. ED
EG=
EF
EH
5. ED
EG=2 and
EF
EH =2
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Similarity is reflexive,
symmetric, and transitive.
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Choose a Problem.
Problem #1
Problem #2
Problem #3
End Slide Show
SSS
SAS
AAC
E
G
F
D
E
DF
G H
PN
L
I
J K
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The End1. Mark the Given.2. Mark …
Shared Angles or Vertical Angles3. Choose a Method. (AA, SSS , SAS)
**Think about what you need for the chosen method and
be sure to include those parts in the proof.