View
240
Download
2
Category
Tags:
Preview:
Citation preview
G.7ProvingTrianglesSimilar
(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~)
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~
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~
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 ~
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~
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 ~
A
B C
D E80
80
ABC ~ ADE by AA ~ Postulate
Slide from MVHS
A B
C
D E
CDE~ CAB by SAS ~ Theorem
6
3
10
5
Slide from MVHS
O
N
L
KM
KLM~ KON by SSS ~ Theorem
63
10
56
6
Slide from MVHS
CB
A
D
ACB~ DCA by SSS ~ Theorem
24
36
20
3016
Slide from MVHS
N
L
AP
LNP~ ANL by SAS ~ Theorem
25 9
15
Slide from MVHS
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
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
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
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
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
Similarity is reflexive,
symmetric, and transitive.
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
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.
Recommended