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.