1. G.7 Proving Triangles Similar
(AA~, SSS~, SAS~)

2. 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.

3. AA Similarity (Angle-Angle or AA~)
If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar.
Given:
and
Conclusion:
by AA~

4. SSS Similarity (Side-Side-Side or SSS~)
If the lengths of the corresponding sides of 2 triangles are proportional, then the triangles are similar.
Given:
Conclusion:
by SSS~

5. Example: SSS Similarity (Side-Side-Side)5 11 22 8 16 10
Given:
Conclusion:
By SSS ~

6. SAS Similarity (Side-Angle-Side or SAS~)
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~

7. Example: SAS Similarity (Side-Angle-Side)5 11 22 10
Given:
Conclusion:
By SAS ~

8. A
80 D E
80 B C
ABC ~ ADE by AA ~ Postulate
Slide from MVHS

9. C 6 10 D E 5 3 A B
CDE~ CAB by SAS ~ Theorem
Slide from MVHS

10. L 5 3 M 6 6 K N 6 10 O
KLM~ KON by SSS ~ Theorem
Slide from MVHS

11. A 20 D 30 24 16 B C 36
ACB~ DCA by SSS ~ Theorem
Slide from MVHS

12. L 15 P A 25 9 N
LNP~ ANL by SAS ~ Theorem
Slide from MVHS

13. Similarity is reflexive, symmetric, and transitive.
Proving Triangles Similar
Similarity is reflexive, symmetric, and transitive.
Steps for proving triangles similar:
1. Mark the Given.
2. Mark …
Reflexive (shared) Angles or Vertical Angles
3. 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.

14. AA Problem #1 Step 1: Mark the given … and what it implies
Step 2: Mark the vertical angles AA
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 C D E G F
Given
Alternate Interior <s
Alternate Interior <s
AA Similarity

15. SSS 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
1. IJ = 3LN ; JK = 3NP ; IK = 3LP
Given
Division Property
Substitution
SSS Similarity

16. SAS Problem #3 Step 1: Mark the given … and what it implies
Step 2: Mark the reflexive angles
SAS
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?

17. Statements
Reasons
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
Reflexive Property
SAS Postulate

18. Similarity is reflexive, symmetric, and transitive.

19. Choose a Problem. Problem #1 AA Problem #2 SSS Problem #3 SAS
End Slide Show
Problem #1 AA
Problem #2
SSS
Problem #3
SAS

20. Problem #1
Given: DE || FG
Prove:
DEC ~
FGC

21. Step 1: Mark the Given Given: DE || FG Prove: DEC ~ FGC
… and what it implies
Step 1: Mark the Given
Given: DE || FG
Prove:
DEC ~
FGC

22. Step 2: Mark . . . Reflexive Angles Vertical Angles Given: DE || FG
Prove:
DEC ~
FGC
… if they exist.

23. Step 3: Choose a Method
Given: DE || FG
Prove:
DEC ~
FGC AA
SSS
SAS

24. Given: DE || FG
Prove:
DEC ~
FGC
STATEMENTS
REASONS 3.
DEC ~
FGC

25. Choose a Problem. Problem #1 AA Problem #2 SSS Problem #3 SAS
End Slide Show
Problem #1 AA
Problem #2
SSS
Problem #3
SAS

26. Problem #2
Choose a Method Based on the given info AA
SSS
SAS

27. 1. Given 2. Division Prop. 3. Substitution 4. SSS Similarity
STATEMENTS
REASONS
1. Given
2. Division Prop.
3. Substitution
4. SSS Similarity

28. Choose a Problem. Problem #1 AA Problem #2 SSS Problem #3 SAS
End Slide Show
Problem #1 AA
Problem #2
SSS
Problem #3
SAS

29. Problem #3 Given: G is the midpoint of ED H is the midpoint of EF
Prove:
EGH~
EDF

30. Midpoint implies =/ @ segments Step 1: Mark the Given Given:
… and what it implies
Step 1: Mark the Given
Given:
G is the midpoint of ED
H is the midpoint of EF
Prove:
EGH~
EDF
Midpoint implies =/ @
segments

31. Reflexive Angles Vertical Angles Step 2: Mark . . . Given:
G is the midpoint of ED
H is the midpoint of EF
Prove:
EGH~
EDF

32. AA SSS SAS Step 3: Choose a Method Given: G is the midpoint of ED
H is the midpoint of EF
Prove:
EGH~
EDF AA
SSS
SAS

33. 1. Given 2. Def. of Midpoint 3. Seg. Add. Post. 4. Substitution Given:
G is the midpoint of ED
H is the midpoint of EF
Prove:
EGH~
EDF
STATEMENTS
REASONS
1. G is the midpoint of ED
H is the midpoint of EF
1. Given
2. Def. of Midpoint
3. Seg. Add. Post.
4. Substitution

34. 4. Substitution 5. Division Prop. = 6. Substitution 7. Reflexive Prop
STATEMENTS
REASONS
4. Substitution
5. Division Prop. =
6. Substitution
7. Reflexive Prop
8. SAS Similarity

35. The End 1. Mark the Given. 2. Mark … Shared Angles or Vertical Angles
3. 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.