G8 Math Q3-Week 7- Proving Triangle Congruence.ppt

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1. PROVING
TRIANGLES
CONGRUENT

2. TRIANGLE CONGRUENCY SHORT-CUTS
If you can prove one of the following short cuts, you have two congruent
triangles
1.SSS (side-side-side)
2.SAS (side-angle-side)
3.ASA (angle-side-angle)
4.AAS (angle-angle-side)
5.HL (hypotenuse-leg) right triangles only!

3. BUILT – IN INFORMATION IN
TRIANGLES

4. Identify the ‘built-in’ part Identify the ‘built-in’ part

5. Shared sideShared side
Parallel lines Parallel lines
-> AIA-> AIA
Shared sideShared side
Vertical anglesVertical angles
SASSAS
SASSAS
SSSSSS

6. SOME REASONS FOR INDIRECT SOME REASONS FOR INDIRECT
INFORMATIONINFORMATION
•Def of midpointDef of midpoint
•Def of a bisectorDef of a bisector
•Vert angles are congruentVert angles are congruent
•Def of perpendicular bisectorDef of perpendicular bisector
•Reflexive property (shared side)Reflexive property (shared side)
•Parallel lines ….. alt int anglesParallel lines ….. alt int angles
•Property of Perpendicular LinesProperty of Perpendicular Lines

7. This is called a common side.This is called a common side.
It is a side for both triangles.It is a side for both triangles.
We’ll use the reflexive property.We’ll use the reflexive property.

8. HLHL ( hypotenuse leg ) is used( hypotenuse leg ) is used
only with right triangles, BUT, only with right triangles, BUT,
not all right triangles. not all right triangles.
HLHL
ASAASA

9. Name That PostulateName That Postulate
(when possible)
SASSAS
SASSAS
SASSAS
Reflexive
Property
Vertical
Angles
Vertical
Angles
Reflexive
PropertySSASSA

10. Let’s PracticeLet’s Practice
Indicate 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

11. DETERMINE IF WHETHER EACH PAIR OF TRIANGLES IS
CONGRUENT BY SSS, SAS, ASA, OR AAS. IF IT IS NOT
POSSIBLE TO PROVE THAT THEY ARE CONGRUENT, WRITE
NOT POSSIBLE.
ΔGIH  ΔJIK by AAS
G
I
H
J
K
Ex 4

12. ΔABC  ΔEDC by ASA
B A
C
ED
Ex 5
DETERMINE IF WHETHER EACH PAIR OF TRIANGLES IS
CONGRUENT BY SSS, SAS, ASA, OR AAS. IF IT IS NOT
POSSIBLE TO PROVE THAT THEY ARE CONGRUENT, WRITE
NOT POSSIBLE.

13. ΔACB  ΔECD by SAS
B
A
C
E
D
Ex 6
DETERMINE IF WHETHER EACH PAIR OF TRIANGLES IS
CONGRUENT BY SSS, SAS, ASA, OR AAS. IF IT IS NOT
POSSIBLE TO PROVE THAT THEY ARE CONGRUENT, WRITE
NOT POSSIBLE.

14. ΔJMK  ΔLKM by SAS or ASA
J K
L
M
Ex 7
DETERMINE IF WHETHER EACH PAIR OF TRIANGLES IS
CONGRUENT BY SSS, SAS, ASA, OR AAS. IF IT IS NOT
POSSIBLE TO PROVE THAT THEY ARE CONGRUENT,
WRITE NOT POSSIBLE.

15. Not possible
K
J
L
T
U
Ex 8
DETERMINE IF WHETHER EACH PAIR OF TRIANGLES IS
CONGRUENT BY SSS, SAS, ASA, OR AAS. IF IT IS NOT
POSSIBLE TO PROVE THAT THEY ARE CONGRUENT,
WRITE NOT POSSIBLE.
V

16. PROBLEM #4
Statements Reasons
37
AAS
Given
Given
Vertical Angles Thm
AAS Postulate
Given: A  C
BE  BD
Prove: ABE  CBD
E
C
D
A
B
4. ABE  CBD

17. PROBLEM #5
3.AC AC
Statements Reasons
38
CB D
A
HL
Given
Given
Reflexive Property
HL Postulate4. ABC  ADC
1. ABC, ADC right s
AB AD
Given ABC, ADC right s,
Prove:

AB AD
ABC ADC 

18. Congruence Proofs
1. Mark the Given.
2. Mark …
Reflexive Sides or Angles / Vertical Angles
Also: 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?
39

19. GIVEN IMPLIES CONGRUENT
PARTS
40
midpoint
parallel
segment bisector
angle bisector
perpendicular
segments
angles
segments
angles
angles

20. Example Problem
CB D
AGiven: AC bisects BAD
AB  AD
Prove: ABC  ADC
41

21. STEP 1: MARK THE GIVEN
42 … and
what it
implies
CB D
AGiven: AC bisects BAD
AB  AD
Prove: ABC  ADC

22. •Reflexive Sides
•Vertical Angles
STEP 2: MARK . . .
43
… if they exist.
CB D
AGiven: AC bisects BAD
AB  AD
Prove: ABC  ADC

23. STEP 3: CHOOSE A METHOD
44
SSS
SAS
ASA
AAS
HL
CB D
AGiven: AC bisects BAD
AB  AD
Prove: ABC  ADC

24. STEP 4: LIST THE PARTS
45
STATEMENTS REASONS
… in the order of the Method
CB D
AGiven: AC bisects BAD
AB  AD
Prove: ABC  ADC
BAC  DAC
AB  AD
AC  AC
S
A
S

25. STEP 5: FILL IN THE REASONS
46
(Why did you mark those parts?)
STATEMENTS REASONS
CB D
AGiven: AC bisects BAD
AB  AD
Prove: ABC  ADC
BAC  DAC
AB  AD
AC  AC
Given
Def. of Bisector
Reflexive (prop.)
S
A
S

26. S
A
S
STEP 6: IS THERE MORE?
47
STATEMENTS REASONS
CB D
AGiven: AC bisects BAD
AB  AD
Prove: 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.

27. Congruent Triangles Proofs
1. Mark the Given and what it implies.
2. Mark … Reflexive Sides / Vertical Angles
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?
53

28. 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.
54

29. 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.
55

30. CORRESPONDING PARTS OF
CONGRUENT TRIANGLES
•A two column proof that sides AD and BC are congruent in the
figure at right is shown below:
Statement Reason
MA @ MB Given
ÐA @ ÐB Given
Ð1 @ Ð2 Vertical angles
DADM @ DBCMASA
AD @ BC CPCTC
56

31. CORRESPONDING PARTS OF
CONGRUENT TRIANGLES
•A two column proof that sides AD and BC are congruent in the
figure at right is shown below:
Statement Reason
MA @ MB Given
ÐA @ ÐB Given
Ð1 @ Ð2 Vertical angles
DADM @ DBCMASA
AD @ BC CPCTC
57

32. 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
Statement Reason
FR @ FO Given
RU @ OU Given
UF @ UF reflexive prop.
DFRU @ DFOUSSS
ÐR @ ÐO CPCTC
58

33. 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
Statement Reason
FR @ FO Given
RU @ OU Given
UF @ UF Same segment
DFRU @ DFOUSSS
ÐR @ ÐO CPCTC
59