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Transcript
G.6
Proving
Triangles
Congruent
1
Visit
www.worldofteaching.com
2
The Idea of Congruence
Two geometric figures with
exactly the same size and
shape.
F
B
A
C
E
D
3
How much do you
need to know. . .
. . . about two triangles
to prove that they
are congruent?
4
Corresponding Parts
Previously we learned that if all six
pairs of corresponding parts (sides and
angles) are congruent, then the
triangles are congruent.
1.
2.
3.
4.
5.
6.
AB ≅ DE
BC ≅ EF
AC ≅ DF
∠ A ≅ ∠ D
∠ B ≅ ∠ E
∠ C ≅ ∠ F
ΔABC ≅ Δ
DEF
5
Do you need all six ?
NO !
SSS
SAS
ASA
AAS
HL
6
Side-Side-Side (SSS)
If the sides of one triangle are congruent to the sides of a
second triangle, then the triangles are congruent.
Side
Side
1. AB ≅ DE
2. BC ≅ EF
3. AC ≅ DF
Side
ΔABC ≅ Δ
DEF
The triangles
are congruent by
SSS.
7
Included Angle
The angle between two sides
∠ HGI
∠ G
∠ GIH
∠ I
∠ GHI
∠ H
This combo is called
side-angle-side, or just SAS.
8
Included Angle
Name the included angle:
E
Y
S
YE and ES
∠ YES or ∠E
ES and YS
∠ YSE or ∠S
YS and YE
∠ EYS or ∠Y
The other two
angles are the
NON-INCLUDED
angles.
9
Side-Angle-Side (SAS)
If two sides and the included angle of one triangle are
congruent to the two sides and the included angle of another
triangle, then the triangles are congruent.
included
angle
Side
Side
1. AB ≅ DE
2. ∠A ≅ ∠ D
3. AC ≅ DF
Angle
ΔABC ≅ Δ
DEF
The triangles
are congruent by
SAS.
10
Included Side
The side between two angles
GI
HI
GH
This combo is called
angle-side-angle, or just ASA.
11
Included Side
Name the included side:
E
Y
S
∠Y and ∠E
YE
∠E and ∠S
ES
∠S and ∠Y
SY
The other two sides
are the NONINCLUDED sides.
Angle-Side-Angle (ASA)
12
If two angles and the included side of one triangle are
congruent to the two angles and the included side of another
triangle, then the triangles are congruent.
included
side
1. ∠A ≅ ∠ D
2. AB ≅ DE
3. ∠ B ≅ ∠ E
Angle
Side
Angle
ΔABC ≅ Δ
DEF
The triangles
are congruent by
ASA.
Angle-Angle-Side (AAS)
13
If two angles and a non-included side of one triangle are
congruent to the corresponding angles and side of another
triangle, then the triangles are congruent.
Non-included
side
1. ∠A ≅ ∠ D
2. ∠ B ≅ ∠ E
3. BC ≅ EF
Angle
Side
Angle
ΔABC ≅ Δ
DEF
The triangles
are congruent by
AAS.
14
Warning: No SSA Postulate
There is no such
thing as an SSA
postulate!
Side
Angle
Side
The triangles are
NOTcongruent!
15
Warning: No SSA Postulate
There is no such
thing as an SSA
postulate!
NOT CONGRUENT!
BUT: SSA DOES work in one
situation!
16
If we know that
the two triangles
are right
triangles!
Side
Side
Side
Angle
17
We call this
HL,
for “Hypotenuse – Leg”
Hypotenuse
Hypotenuse
Leg
RIGHT Triangles!
These triangles ARE CONGRUENT by HL!
Remember!
The
triangles
must be
RIGHT!
Hypotenuse-Leg (HL)
18
If the hypotenuse and a leg of a right triangle are congruent
to the hypotenuse and a leg of another right triangle, then the
triangles are congruent.
Right Triangle
Leg
1.AB ≅ HL
2.CB ≅ GL
3.∠C and ∠G
are rt. ∠ ‘s
ΔABC ≅ Δ
DEFThe triangles
are congruent
by HL.
19
Warning: No AAA Postulate
There is no such
thing as an AAA
postulate!
Same
Shapes!
E
B
A
C
D
NOT CONGRUENT!
Different
Sizes!
F
Congruence Postulates
and Theorems
20
• SSS
• SAS
• ASA
• AAS
• AAA?
• SSA?
• HL
21
Name That Postulate
(when possible)
SAS
SSA
Not enough
info!
ASA
AAS
22
Name That Postulate
(when possible)
AAA
SSA
Not enough
info!
Not enough
info!
SSS
SSA
HL
23
Name That Postulate
(when possible)
Not enough
info!
Not enough
info!
SSA
SSA
HL
Not enough
info!
AAA
Vertical Angles,
Reflexive Sides and Angles
24
When two triangles touch, there may be
additional congruent parts.
Vertical Angles
Reflexive Side
side shared by two
triangles
25
Name That Postulate
(when possible)
Reflexive
Property
SAS
Vertical
Angles
AAS
Vertical
Angles
SAS
Reflexive
Property
SSA
Not enough
info!
26
Reflexive Sides and Angles
When two triangles overlap, there may be
additional congruent parts.
Reflexive Side
side shared by two
triangles
Reflexive Angle
angle shared by two
triangles
Let’s Practice
27
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
∠B ≅ ∠D
For SAS:
AC ≅
FE
∠A ≅ ∠F
For AAS:
28
What’s Next
Try Some Proofs
End Slide Show
29
Choose a Problem.
Problem
#1SSS
Problem
#2SAS
Problem
#3ASA
End Slide
Show
30
Problem #1
31
Step 1: Mark the Given
32
Step 2: Mark . . .
•Reflexive Sides
•Vertical Angles
… if they exist.
33
Step 3: Choose a Method
SSS
SAS
ASA
AAS
HL
34
Step 4: List the Parts
STATEMENTS
REASONS
S
S
S
… in the order of the Method
35
Step 5: Fill in the Reasons
STATEMENTS
REASONS
S
S
S
(Why did you mark those parts?)
36
The “Prove”
Statement
is always last !
Step 6: Is there more?
STATEMENTS
S
S
S
REASONS
37
Choose a Problem.
Problem
#1SSS
Problem
#2SAS
Problem
#3ASA
End Slide
Show
38
Problem #2
39
Step 1: Mark the Given
40
Step 2: Mark . . .
•Reflexive Sides
•Vertical Angles
… if they exist.
41
Step 3: Choose a Method
SSS
SAS
ASA
AAS
HL
42
Step 4: List the Parts
STATEMENTS
REASONS
S
A
S
… in the order of the Method
43
Step 5: Fill in the Reasons
STATEMENTS
REASONS
S
A
S
(Why did you mark those parts?)
44
The “Prove”
Statement
is always last !
Step 6: Is there more?
STATEMENTS
S
A
S
REASONS
45
Choose a Problem.
Problem
#1SSS
Problem
#2SAS
Problem
#3ASA
End Slide
Show
46
Problem #3
47
Step 1: Mark the Given
48
Step 2: Mark . . .
•Reflexive Sides
•Vertical Angles
… if they exist.
49
Step 3: Choose a Method
SSS
SAS
ASA
AAS
HL
50
Step 4: List the Parts
STATEMENTS
REASONS
A
S
A
… in the order of the Method
51
Step 5: Fill in the Reasons
STATEMENTS
REASONS
A
S
A
(Why did you mark those parts?)
52
The “Prove”
Statement
is always last !
Step 6: Is there more?
STATEMENTS
A
S
A
REASONS
53
Choose a Problem.
Problem
#1SSS
Problem
#2SAS
Problem
#3ASA
End Slide
Show
54
Choose a Problem.
Problem
#4AAS
Problem
#5HL
End Slide
Show
Problem #4
AAS
Statements
Reasons
Given
Vertical Angles Thm
Given
AAS Postulate
55
56
Choose a Problem.
Problem
#4AAS
Problem
#5HL
End Slide
Show
Problem #5
HL
Given ΔABC, ΔADC right Δs,
Prove:
Statements
1. ΔABC, ΔADC right Δs
Reasons
Given
Given
Reflexive Property
HL Postulate
57
58
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?
59
Given implies Congruent Parts
midpoin
t
paralle
l
segment
bisector
angle
bisector
perpendicula
r
segments
angles
segments
angles
angles
60
Example Problem
61
Step 1: Mark the Given
… and
what
it
implies
62
Step 2: Mark . . .
•Reflexive Sides
•Vertical Angles
… if they exist.
63
Step 3: Choose a Method
SSS
SAS
ASA
AAS
HL
64
Step 4: List the Parts
STATEMENTS
REASONS
S
A
S
… in the order of the Method
65
Step 5: Fill in the Reasons
STATEMENTS
REASONS
S
A
S
(Why did you mark those parts?)
66
Step 6: Is there more?
STATEMENTS
S 1.
2.
A 3.
S 4.
5.
REASONS
1.
2.
3.
4.
5.
67
Back
Midpoint implies
STATEMENTS
S
3.
…
segments.
REASONS
3. Given
68
Back
Parallel implies
STATEMENTS
A
A
angles.
REASONS
69
Back
Seg. bisector implies
STATEMENTS
S
S
…
segments.
REASONS
70
Back
Angle bisector implies
STATEMENTS
A
…
angles.
REASONS
71
Back
implies right ( ) angles.
STATEMENTS
A
S 4.
…
REASONS
4. Given
72
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?
73
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.
74
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.
75
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 @ DBCM ASA
AD @ BC
CPCTC
76
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 @ DBCM ASA
AD @ BC
CPCTC
77
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 @ DFOU SSS
ÐR @ ÐO
CPCTC
78
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 @ DFOU SSS
ÐR @ ÐO
CPCTC