G.6 Visit www.worldofteaching.com For 100’s of free powerpoints. Proving Triangles Congruent 1
G.6
Visit www.worldofteaching.com
For 100’s of free powerpoints.
Proving Triangles Congruent
1
Two geometric figures with exactly the same size and shape.
The Idea of Congruence
A C
B
D E
F
2
How much do you need to know. . . . . . about two triangles to prove that they are congruent?
3
Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.
Corresponding Parts
ABC DEF
1. AB DE
2. BC EF
3. AC DF
4. A D
5. B E
6. C F
4
Do you need all six ?
NO !
SSS SAS ASA AAS HL
5
Side-Side-Side (SSS)
1. AB DE
2. BC EF
3. AC DF
ABC DEF
Side
Side
Side
The triangles are congruent by
SSS.
If the sides of one triangle are congruent to the sides of a
second triangle, then the triangles are congruent.
6
The angle between two sides
Included Angle
HGI G
GIH I
GHI H
This combo is called side-angle-side, or just SAS.
7
Name the included angle:
YE and ES
ES and YS
YS and YE
Included Angle
S Y
E
YES or E
YSE or S EYS or Y The other two
angles are the NON-INCLUDED
angles.
8
Side-Angle-Side (SAS)
1. AB DE
2. A D
3. AC DF
ABC DEF
included angle
Side
Angle
Side
The triangles are congruent by
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.
9
The side between two angles
Included Side
GI HI GH
This combo is called angle-side-angle, or just ASA.
10
Name the included side:
Y and E
E and S
S and Y
Included Side
S Y
E
YE
ES
SY
The other two sides are the
NON-INCLUDED sides.
11
Angle-Side-Angle (ASA)
1. A D
2. AB DE
3. B E
ABC DEF
included side
Angle
Side
Angle
The triangles are congruent by
ASA.
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.
12
Angle-Angle-Side (AAS)
1. A D
2. B E
3. BC EF
ABC DEF
Non-included side Side
Angle
Angle
The triangles are congruent by
AAS.
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.
13
Warning: No SSA Postulate
There is no such
thing as an SSA
postulate!
The triangles are NOTcongruent!
Side
Side
Angle
14
Warning: No SSA Postulate
NOT CONGRUENT!
There is no such
thing as an SSA
postulate!
15
BUT: SSA DOES work in one
situation!
If we know that
the two triangles
are right
triangles!
Side
Side
Side
Angle
16
We call this
These triangles ARE CONGRUENT by HL!
HL,
for “Hypotenuse – Leg”
Hypotenuse
Leg
Hypotenuse
RIGHT Triangles!
Remember! The
triangles must be RIGHT!
17
Hypotenuse-Leg (HL)
1.AB HL
2.CB GL
3.C and G are rt. ‘s
ABC DEF
The triangles are congruent
by HL.
Right Triangle
Leg
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.
18
Warning: No AAA Postulate
A C
B
D
E
F
There is no such
thing as an AAA
postulate!
NOT CONGRUENT!
Same Shapes!
Different Sizes!
19
Congruence Postulates
and Theorems
• SSS • SAS • ASA • AAS • AAA? • SSA? • HL
20
Name That Postulate
SAS ASA
AAS SSA
(when possible)
Not enough info!
21
Name That Postulate (when possible)
SSS AAA
SSA
Not enough info!
Not enough info!
SSA HL
22
Name That Postulate (when possible)
SSA
AAA
Not enough info!
Not enough info!
HL
SSA
Not enough info!
23
Vertical Angles,
Reflexive Sides and Angles When two triangles touch, there may be additional congruent parts.
Vertical Angles
Reflexive Side
side shared by two
triangles
24
Name That Postulate (when possible)
SAS
AAS
SAS Reflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SSA
Not enough info!
25
When two triangles overlap, there may be additional congruent parts.
Reflexive Side side shared by two
triangles
Reflexive Angle angle shared by two
triangles
Reflexive Sides and Angles 26
Let’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
27
Try Some Proofs
End Slide Show
What’s Next
28
Choose a
Problem.
Problem #1
Problem #2
Problem #3
End Slide Show
D
A B
C
E
C
D
AB
Z
W Y
X
SSS
SAS
ASA
29
D
A B
C
Given: AB CD
BC DAProve: ABC CDA
Problem #1
30
Step 1: Mark the Given
D
A B
C
Given: AB CD
BC DAProve: ABC CDA
31
•Reflexive Sides
•Vertical Angles
D
A B
C
Given: AB CD
BC DAProve: ABC CDA
Step 2: Mark . . .
… if they exist.
32
Step 3: Choose a Method
SSS
SAS
ASA
AAS
HL
D
A B
C
Given: AB CD
BC DAProve: ABC CDA
33
Step 4: List the Parts
D
A B
C
Given: AB CD
BC DAProve: ABC CDA
STATEMENTS REASONS
1. AB CD
2. BC DA
3. AC AC
… in the order of the Method
S
S
S
34
Step 5: Fill in the Reasons
(Why did you mark those parts?)
D
A B
C
Given: AB CD
BC DAProve: ABC CDA
STATEMENTS REASONS
1. AB CD
2. BC DA
3. AC AC
1. Given2. Given3. Reflexive Prop.
S
S
S
35
S
S
S
Step 6: Is there more?
D
A B
C
Given: AB CD
BC DAProve: ABC CDA
STATEMENTS REASONS
1. AB CD
2. BC DA
3. AC AC
1. Given2. Given3. Reflexive Prop.
4. ABC CDA 4. SSS (pos.)
The “Prove” Statement
is always last !
36
Choose a
Problem.
Problem #1
Problem #2
Problem #3
End Slide Show
D
A B
C
E
C
D
AB
Z
W Y
X
SSS
SAS
ASA
37
Problem #2
E
C
D
AB
Given: AB CB
EB DBProve: ABE CBD
38
Step 1: Mark the Given
E
C
D
AB
Given: AB CB
EB DBProve: ABE CBD
39
•Reflexive Sides
•Vertical Angles
… if they exist.
Step 2: Mark . . .
E
C
D
AB
Given: AB CB
EB DBProve: ABE CBD
40
Step 3: Choose a Method
SSS
SAS
ASA
AAS
HL
E
C
D
AB
Given: AB CB
EB DBProve: ABE CBD
41
Step 4: List the Parts
… in the order of the Method
STATEMENTS REASONS
E
C
D
AB
Given: AB CB
EB DBProve: ABE CBD
1. AB CB2. ABE CBD
3. EB DB
S
A
S
42
Step 5: Fill in the Reasons
(Why did you mark those parts?)
STATEMENTS REASONS
E
C
D
AB
Given: AB CB
EB DBProve: ABE CBD
1. AB CB2. ABE CBD
3. EB DB
1. Given2. Vertical s (thm.)
3. Given
S
A
S
43
Step 6: Is there more?
STATEMENTS REASONS
E
C
D
AB
Given: AB CB
EB DBProve: ABE CBD
4. ABE CBD
1. AB CB2. ABE CBD
3. EB DB
1. Given2. Vertical s (thm.)
3. Given4. SAS (pos.)
The “Prove” Statement
is always last !
S
A
S
44
Choose a
Problem.
Problem #1
Problem #2
Problem #3
End Slide Show
D
A B
C
E
C
D
AB
Z
W Y
X
SSS
SAS
ASA
45
Problem #3
Z
W Y
XGiven: XWY ZWY XYW ZYWProve: WXY WZY
46
Step 1: Mark the Given
Z
W Y
XGiven: XWY ZWY XYW ZYWProve: WXY WZY
47
•Reflexive Sides
•Vertical Angles
… if they exist.
Step 2: Mark . . .
Z
W Y
XGiven: XWY ZWY XYW ZYWProve: WXY WZY
48
Step 3: Choose a Method
SSS
SAS
ASA
AAS
HL
Z
W Y
XGiven: XWY ZWY XYW ZYWProve: WXY WZY
49
Step 4: List the Parts
… in the order of the Method
STATEMENTS REASONS
Z
W Y
XGiven: XWY ZWY XYW ZYWProve: WXY WZY
1. XWY ZWY
2. WY WY3. XYW ZYW
A
S
A
50
Step 5: Fill in the Reasons
(Why did you mark those parts?)
STATEMENTS REASONS
Z
W Y
XGiven: XWY ZWY XYW ZYWProve: WXY WZY
1. XWY ZWY
2. WY WY3. XYW ZYW
1. Given2. Reflexive (pos.)
3. Given
A
S
A
51
Step 6: Is there more?
STATEMENTS REASONS
Z
W Y
XGiven: XWY ZWY XYW ZYWProve: WXY WZY
1. XWY ZWY
2. WY WY3. XYW ZYW
4. WXY WZY
1. Given2. Reflexive (pos.)
3. Given
4. ASA (pos.)
The “Prove” Statement
is always last !
A
S
A
52
Choose a
Problem.
Problem #1
Problem #2
Problem #3
End Slide Show
D
A B
C
E
C
D
AB
Z
W Y
X
SSS
SAS
ASA
53
Choose a
Problem.
Problem #4
Problem #5
End Slide Show
AAS
HL
E
C
D
AB
CB D
A
54
Problem #4
Statements Reasons
AAS
Given
Given
Vertical Angles Thm
AAS Postulate
Given: A C
BE BDProve: ABE CBD
E
C
D
AB
4. ABE CBD
55
Choose a
Problem.
Problem #4
Problem #5
End Slide Show
AAS
HL
E
C
D
AB
CB D
A
56
Problem #5
3. AC AC
Statements Reasons
CB D
AHL
Given
Given
Reflexive Property
HL Postulate 4. ABC ADC
1. ABC, ADC right s
AB AD
Given ABC, ADC right s,
Prove:
AB AD
ABC ADC
57
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?
58
Given implies Congruent Parts
midpoint
parallel
segment bisector
angle bisector
perpendicular
segments
angles
segments
angles
angles
59
Example Problem
CB D
AGiven: AC bisects BAD
AB ADProve: ABC ADC
60
Step 1: Mark the Given
… and
what it
implies
CB D
AGiven: AC bisects BAD
AB ADProve: ABC ADC
61
•Reflexive Sides
•Vertical Angles Step 2: Mark . . .
… if they exist.
CB D
AGiven: AC bisects BAD
AB ADProve: ABC ADC
62
Step 3: Choose a Method
SSS
SAS
ASA
AAS
HL
CB D
AGiven: AC bisects BAD
AB ADProve: ABC ADC
63
Step 4: List the Parts
STATEMENTS REASONS
… in the order of the Method
CB D
AGiven: AC bisects BAD
AB ADProve: ABC ADC
BAC DAC
AB AD
AC AC
S
A
S
64
Step 5: Fill in the Reasons
(Why did you mark those parts?)
STATEMENTS REASONS
CB D
AGiven: AC bisects BAD
AB ADProve: ABC ADC
BAC DAC
AB AD
AC AC
Given
Def. of Bisector
Reflexive (prop.)
S
A
S
65
S
A
S
Step 6: Is there more?
STATEMENTS REASONS
CB D
AGiven: AC bisects BAD
AB ADProve: 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.
66
Midpoint implies segments.
Back
CB D
A
STATEMENTS REASONS
1. C is the midpoint
of BD
S
Given: C is the midpoint of BD
AB AD Prove: ABC ADC
2. BC CD 2. Def. of Midpoint
1. Given
… AB AD3. 3. Given
67
Parallel implies angles.
Back
STATEMENTS REASONS D
A B
C
Given: AB DC
AD BC Prove: ABC CDA
2. BAC DCA 2. Alt. Int. s (thm.)1. AB DC
3. AD BC 3. Given
4. DAC BCA 4. Alt. Int. s (thm.)
A
A
1. Given
68
Seg. bisector implies segments. Back
STATEMENTS REASONS
CB
A
DE
2. EB BC
4. AB BD
1. AD bisects EC 1. Given 2. Def. of bisect
3. EC bisects AD 3. Given 4. Def. of bisect
S
S
Given: AD bisects EC
EC bisects ADProve: ABE DBC
…
69
Angle bisector implies angles.
Back
STATEMENTS REASONS
CB D
AGiven: AC bisects BAD
AB ADProve: ABC ADC
2. BAC DAC
1. AC bisects BAD 1. Given
A
…
2. Def. of bisect
70
implies right ( ) angles. Back
STATEMENTS REASONS
CB D
AGiven: AC BD
BC DCProve: ABC ADC
2. ACB and ACD are right s
3. ACB ACD
1. AC BD 1. Given
2. lines form 4 rt. s (thm)
3. All rt. s are (thm)
A
… S 2. BC CD 4. Given 4.
71
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?
72
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.
73
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.
74
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
A B Given
1 2 Vertical angles
ADM BCM ASA
AD BC 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
A B Given
1 2 Vertical angles
ADM BCM ASA
AD BC CPCTC
76
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
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 Same segment
DFRU @ DFOU SSS
ÐR @ ÐO CPCTC
78