Lesson 4.4 - 4.5 Proving Triangles Congruent

1
Lesson 4.4 - 4.5Proving Triangles Congruent
2
Triangle Congruency Short-Cuts

• If you can prove one of the following short cuts,
  you have two congruent triangles
• SSS (side-side-side)
• SAS (side-angle-side)
• ASA (angle-side-angle)
• AAS (angle-angle-side)
• HL (hypotenuse-leg) right triangles only!

3
Built In Information in Triangles

•  

4
Identify the built-in part
5
Shared side
SSS
Vertical angles
SAS
Parallel lines -gt AIA
Shared side
SAS
6
SOME REASONS For Indirect Information

• Def of midpoint
• Def of a bisector
• Vert angles are congruent
• Def of perpendicular bisector
• Reflexive property (shared side)
• Parallel lines .. alt int angles
• Property of Perpendicular Lines

7
Side-Side-Side (SSS)
8
Side-Angle-Side (SAS)


B

E






F
A



C
D

1. AB ? DE
2. ?A ? ? D
3. AC ? DF

?ABC ? ? DEF
included angle
9
Angle-Side-Angle (ASA)


B

E






F
A



C
D

1. ?A ? ? D
2. AB ? DE
3. ? B ? ? E

?ABC ? ? DEF
included side
10
Angle-Angle-Side (AAS)


B

E






F
A



C
D

1. ?A ? ? D
2. ? B ? ? E
3. BC ? EF

?ABC ? ? DEF
Non-included side
11
Warning No AAA Postulate
There is no such thing as an AAA postulate!
E
B
A
C
F
D
NOT CONGRUENT
12
Warning No SSA Postulate
There is no such thing as an SSA postulate!
E
B
F
A
C
D
NOT CONGRUENT
13
Name That Postulate
(when possible)
SAS
ASA
SSA
SSS
14
This is called a common side. It is a side for
both triangles.
Well use the reflexive property.
15
HL ( hypotenuse leg ) is used only with right
triangles, BUT, not all right triangles.
ASA
HL
16
Name That Postulate
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Reflexive Property
Vertical Angles
SSA
SAS
17
Name That Postulate
(when possible)
18
Name That Postulate
(when possible)
19
Closure Question
20
Lets Practice
Indicate the additional information needed to
enable us to apply the specified congruence
postulate.
For ASA
?B ? ?D
For SAS
?A ? ?F
For AAS
21
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.
Ex 4
?GIH ? ?JIK by AAS
22
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.
Ex 5
?ABC ? ?EDC by ASA
23
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.
E
Ex 6
A
C
B
D
?ACB ? ?ECD by SAS
24
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.
Ex 7
J
K
L
M
?JMK ? ?LKM by SAS or ASA
25
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.
J
T
Ex 8
L
K
V
U
Not possible
26
SSS (Side-Side-Side) Congruence Postulate

• If three sides of one triangle are congruent to
  three sides of a second triangle, the two
  triangles are congruent.

If Side
Side
Side
Then
?ABC ? ?PQR
27
Example 1
Prove ?DEF ? ?JKL
From the diagram,
SSS Congruence Postulate.
?DEF ? ?JKL
28
SAS (Side-Angle-Side) Congruence Postulate

• If two sides and the included angle of one
  triangle are congruent to two sides and the
  included angle of a second triangle, then the two
  triangles are congruent.

29
Angle-Side-Angle (ASA) Congruence Postulate

• If two angles and the included side of one
  triangle are congruent to two angles and the
  included side of a second triangle, the two
  triangles are congruent.

If Angle
?A ? ?D
Side
Angle
?C ? ?F
Then
?ABC ? ?DEF
30
Example 2
Prove ?SYT ? ?WYX
31
Side-Side-Side Postulate

• SSS postulate If two triangles have three
  congruent sides, the triangles are congruent.

32
Angle-Angle-Side Postulate

• If two angles and a non included side are
  congruent to the two angles and a non included
  side of another triangle then the two triangles
  are congruent.

33
Angle-Side-Angle Postulate

• If two angles and the side between them are
  congruent to the other triangle then the two
  angles are congruent.

34
Side-Angle-Side Postulate

• If two sides and the adjacent angle between them
  are congruent to the other triangle then those
  triangles are congruent.

35
Which Congruence Postulate to Use?

• 1. Decide whether enough information is given in
  the diagram to prove that triangle PQR is
  congruent to triangle PQS. If so give a
  two-column proof and state the congruence
  postulate.

36
ASA

• If 2 angles and the included side of one triangle
  are congruent to two angles and the included side
  of a second triangle, then the 2 triangles are
  congruent.

A
Q
S
C
R
B
37
AAS

• If 2 angles and a nonincluded side of one
  triangle are congruent to 2 angles and the
  corresponding nonincluded side of a second
  triangle, then the 2 triangles are congruent.

A
Q
S
C
R
B
38
AAS Proof

• If 2 angles are congruent, so is the 3rd
• Third Angle Theorem
• Now QR is an included side, so ASA.

A
Q
S
C
R
B
39
Example

• Is it possible to prove these triangles are
  congruent?

40
Example

• Is it possible to prove these triangles are
  congruent?
• Yes - vertical angles are congruent, so you have
  ASA

41
Example

• Is it possible to prove these triangles are
  congruent?

42
Example

• Is it possible to prove these triangles are
  congruent?
• No. You can prove an additional side is
  congruent, but that only gives you SS

43
Example

• Is it possible to prove these triangles are
  congruent?

2
1
3
4
44
Example

• Is it possible to prove these triangles are
  congruent?
• Yes. The 2 pairs of parallel sides can be used
  to show Angle 1 Angle 3 and Angle 2 Angle
  4. Because the included side is congruent to
  itself, you have ASA.

2
1
3
4
45
Included Angle
The angle between two sides
? H
? G
? I
46
Included Angle
Name the included angle YE and ES ES and
YS YS and YE
? E
? S
? Y
47
Included Side
The side between two angles
GI
GH
HI
48
Included Side
Name the included side ?Y and ?E ?E and ?S
?S and ?Y
YE
ES
SY
49
Side-Side-Side Congruence Postulate
SSS Post. - If three sides of one triangle are
congruent to three sides of a second triangle,
then the two triangles are congruent.
If
then,
50
Using SSS Congruence Post.
Prove

• 1)
• 2)

• 1) Given
• 2) SSS

51
Side-Angle-Side Congruence Postulate
SAS Post. If two sides and the included angle
of one triangle are congruent to two sides and
the included angle of a second triangle, then the
two triangles are congruent.
If
then,
52
Included Angle
The angle between two sides
? H
? G
? I
53
Included Angle
Name the included angle YE and ES ES and
YS YS and YE
E
? E
? S
? Y
S
Y
54
Included Side
The side between two angles
GI
GH
HI
55
Included Side
Name the included side ?Y and ?E ?E and ?S
?S and ?Y
YE
ES
SY
56
Triangle congruency short-cuts

•  

57

• Given HJ ? GI, GJ ? JI
• Prove ?GHJ ? ?IHJ
• HJ ? GI Given
• ?GJH ?IJH are Rt lts
• Def. - lines
• ?GJH ? ?IJH
• Rt lts are ?
• GJ ? JI Given
• HJ ? HJ Reflexive Prop
• ??GHJ ? ?IHJ SAS

58

• Given ?1 ? ?2, ?A ? ?E and AC ? EC
• Prove ?ABC ? ?EDC
• ?1 ? ?2 Given
• ?A ? ?E Given
• AC ? EC Given
• ??ABC ? ?EDC ASA

59

• Given ?ABD, ?CBD, AB ? CB,
• and AD ? CD
• Prove ?ABD ? ?CBD
• AB ? CB Given
• AD ? CD Given
• BD ? BD Reflexive Prop
• ? ?ABD ? ?CBD SSS

60

• Given LJ bisects ?IJK,
• ?ILJ ? ? JLK
• Prove ?ILJ ? ?KLJ
• LJ bisects ?IJK Given
• ?IJL ? ?IJH Definition of bisector
• ?ILJ ? ? JLK Given
• JL ? JL Reflexive Prop
• ??ILJ ? ?KLJ ASA

61

• Given TV ? VW, UV ?VX
• Prove ?TUV ? ?WXV
• TV ? VW Given
• UV ? VX Given
• ?TVU ? ?WVX Vertical angles
• ? ?TUV ? ?WXV SAS

62

• Given Given HJ ? JL, ?H ??L
• Prove ?HIJ ? ?LKJ
• HJ ? JL Given
• ?H ??L Given
• ?IJH ? ?KJL Vertical angles
• ? ?HIJ ? ?LKJ ASA

63

• Given Quadrilateral PRST with PR ? ST,
• ?PRT ? ?STR
• Prove ?PRT ? ?STR
• PR ? ST Given
• ?PRT ? ?STR Given
• RT ? RT Reflexive Prop
• ??PRT ? ?STR SAS

64

• Given Quadrilateral PQRS, PQ ? QR,
• PS ? SR, and QR ? SR
• Prove ?PQR ? ?PSR
• PQ ? QR Given
• ?PQR 90 PQ ? QR
• PS ? SR Given
• ?PSR 90 PS ? SR
• QR ? SR Given
• PR ? PR Reflexive Prop
• ?PQR ? ?PSR HL

65
Prove it!

• NOT triangle congruency short cuts

66
NOT triangle congruency short-cuts

• The following are NOT short cuts
• AAA (angle-angle-angle)
• Triangles are similar but not necessarily
  congruent

67
NOT triangle congruency short-cuts

• The following are NOT short cuts
• SSA (side-side-angle)
• SAS is a short cut but the angle is in between
  both sides!

68
Prove it!

• CPCTC (Corresponding Parts of Congruent Triangles
  are Congruent)

69
CPCTC

• Once you have proved two triangles congruent
  using one of the short cuts, the rest of the
  parts of the triangle you havent proved directly
  are also congruent!
• We say Corresponding Parts of Congruent
  Triangles are Congruent or CPCTC for short

70
CPCTC example

• Given ?TUV, ?WXV, TV ? WV,
• TW bisects UX
• Prove TU ? WX
• Statements Reasons
• TV ? WV Given
• UV ? VX Definition of bisector
• ?TVU ? ?WVX Vertical angles are congruent
• ?TUV ? ?WXV SAS
• ?TU ? WX CPCTC

71
Side Side Side

• If 2 triangles have 3 corresponding pairs of
  sides that are congruent, then the triangles are
  congruent.

72
Side Angle Side

• If two sides and the INCLUDED ANGLE in one
  triangle are congruent to two sides and INCLUDED
  ANGLE in another triangle, then the triangles are
  congruent.

73
Angle Side Angle

• If two angles and the INCLUDED SIDE of one
  triangle are congruent to two angles and the
  INCLUDED SIDE of another triangle, the two
  triangles are congruent.

X
60
A
3 inches
3 inches
70
70
P
N
60
C
B
74
Side Angle Angle

• Triangle congruence can be proved if two angles
  and a NON-included side of one triangle are
  congruent to the corresponding angles and
  NON-included side of another triangle, then the
  triangles are congruent.

60
70
5 m
5 m
These two triangles are congruent by SAA
75
Remembering our shortcuts

• SSS
• ASA
• SAS
• SAA

76
Corresponding parts

• When you use a shortcut (SSS, AAS, SAS, ASA, HL)
  to show that 2 triangles are ?,
• that means that ALL the corresponding parts are
  congruent.
• EX If a triangle is congruent by ASA (for
  instance), then all the other corresponding parts
  are ?.

77
Corresponding parts of congruent triangles are
congruent.
Corresponding parts of congruent triangles are
congruent.
Corresponding parts of congruent triangles are
congruent.
Corresponding parts of congruent triangles are
congruent.
78
Corresponding Parts of Congruent Triangles are
Congruent.
CPCTC

• If you can prove congruence using a shortcut,
  then you KNOW that the remaining corresponding
  parts are congruent.

You can only use CPCTC in a proof AFTER you have
proved congruence.
79
For example
A

• Prove AB ? DE

B
C
D
F
E
80
(No Transcript)
81
Using SAS Congruence
Prove ? VWZ ? ? XWY
PROOF
Given
? VWZ ? ? XWY
Vertical Angles

• SAS

82
Proof
Given MB is perpendicular bisector of AP Prove

• 1) MB is perpendicular bisector of AP
• 2) ltABM and ltPBM are right lts
• 3)
• 4)
• 5)
• 6)

• 1) Given
• 2) Def of Perpendiculars
• 3) Def of Bisector
• 4) Def of Right lts
• 5) Reflexive Property
• 6) SAS

83
Proof
Given O is the midpoint of MQ and NP Prove

• 1) O is the midpoint of MQ and NP
• 2)
• 3)
• 4)

• 1) Given
• 2) Def of midpoint
• 3) Vertical Angles
• 4) SAS

84
Proof
Given Prove

• 1)
• 2)
• 3)

• 1) Given
• 2) Reflexive Property
• 3) SSS

85
Proof
Given Prove

• 1)
• 2)
• 3)
• 4)

• 1) Given
• 2) Alt. Int. lts Thm
• 3) Reflexive Property
• 4) SAS

86
Checkpoint

• Decide if enough information is given to prove
  the triangles are congruent. If so, state the
  congruence postulate you would use.

87
Congruent Triangles in the Coordinate Plane
Use the SSS Congruence Postulate to show that
?ABC ? ?DEF
Which other postulate could you use to prove the
triangles are congruent?
88
Congruent Triangles
89
EXAMPLE 2
Standardized Test Practice
SOLUTION
By counting, PQ 4 and QR 3. Use the Distance
Formula to find PR.
90
(No Transcript)
91
for Example 1
GUIDED PRACTICE
Decide whether the congruence statement is true.
Explain your reasoning.
SOLUTION
Three sides of one triangle are congruent to
three sides of second triangle then the two
triangle are congruent.
Yes. The statement is true.
92
Included Angle
The angle between two sides
? H
? G
? I
93
Included Angle
Name the included angle YE and ES ES and
YS YS and YE
? E
? S
? Y
94
? RST ? ? VUT SAA
95

• Now For The Fun Part

Proofs!
96
Given JO ? SH O is the midpoint of SH
Prove ? SOJ ? HOJ
97

• Given BC bisects AD
• ?A ? ? D
  
  
• Prove AB ? DC

A C E
B D
98
WORK