Transcript Lesson 4.4 - 4.5 Proving Triangles Congruent

Lesson 4.4 - 4.5
Proving Triangles
Congruent
Triangle Congruency Short-Cuts
If you can prove one of the following short
cuts, you have two congruent triangles
1.
2.
3.
4.
5.
SSS (side-side-side)
SAS (side-angle-side)
ASA (angle-side-angle)
AAS (angle-angle-side)
HL (hypotenuse-leg) right triangles only!
Built – In Information in
Triangles
Identify the ‘built-in’ part
Shared side
SSS
Parallel lines
-> AIA
Vertical angles
SAS
Shared side
SAS
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
Side-Side-Side (SSS)
1. AB  DE
2. BC  EF
3. AC  DF
ABC   DEF
Side-Angle-Side (SAS)
1. AB  DE
2. A   D
3. AC  DF
ABC   DEF
included
angle
Angle-Side-Angle (ASA)
1. A   D
2. AB  DE
ABC   DEF
3.  B   E
include
d
side
Angle-Angle-Side (AAS)
1. A   D
2.  B   E
ABC   DEF
3. BC  EF
Non-included
side
Warning: No AAA Postulate
There is no such
thing as an AAA
postulate!
E
B
A
C
D
NOT CONGRUENT
F
Warning: No SSA Postulate
There is no such
thing as an SSA
postulate!
E
B
F
A
C
D
NOT CONGRUENT
Name That Postulate
(when possible)
SAS
SSA
ASA
SSS
This is called a common side.
It is a side for both triangles.
We’ll use the reflexive property.
HL ( hypotenuse leg ) is used
only with right triangles, BUT,
not all right triangles.
HL
ASA
Name That Postulate
(when possible)
Reflexive
Property
SAS
Vertical
Angles
SAS
Vertical
Angles
SAS
Reflexive
Property
SSA
Name That Postulate
(when possible)
Name That Postulate
(when possible)
Closure Question
Let’s Practice
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
For AAS:
A  F
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
G
K
I
H
J
ΔGIH  ΔJIK by AAS
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
B
A
C
D
E
ΔABC  ΔEDC by ASA
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 6
E
A
C
B
D
ΔACB  ΔECD by SAS
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
M
K
L
ΔJMK  ΔLKM by SAS or ASA
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
K
L
V
Not possible
U
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 AB 
Side
PQ
BC  QR
Side
AC  PR
Then
∆ABC ≅ ∆PQR
Example 1
Prove: ∆DEF ≅ ∆JKL
From the diagram,
DE  JK , DF  JL, and EF  KL.
∆DEF ≅ ∆JKL
SSS Congruence Postulate.
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.
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
AC  DF
Angle ∠C ≅ ∠F
Then
∆ABC ≅ ∆DEF
Example 2
Prove: ∆SYT ≅ ∆WYX
Side-Side-Side Postulate
• SSS postulate: If two triangles have
three congruent sides, the triangles are
congruent.
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.
Angle-Side-Angle
Postulate
• If two angles and the side between
them are congruent to the other
triangle then the two angles are
congruent.
Side-Angle-Side
Postulate
• If two sides and the adjacent angle
between them are congruent to the other
triangle then those triangles are
congruent.
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.
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
A
areQ congruent.
R
S
B
C
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
A
Q
congruent.
R
S
B
C
AAS Proof
• If 2 angles are congruent, so is the
3rd
•
Third Angle Theorem
A
• NowQ QR is an included side,
so ASA.
R
S
B
C
Example
• Is it possible to prove these
triangles are congruent?
Example
• Is it possible to prove these
triangles are congruent?
• Yes - vertical angles are congruent,
so you have ASA
Example
• Is it possible to prove these
triangles are congruent?
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
Example
• Is it possible to prove these
triangles are congruent?
1
2
4
3
Example
• Is it possible to prove these triangles are
congruent?
1
2
4
3
• 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.
Included Angle
The angle between two sides
G
I
H
Included Angle
Name the included angle:
E
Y
S
YE and ES
E
ES and YS
S
YS and YE
Y
Included Side
The side between two angles
GI
HI
GH
Included Side
Name the included side:
E
Y
S
Y and E
YE
E and S
ES
S and Y
SY
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
MN  QR, NP  RS , PM  SQ
then,
MNP  QRS
Using SSS Congruence Post.
Prove:
PQW  TSW
• 1) PQ  ST , QW  WS , PW  TW
• 2) PQW  TSW
• 1) Given
• 2) SSS
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 PQ
 WX , QS  XY ,  Q  X
then,
PQS  WXY
Included Angle
The angle between two sides
G
I
H
Included Angle
Name the included angle:
E
Y
S
YE and ES
E
ES and YS
S
YS and YE
Y
Included Side
The side between two angles
GI
HI
GH
Included Side
Name the included side:
E
Y
S
Y and E
YE
E and S
ES
S and Y
SY
Triangle congruency shortC
cuts
B
A
D
F
E
H
Given: HJ  GI, GJ  JI
Prove: ΔGHJ  ΔIHJ
HJ  GI
Given
GJH & IJH are Rt <‘s
Def. ┴ lines
GJH  IJH
Rt <‘s are ≅
GJ  JI
Given
HJ  HJ
Reflexive Prop
ΔGHJ  ΔIHJ
G
SAS
J
I
Given: 1  2, A  E and AC  EC
B
Prove: ΔABC  ΔEDC
1  2
A  E
AC  EC
Given
Given
Given
ΔABC  ΔEDC
D
1
A
ASA
2
C
E
Given: ΔABD, ΔCBD, AB  CB,
and AD  CD
Prove: ΔABD  ΔCBD
A
AB  CB
AD  CD
BD  BD
C
Given
Given
Reflexive Prop
 ΔABD  ΔCBD
B
SSS
D
Given: LJ bisects IJK,
ILJ   JLK
Prove: ΔILJ  ΔKLJ
I
J
L
LJ bisects IJK
IJL  IJH
ILJ   JLK
JL  JL
Given
K
Definition of bisector
Given
Reflexive Prop
ΔILJ  ΔKLJ
ASA
Given: TV  VW, UV VX
Prove: ΔTUV  ΔWXV
TV  VW
UV  VX
TVU  WVX
U
Given
T
Given
Vertical angles
W
V
X
 ΔTUV  ΔWXV
SAS
Given: Given: HJ  JL, H L
Prove: ΔHIJ  ΔLKJ
I
HJ  JL
H L
IJH  KJL
K
Given
Given
Vertical angles
J
H
 ΔHIJ  ΔLKJ ASA
L
Given: Quadrilateral PRST with PR  ST,
PRT  STR
Prove: ΔPRT  ΔSTR
PR  ST
PRT  STR
RT  RT
R
S
Given
Given
Reflexive Prop
P
ΔPRT  ΔSTR
SAS
T
Given: Quadrilateral PQRS, PQ  QR,
PS  SR, and QR  SR
Prove: ΔPQR  ΔPSR
PQ  QR
PQR = 90°
PS  SR
PSR = 90°
QR  SR
PR  PR
Given
P
PQ  QR
Given
PS  SR
Given
Reflexive Prop
ΔPQR  ΔPSR
HL
Q
R
S
Prove it!
NOT triangle congruency short
cuts
NOT triangle congruency
short-cuts A
•
•
•
The following are
NOT short cuts:
AAA (angle-angleangle)
Triangles are similar
but not necessarily
congruent
60
60
C
60
D
B
60
60
F
60
E
NOT triangle congruency
B
short-cuts
•
•
•
The following are
NOT short cuts
SSA (side-sideangle)
SAS is a short cut
but the angle is in
between both sides!
8 cm
5 cm
34
A
E
8 cm
5 cm
34
F
D
C
Prove it!
CPCTC (Corresponding Parts of
Congruent Triangles are
Congruent)
CPCTC
• Once you have proved two triangles
congruent using one of the short
cuts, the rest of the parts of the
triangle you haven’t proved directly
are also congruent!
• We say: Corresponding Parts of
Congruent Triangles are Congruent or
CPCTC for short
CPCTC example
U
Given: ΔTUV, ΔWXV, TV  WV,
TW bisects UX
T
V
Prove: TU  WX
Statements:
Reasons:
X
1. TV  WV
Given
2. UV  VX
Definition of bisector
3. TVU  WVX
Vertical angles are congruent
4. ΔTUV  ΔWXV
SAS
5. TU  WX
CPCTC
W
Side Side Side
If 2 triangles have 3 corresponding pairs
of sides that are congruent, then the
X
~
AC = PX
triangles are congruent.
AB ~
= PN
~
CB = XN
A
P
C
N
B
Therefore, using SSS,
~
∆ABC = ∆PNX
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.
X
60°
A
C
60°
~
CA = XP
~
CB = XN
~
C =  X
P
N
B
Therefore, by
SAS, ~
=
∆ABC ∆PNX
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
~
A
70
°
C
60°
CA = XP
~
A = P
~
C = X
60°
70
°
N
B
P
Therefore, by ASA,
~
∆ABC= ∆PNX
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°
60°
70°
These two triangles are congruent by SAA
Remembering our shortcuts
SSS
ASA
SAS
SAA
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 .
B
F
That means that EG  CB
A
C
E
G
What is AC congruent
to?
FE
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.
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.
For example:
A
Prove: AB  DE
Statements
B
C
D
AC  DF
Given
C  F
Given
CB  FE
Given
ΔABC  ΔDEF
AB  DE
F
E
Reasons
SAS
CPCTC
Using SAS Congruence
Prove: Δ VWZ ≅ Δ XWY
PROOF
WZ  WY ,VW  XW
Given
 VWZ  XWY
Vertical Angles
Δ VWZ ≅ Δ XWY
SAS
Proof
Given: MB is perpendicular bisector of AP
Prove:
•
•
•
•
•
•
ABM  PBM
1) MB is perpendicular bisector of AP
2) <ABM and <PBM are right <‘s
3) AB  BP
4)  ABM  PBM
5) BM  BM
6) ABM  PBM
•
•
•
•
•
•
1) Given
2) Def of Perpendiculars
3) Def of Bisector
4) Def of Right <‘s
5) Reflexive Property
6) SAS
Proof
Given: O is the midpoint of MQ and NP
Prove:
•
•
•
•
MON  POQ
1) O is the midpoint of MQ and NP
2) MO  OQ, NO  OP
3)  MON  POQ
4) MON  POQ
•
•
•
•
1) Given
2) Def of midpoint
3) Vertical Angles
4) SAS
Proof
Given:
Prove:
AB  CD, BC  AD
ABC  ADC
• 1) AB  CD, BC  AD
• 2) AC  AC
• 3) ABC  ADC
• 1) Given
• 2) Reflexive Property
• 3) SSS
Proof
Given:
Prove:
•
•
•
•
AD  CB, AD || CB
ABD  CDB
1) AD  CB, AD || CB
2)  ADB  CBD
3) DB  DB
4) ABD  CDB
•
•
•
•
1) Given
2) Alt. Int. <‘s Thm
3) Reflexive Property
4) SAS
Checkpoint
Decide if enough information is given to prove
the triangles are congruent. If so, state the
congruence postulate you would use.
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?
EXAMPLE 2
Standardized Test Practice
SOLUTION
By counting, PQ = 4 and QR = 3. Use the Distance Formula
to find PR.
d
=
( x2 – x1 ) 2 + ( y2 – y1 ) 2
Write a proof.
GIVEN
PROVE
Proof
KL
NL, KM
KLM
NM
NLM
It is given that KL
NL and KM
By the Reflexive Property, LM
So, by the SSS Congruence
Postulate,
KLM
NLM
NM
LM.
GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true.
Explain your reasoning.
1.
DFG
HJK
SOLUTION
Three sides of one triangle are congruent to three
sides of second triangle then the two triangle are
congruent.
Side DG
HK, Side DF
JH,and Side FG JK.
So by the SSS Congruence postulate,
Yes. The statement is true.
DFG
HJK.
Included Angle
The angle between two sides
G
I
H
Included Angle
Name the included angle:
E
Y
S
YE and ES
E
ES and YS
S
YS and YE
Y
In the diagram at the right, what
postulate or theorem can you
use to prove that
RST
S
RS
RTS
VUT
Given
U
Given
UV
UTV
Vertical angles
Δ RST ≅ Δ VUT
SAA
Now For The Fun Part…
Given: JO  SH;
O is the midpoint of SH
Prove:  SOJ  HOJ
J
S
0
H
Given: BC bisects AD
A   D
Prove: AB  DC
A
C
E
B
D