Geometry unit 4.1 4.3
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Transcript of Geometry unit 4.1 4.3
Two geometric figures with exactly the same size and shape.
The Idea of a CongruenceThe Idea of a Congruence
A C
B
DE
F
How much do you How much do you need to know. . .need to know. . .
. . . about two triangles to prove that they are congruent?
You learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.
Corresponding PartsCorresponding Parts
∆ABC ≅ ∆ DEF
B
A C
E
D
F
1. AB ≅ DE2. BC ≅ EF3. AC ≅ DF4. ∠ A ≅ ∠ D5. ∠ B ≅ ∠ E6. ∠ C ≅ ∠ F
Do you need Do you need all six ?all six ?
NO !
SSSSASASAAAS
Side-Side-Side (SSS)Side-Side-Side (SSS)
1. AB ≅ DE2. BC ≅ EF3. AC ≅ DF
∆ABC ≅ ∆ DEF
B
A
C
E
D
F
Side-Angle-Side (SAS)Side-Angle-Side (SAS)
1. AB ≅ DE2. ∠A ≅ ∠ D3. AC ≅ DF
∆ABC ≅ ∆ DEF
B
A
C
E
D
F
included angle
The angle between two sides
Included AngleIncluded Angle
∠ G ∠ I ∠ H
Name the included angle:
YE and ES
ES and YS
YS and YE
Included AngleIncluded Angle
SY
E
∠ E∠ S∠ Y
Angle-Side-Angle-Side-AngleAngle (ASA) (ASA)
1. ∠A ≅ ∠ D2. AB ≅ DE3. ∠ B ≅ ∠ E
∆ABC ≅ ∆ DEF
B
A
C
E
D
F
included side
The side between two angles
Included SideIncluded Side
GI HI GH
Name the included side:
∠Y and ∠E
∠E and ∠S
∠S and ∠Y
Included SideIncluded Side
SY
E
YEESSY
Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)
1. ∠A ≅ ∠ D2. ∠ B ≅ ∠ E3. BC ≅ EF
∆ABC ≅ ∆ DEF
B
A
C
E
D
F
Non-included side
Warning:Warning: No SSA Postulate No SSA Postulate
A C
B
D
E
F
NOT CONGRUENT
There is no such thing as an SSA
postulate!
Warning:Warning: No AAA Postulate No AAA Postulate
A C
B
D
E
F
There is no such thing as an AAA
postulate!
NOT CONGRUENT
The Congruence PostulatesThe Congruence Postulates
SSS correspondence
ASA correspondence
SAS correspondence
AAS correspondence
SSA correspondence
AAA correspondence
Name That PostulateName That Postulate
SASSAS ASAASA
SSSSSSSSASSA
(when possible)
Name That PostulateName That Postulate(when possible)
ASAASA
SASSAS
AAAAAA
SSASSA
Name That PostulateName That Postulate(when possible)
SASSAS
SASSAS
SASSASReflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SSASSA
HW: Name That PostulateHW: Name That Postulate(when possible)
(when possible)HW: Name That PostulateHW: Name That Postulate
Let’s PracticeLet’s PracticeIndicate 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
HWHWIndicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
For SAS:
For AAS:
A
B
FC
E
D
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Slide 1 of 2
Before we start…let’s get a few things straight
INCLUDED SIDE
A B
C
X Z
Y
Angle-Side-Angle (ASA) Congruence Postulate
Two angles and the INCLUDED side
Angle-Angle-Side (AAS) Congruence Postulate
Two Angles and One Side that is NOT included
}Your Only Ways To Prove
Triangles Are Congruent
Overlapping sides are congruent in each
triangle by the REFLEXIVE property
Vertical Angles are congruent
Alt Int Angles are congruent
given parallel lines
Things you can mark on a triangle when they aren’t marked.
statement. congruence a Write.
and ,, and In
LE
NLDENDΔLMNΔDEF
∠≅∠≅∠≅∠
∆≅∆ DEF NLM
What other pair of angles needs to be marked so that the two triangles are congruent by AAS?
F
D
E
M
L
N
NE ∠≅∠
What other pair of angles needs to be marked so that the two triangles are congruent by ASA?
F
D
E
M
L
N
LD ∠≅∠
ΔGIH ≅ ΔJIK by AAS
G
I
H J
KEx 4
ΔABC ≅ ΔEDC by ASA
B A
C
ED
Ex 5
ΔACB ≅ ΔECD by SAS
B
A
C
E
D
Ex 6
ΔJMK ≅ ΔLKM by SAS or ASA
J K
LM
Ex 7
Not possible
K
J
L
T
U
Ex 8
V
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Slide 1 of 2
(over Lesson 5-5)
Slide 1 of 2
(over Lesson 5-5)
Slide 1 of 2
Slide 1 of 2
Slide 1 of 2
Slide 1 of 2
Slide 2 of 2
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