geometry unit 4.1 4.3
TRANSCRIPT
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Two geometric figures with exactly the same size and shape.
The Idea of a CongruenceThe Idea of a Congruence
A C
B
DE
F
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How much do you How much do you need to know. . .need to know. . .
. . . about two triangles to prove that they are congruent?
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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
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Do you need Do you need all six ?all six ?
NO !
SSSSASASAAAS
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Side-Side-Side (SSS)Side-Side-Side (SSS)
1. AB ≅ DE2. BC ≅ EF3. AC ≅ DF
∆ABC ≅ ∆ DEF
B
A
C
E
D
F
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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
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The angle between two sides
Included AngleIncluded Angle
∠ G ∠ I ∠ H
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Name the included angle:
YE and ES
ES and YS
YS and YE
Included AngleIncluded Angle
SY
E
∠ E∠ S∠ Y
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Angle-Side-Angle-Side-AngleAngle (ASA) (ASA)
1. ∠A ≅ ∠ D2. AB ≅ DE3. ∠ B ≅ ∠ E
∆ABC ≅ ∆ DEF
B
A
C
E
D
F
included side
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The side between two angles
Included SideIncluded Side
GI HI GH
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Name the included side:
∠Y and ∠E
∠E and ∠S
∠S and ∠Y
Included SideIncluded Side
SY
E
YEESSY
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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
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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!
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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
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The Congruence PostulatesThe Congruence Postulates
SSS correspondence
ASA correspondence
SAS correspondence
AAS correspondence
SSA correspondence
AAA correspondence
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Name That PostulateName That Postulate
SASSAS ASAASA
SSSSSSSSASSA
(when possible)
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Name That PostulateName That Postulate(when possible)
ASAASA
SASSAS
AAAAAA
SSASSA
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Name That PostulateName That Postulate(when possible)
SASSAS
SASSAS
SASSASReflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SSASSA
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HW: Name That PostulateHW: Name That Postulate(when possible)
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(when possible)HW: Name That PostulateHW: Name That Postulate
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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
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HWHWIndicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
For SAS:
For AAS:
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A
B
FC
E
D
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Slide 1 of 2
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Slide 1 of 2
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Before we start…let’s get a few things straight
INCLUDED SIDE
A B
C
X Z
Y
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Angle-Side-Angle (ASA) Congruence Postulate
Two angles and the INCLUDED side
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Angle-Angle-Side (AAS) Congruence Postulate
Two Angles and One Side that is NOT included
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}Your Only Ways To Prove
Triangles Are Congruent
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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.
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statement. congruence a Write.
and ,, and In
LE
NLDENDΔLMNΔDEF
∠≅∠≅∠≅∠
∆≅∆ DEF NLM
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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 ∠≅∠
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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 ∠≅∠
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ΔGIH ≅ ΔJIK by AAS
G
I
H J
KEx 4
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ΔABC ≅ ΔEDC by ASA
B A
C
ED
Ex 5
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ΔACB ≅ ΔECD by SAS
B
A
C
E
D
Ex 6
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ΔJMK ≅ ΔLKM by SAS or ASA
J K
LM
Ex 7
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Not possible
K
J
L
T
U
Ex 8
V
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Slide 2 of 2
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Slide 2 of 2
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Slide 2 of 2
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Slide 2 of 2
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Slide 1 of 2
(over Lesson 5-5)
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Slide 1 of 2
(over Lesson 5-5)
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Slide 1 of 2
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Slide 1 of 2
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Slide 1 of 2
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Slide 1 of 2
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Slide 2 of 2
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Slide 2 of 2
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Slide 2 of 2
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Slide 2 of 2
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