Download proving triangles congruent

Proving Triangles
Congruent
The Idea of a Congruence
Two geometric figures with
exactly the same size and shape.
F
B
A
C
E
D
How much do you
need to know. . .
. . . about two triangles
to prove that they
are congruent?
Corresponding Parts
You learned that if all six pairs of
corresponding parts (sides and angles)
are congruent, then the triangles are
congruent.
B
1. AB ≅ DE
2. BC ≅ EF
3. AC ≅ DF
4. ∠ A ≅ ∠ D
5. ∠ B ≅ ∠ E
A
C
ΔABC ≅ Δ DEF
E
F
6. ∠ C ≅ ∠ F
D
Do you need all six ?
NO !
SSS
SAS
ASA
AAS
Side-Side-Side (SSS)
B
E
F
A
C
D
1. AB ≅ DE
2. BC ≅ EF
3. AC ≅ DF
ΔABC ≅ Δ DEF
Side-Angle-Side (SAS)
B
E
F
A
C
D
1. AB ≅ DE
2. ∠A ≅ ∠ D
3. AC ≅ DF
ΔABC ≅ Δ DEF
included
angle
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
Angle-Side-Angle (ASA)
B
E
F
A
C
D
1. ∠A ≅ ∠ D
2. AB ≅ DE
ΔABC ≅ Δ DEF
3. ∠ B ≅ ∠ E
included
side
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
Angle-Angle-Side (AAS)
B
E
F
A
C
D
1. ∠A ≅ ∠ D
2. ∠ B ≅ ∠ E
ΔABC ≅ Δ DEF
3. BC ≅ EF
Non-included
side
Warning: No SSA Postulate
There is no such
thing as an SSA
postulate!
E
B
F
A
C
D
NOT CONGRUENT
Warning: No AAA Postulate
There is no such
thing as an AAA
postulate!
E
B
A
C
D
NOT CONGRUENT
F
Name That Postulate
(when possible)
SAS
SSA
ASA
SSS
Name That Postulate
(when possible)
AAA
ASA
SAS
SSA
Things you can mark on a triangle when they aren’t
marked.
Overlapping sides are
congruent in each
triangle by the
REFLEXIVE property
Vertical
Angles are
congruent
Alt Int Angles
are
congruent
given parallel
lines
Name That Postulate
(when possible)
Reflexive
Property
Vertical
Angles
SAS
Vertical
Angles
SAS
SAS
Reflexive
Property
SSA
HW: Name That Postulate
(when possible)
HW: Name That Postulate
(when possible)
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
HW
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
For SAS:
For AAS:
Write a congruence statement for
each pair of triangles represented.
D
B
A
E
C
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.
Ex 8
J
T
K
L
V
Not possible
U
(over Lesson 5-5)
Slide 1 of 2
(over Lesson 5-5)
Slide 1 of 2