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Proving Triangles Congruent powerpoints at http://www.worldofteaching.com
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Proving Triangles Congruent
Proving Triangles Congruent
Free powerpoints at http://www.worldofteaching.com
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?
In Lesson 4.2, 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 DE
2. BC EF
3. AC DF
4. A D
5. B E
6. C F
Do you need Do you need all six ?all six ?
NO !
Side-Side-Side (SSS)Side-Side-Side (SSS)
1. AB DE
2. BC EF
3. AC DF
ABC DEF
B
A
C
E
D
F
Side-Angle-Side (SAS)Side-Angle-Side (SAS)
1. AB DE
2. A D
3. 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 D
2. AB DE
3. 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
YE
ES
SY
Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)
1. A D
2. B E
3. 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
SASSASASAASA
SSSSSSSSASSA
(when possible)
Name That PostulateName That Postulate(when possible)
ASAASA
SASASS
AAAAAA
SSASSA
Name That PostulateName That Postulate(when possible)
SASASS
SASSAS
SASASS
Reflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SSSS
AA
Name That PostulateName That Postulate(when possible)
(when possible)Name That PostulateName 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
Indicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
For SAS:
For 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.
ΔGIH ΔJIK by AAS
G
I
H J
KEx 4
ΔABC ΔEDC by ASA
B A
C
ED
Ex 5
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.
ΔACB ΔECD by SASB
A
C
E
D
Ex 6
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.
ΔJMK ΔLKM by SAS or ASA
J K
LM
Ex 7
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.
Not possible
K
J
L
T
U
Ex 8
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.
V
Hypotenuse-Leg Congruence Theorem: HL• Hypotenuse-Leg Congruence Theorem
(HL)– If the hypotenuse and a leg of a right triangle
are congruent to the hypotenuse and the leg of a second right triangle, then the two triangles are congruent.
– If ABC and DEF are right triangles, and
AC = DF, and BC = EF, then
ABC = DEF
~
~
~
A
B C
D
E F
Determine When to Use HL• Is it possible to show that the two
triangles are congruent using the HL Congruence Theorem? Explain your reasoning.
• The two triangles are right triangles. You know that JH = JH (Hypotenuse). You know that JG = HK (Leg). So, you can use the HL Congruence theorem to prove that JGH = HKJ.
G H
J K
~
~ ~
Triangle Congruence Postulates and Theorems1. SSS (Side-Side-Side)2. SAS (Side-Angle-Side)3. ASA (Angle-Side-Angle)4. AAS (Angle-Angle-Side)5. HL (Hypotenuse-Leg)
