Download Triangle Congruence

Proving Triangles
Congruent
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Why do we study
triangles so much?
• Because they are the only “rigid”
shape.
• Show pictures of triangles
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
If all six pairs of corresponding parts of
two triangles (sides and angles) are
congruent, then the triangles are
congruent.
1. AB  DE
2. BC  EF
3. AC  DF
4.  A   D
5.  B   E
6.  C   F
ABC   DEF
Do you need all six ?
NO !
SSS
SAS
ASA
AAS
Side-Side-Side (SSS)
1. AB  DE
2. BC  EF
3. AC  DF
ABC   DEF
Order
matters!!
Distance Formula
• The Cartesian coordinate system is
really two number lines that are
perpendicular to each other.
• We can find distance between two
points via………..
• Mr. Moss’s favorite formula
a2 + b2 = c2
Distance Formula
• The distance formula is from
Pythagorean’s Theorem.
• Remember distance on the number
line is the difference of the two
numbers, so a  x2  x1
• And b   y2  y1 
• So


d  x2  x1    y2  y1 
2
2
To find distance:
• You can use the Pythagorean theorem
or the distance formula.
• You can get the lengths of the legs
by subtracting or counting.
• A good method is to align the points
vertically and then subtract them to
get the distance between them
• Find the distance between (-3, 7) and
(4, 3)
 3,7 



4
,
3
_____
 7,4
d
 7  4
2
2
 49  16  65
• Do some examples via Geo Sketch
Class Work
• Page 240, # 1 – 13 all
Warm Up
• Are these triangle pairs congruent by
SSS? Give congruence statement or
reason why not.
A
F
B
E
G
D
H
C
Q
N
J
R
M
I
L
K
O P
Side-Angle-Side (SAS)
1. AB  DE
2. A   D
3. AC  DF
ABC   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
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
Why no SSA Postulate?
• We would get in trouble if someone
spelled it backwards. (bummer!)
• Really – SSA allows the possibility of
having two different solutions.
• Remember: It only takes one counter
example to prove a conjecture wrong.
• Show problem on Geo Sketch
* A Right Δ has one Right .
Hypotenuse –
Δ
Leg
- Longest side of a Rt.
- Opp. of Right 
Right 
Leg – one of the sides that
form the right  of the Δ.
Shorter than the hyp.
Th(4-6) Hyp – Leg Thm.
(HL)
• If the hyp. & a leg of 1 right Δ are 
to the hyp & a leg of another rt. Δ,
then the Δ’s are .
Group Class Work
• Do problem 18 on page 245
Class Work
• Pg 244 # 1 – 17 all
Warm Up
• Are these triangle pairs congruent?
State postulate or theorem and
congruence statement or reason why
F
G
not.
A
J
B
I
E
M
H
C
D
K
L
N
Q
R
U
V
W
O
P
T
S
Y
X
Angle-Side-Angle (ASA)
1.A   D
2.AB  DE
3.B   E
ABC   DEF
included
side
Included Side
The side between two angles
GI
HI
GH
Included Side
Name the included angle:
E
Y
S
Y and E
YE
E and S
ES
S and Y
SY
Angle-Angle-Side (AAS)
1. A   D
2.  B   E
ABC   DEF
3. BC  EF
Non-included
side
Angle-Angle-Side (AAS)
• The proof of this is based on ASA.
• If you know two angles, you really
know all three angles.
• The difference between ASA and
AAS is just the location of the
angles and side.
• ASA – Side is between the angles
• AAS –Side is not between the angles
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
Triangle Congruence
(5 of them)
 SSS
postulate
 ASA
postulate
 SAS postulate
 AAS
 HL
theorem
Theorem
Right Triangle Congruence
When dealing with right triangles, you will
sometimes see the following definitions of
congruence:
HL - Hypotenuse Leg
HA – Hypotenuse Angle (AAS)
LA - Leg Angle (AAS of ASA)
LL – Leg Leg Theorem (SAS)
Name That Postulate
(when possible)
SAS
SSA
ASA
SSS
Name That Postulate
(when possible)
AAA
SAS
ASA
SSA
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)
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
Review
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
For SAS:
For AAS:
Class Work
• Pg 251, # 1 – 17 all
CPCTC
• Proving shapes are congruent proves
ALL corresponding dimensions are
congruent.
• For Triangles: Corresponding Parts of
Congruent Triangles are Congruent
• This includes, but not limited to,
corresponding sides, angles,
altitudes, medians, centroids,
incenters, circumcenters, etc.
They are ALL CONGRUENT!
Warm Up
Given that triangles ABC and DEF are congruent, and
G and H are the centroids, find AH.
AG = 7x + 8
JH = 5x - 8
Find DJ
B
I
G
A
E
C
J
H
D
F