proving triangles congruent
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Proving Triangles Congruent. Powerpoint hosted on www.worldofteaching.com Please visit for 100’s more free powerpoints. Why do we study triangles so much?. Because they are the only “rigid” shape. Show pictures of triangles. F. B. A. C. E. D. The Idea of a Congruence. - PowerPoint PPT PresentationTRANSCRIPT
Proving Triangles Congruent
Powerpoint hosted on www.worldofteaching.comPlease visit for 100’s more free powerpoints
Why do we study triangles so much?
• Because they are the only “rigid” shape.
• Show pictures of triangles
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?
If all six pairs of corresponding parts of two triangles (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
1. AB DE2. BC EF3. AC DF
ABC DEF
Order matters!!
B
A
C
E
D
F
Side-Side-Side (SSS)Side-Side-Side (SSS)
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 formulaa2 + 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
• And• So
212
212 yyxxd
12 xxa 12 yyb
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)
_____
4,73,47,3
65164947 22 d
• 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.
D
BA
C
E G
F
H
I K
J
N
MO
Q
P
R
L
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 sidesIncluded 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
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
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!
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 .
Right
Leg
Leg – one of the sides that form the right of the Δ. Shorter than the hyp.
Hypotenuse –- Longest side of a Rt.
Δ - Opp. of Right
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 not.
IA B
CD
R
S T
J
KM
L
G
E
F
H
U V
WX
N
O
Q
P Y
Angle-Side-Angle (ASA)Angle-Side-Angle (ASA)
1. A D2.AB DE3. B E
ABC DEF
B
A
C
E
D
F
included side
The side between two anglesIncluded SideIncluded Side
GI HI GH
Name the included angle:
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
Angle-Angle-Side (AAS)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: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
Triangle Congruence Triangle Congruence (5 of them)(5 of them)
SSS postulate ASA postulate SAS postulate AAS theorem HL Theorem
Right Triangle CongruenceRight Triangle CongruenceWhen dealing with right triangles, you
will sometimes see the following definitions of congruence:
HL - Hypotenuse LegHA – Hypotenuse Angle
(AAS)LA - Leg Angle (AAS of ASA)LL – Leg Leg Theorem (SAS)
Name That PostulateName That Postulate
SASSAS ASAASA
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
ReviewReviewIndicate 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 UpGiven that triangles ABC and DEF are congruent, andG and H are the centroids, find AH.
AG = 7x + 8JH = 5x - 8Find DJ
JI
A
B
C F
E
D
G H