Proving Triangles Congruent

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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

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The Idea of a Congruence
Two geometric figures with exactly the same
size and shape.
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How much do you need to know. . .
. . . about two triangles to
prove that they are congruent?
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Corresponding Parts
If all six pairs of corresponding parts of two
triangles (sides and angles) are congruent, then
the triangles are congruent.
?ABC ? ? DEF
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Do you need all six ?
NO !
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Side-Side-Side (SSS)
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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. Mosss favorite formula
• a2 b2 c2

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Distance Formula

• The distance formula is from Pythagoreans
  Theorem.
• Remember distance on the number line is the
  difference of the two numbers, so
• And
• So

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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

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• Find the distance between (-3, 7) and (4, 3)
• _____

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• Do some examples via Geo Sketch

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Class Work

• Page 240, 1 13 all

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Warm Up

• Are these triangle pairs congruent by SSS? Give
  congruence statement or reason why not.

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Side-Angle-Side (SAS)


B

E






F
A



C
D

1. AB ? DE
2. ?A ? ? D
3. AC ? DF

?ABC ? ? DEF
included angle
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Included Angle
The angle between two sides
? H
? G
? I
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Included Angle
Name the included angle YE and ES ES and
YS YS and YE
? E
? S
? Y
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Warning No AAA Postulate
There is no such thing as an AAA postulate!
E
B
A
C
F
D
NOT CONGRUENT
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Warning No SSA Postulate
There is no such thing as an SSA postulate!
E
B
F
A
C
D
NOT CONGRUENT
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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

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A Right ? has one Right ?.
Hypotenuse - Longest side of a Rt. ? -
Opp. of Right ?
Leg
Leg one of the sides that form the right ? of
the ?. Shorter than the hyp.
Right ?
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Th(4-6) Hyp Leg Thm. (HL)

• If the hyp. a leg of 1 right ? are ? to the hyp
  a leg of another rt. ?, then the ?s are ?.

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Group Class Work

• Do problem 18 on page 245

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Class Work

• Pg 244 1 17 all

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Warm Up

• Are these triangle pairs congruent? State
  postulate or theorem and congruence statement or
  reason why not.

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Angle-Side-Angle (ASA)


B

E






F
A



C
D

1. ?A ? ? D
2. AB ? DE
3. ?B ? ? E

?ABC ? ? DEF
included side
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Included Side
The side between two angles
GI
GH
HI
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Included Side
Name the included angle ?Y and ?E ?E and
?S ?S and ?Y
YE
ES
SY
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Angle-Angle-Side (AAS)


B

E






F
A



C
D

1. ?A ? ? D
2. ? B ? ? E
3. BC ? EF

?ABC ? ? DEF
Non-included side
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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

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Warning No SSA Postulate
There is no such thing as an SSA postulate!
E
B
F
A
C
D
NOT CONGRUENT
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Warning No AAA Postulate
There is no such thing as an AAA postulate!
E
B
A
C
F
D
NOT CONGRUENT
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Triangle Congruence (5 of them)

• SSS postulate
• ASA postulate
• SAS postulate
• AAS theorem
• HL Theorem

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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)
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Name That Postulate
(when possible)
SAS
ASA
SSA
SSS
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Name That Postulate
(when possible)
AAA
ASA
SSA
SAS
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Name That Postulate
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Reflexive Property
Vertical Angles
SSA
SAS
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Name That Postulate
(when possible)
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Name That Postulate
(when possible)
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Lets Practice
Indicate the additional information needed to
enable us to apply the specified congruence
postulate.
For ASA
?B ? ?D
For SAS
?A ? ?F
For AAS
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Review
Indicate the additional information needed to
enable us to apply the specified congruence
postulate.
For ASA
For SAS
For AAS
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Class Work

• Pg 251, 1 17 all

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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!

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Warm Up