4.1 Classifying Triangles

1
4.1 Classifying Triangles
2
Classifying Triangles

• By Sides
• Scalene ?
• A triangle with no two sides congruent.
• Isosceles ?
• A triangle with two or mores sides congruent.
• Equilateral ?
• A triangle with all sides congruent.

• By Angles
• Acute ?
• A triangle with all acute angles.
• Obtuse ?
• A triangle with one obtuse angle.
• Right ?
• A triangle with one right angle.
• Equiangular?
• A triangle with all angles congruent.

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Combinations of Classifications
Acute ?
Scalene ?
Obtuse ?
Isosceles ?
Right ?
Equilateral ?
Equiangular ?
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What Can You Determine?

• Distance Formula?
• Lengths of segments, thus you can determine if
  segments are congruent or not.
• Helps with classification by sides?
• Can it help with classification by angles?
• Pythagorean Theorem
• If leg2 leg2 Hypt2, then it is a Rt ?
• What can slopes help you with?
• Slopes can help determine if a Rt ?

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4.2 Angles of Triangles
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Angle Sum Theorem

• Angle Sum Theorem The sum of the measures of
  the angles of a triangle equals 180.

Three adjacent angles the sum of their
measurements is 180
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Third Angle Theorem

• The Third Angle Theorem (No Choice Theorem)
  states if two angles in one triangle are
  congruent to two angles in another triangle, then
  the third set of angles are also congruent.

90
90
30
60
30
60
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Exterior Angles

• Exterior Angle An angle made between one side
  of a triangle and the extension of the other side.

5
6
1
3
2
4

• There six exterior angles in a triangle, two per
  vertex.

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Exterior Angle Theorem

• Exterior Angle Theorem The measure of the
  exterior angle of a triangle is equal to the sum
  of the two remote interior angles.

C
Triangle Sum Theorem mlt2 mltA mltC 180
lt1 and lt2 are LP, therefore Supplementary mlt2
mlt1 180
1
2
A
B
Substitution mlt2 mlt1 mlt2 mltA mltC
Add/Subt mlt1 mltA mltC
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Triangle Corollaries

• Corollaries are just like theorems but are easily
  proved.
• Two Triangle Corollaries are
• The acute angles of a right triangle are
  complementary.
• There can be at most one right or obtuse angle in
  a triangle (Euclidian Geometry)

A
mltA mltB mltC 180 ?Sum Thrm
mltA 90 mltC 180 Subst.
mltA mltC 90 Add/Sub
B
C
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Flow Proofs

• Flow proofs are the 2nd and last formal proof
  that we will study.
• Similarities with the two column proof is that
  each element has a statement and reason.
• Two column proofs work well for linear type
  proofs in other words, one step follows
  another, etc.
• Flow proofs work better for non-linear proofs
  in other words, the order is less defined.

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Old Proof
A B C D
AB CD Given
BC BC Reflexive
Given AB CD Prove AC BD
AB BC AC BC CD BD SAP
AB BC BC CD Add/Subt
AC BDSubstitution
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4.3 Congruent Triangles
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Congruent Triangles

• Definition of Congruent Segments Are Segments
  that have the same measurement.
• There is only one measurement for a segment.
• How many parts are in a triangle?
• Six? Three segments and three angles.
• So, Congruent Triangles are triangles where all
  SIX corresponding parts are congruent.
• CPCTC Corresponding Parts of Congruent
  Triangles are Congruent.

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Congruent Triangles
A
T
C
O
D
G
? congruency statement
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Important Concepts

• Unlike the order of the letters of an angle, the
  order of the letters of the triangles matters.
• ltABC is congruent to ltCBA
• ?ABC may or may not be congruent to ?CBA b/c ltA
  may not be congruent to ltC etc.
• When you write a triangle congruency statement
  make sure the corresponding parts in fact are
  congruent.

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Properties of Triangle Congruence

• Reflexive
• ?ABC is congruent to ?ABC
• Symmetric
• If ?ABC is congruent to ?XYZ, then ?XYZ is
  congruent to ?ABC.
• Transitive
• If ?ABC is congruent to ?XYZ, and ?XYZ is
  congruent to ?LMN, then ?ABC is congruent to
  ?LMN.

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4.4 Proving Congruence SSS and SAS
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Shortcuts

• Previously in order to prove triangles were
  congruent to each other you needed to prove all
  three sets of angles and all three sets of sides
  were congruent.
• There are 8 shortcuts that can be used to prove
  triangles congruent.
• Today were going to use two of them
• SSS (Side Side Side)
• SAS (Side Angle Side)

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SSS

• SSS If the three sides of one triangle are
  congruent to the three sides of another triangle,
  then the triangles are congruent.

N
T
L
M
R
S
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SAS

• SAS If two sides and one included angle of one
  triangle are congruent to the two sides and one
  included angle of the other triangle, then the
  triangles are congruent.

N
T
L
M
R
S
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Important Concepts

• In the order of the proof you must have three
  sets of congruent marks (for sides and/or angles)
  BEFORE you can say that the triangles are
  congruent.
• Once you say that the triangles are congruent,
  then you can say that any other part of the
  triangle can be congruent by CPCTC.

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Example
N
T
L
M
R
S
Given
SSS
CPCTC
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Flow Proof

• Hints Make each piece of the given its own
  line down.
• Make the stuff you can get from the pictures
  (Vert Angles, LP, SAP, AAP, etc..) their own line
  down too.
• See example on next slide.

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Example
D
Given
Prove
A
B
C
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4.5 Proving Triangles Congruent by ASA and AAS.
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ASA and AAS

• There are two more ways to prove triangles are
  congruent.
• ASA (Angle Side Angle) If two angles and
  the included side of one triangle are congruent
  to two angles and an included side of another
  triangle, then the triangles are congruent.
• AAS (Angle Angle Side) If two angles and a
  non included side of one triangle are congruent
  to two angles and a non included side in another
  triangle, then the triangles are congruent.

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Example ASA
Z
X
Y
Here we have two sets of congruent angles that
are congruent along with the included sides that
are congruent, therefore the two triangles are
congruent by ASA.
N
L
M
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Example AAS
Z
X
Y
Here we have two sets of congruent angles that
are congruent along with the non included sides
that are congruent, therefore the two triangles
are congruent by AAS.
N
L
M
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Important Reminders

• If youre trying to prove triangles congruent,
  you MUST have three sets of corresponding parts
  that are congruent BEFORE you can say that the
  triangles are congruent. (SSS, SAS, ASA and AAS)
• If you dont have three sets of parts that are
  congruent, you cant prove the triangles
  congruent.
• After you prove the triangles congruent, you can
  use CPCTC to prove any of the unused parts
  congruent.

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General Flow Proof
?
?
?
ASA
CPCTC
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Proving Right Triangles Congruent
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Four Additional Ways

• I told you there were 8 short cuts to proving
  triangles congruent.
• Four ways that work for all triangles are SSS,
  SAS, ASA and AAS.
• The other four ways work for Right Triangles
  only.
• They are HA, LL, LA, and HL.
• S was for sides, and A was for angles.
• H is for Hypotenuse, L is for Leg and A is for
  ACUTE angle.

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Process

• Notice that these four ways, HA, LL, LA, and HL
  only have two letters.
• That means you only need two sets of congruent
  marks to prove Right Triangles congruent.
• However, you need to tell me that theyre right
  triangles too.
• So, you still need three things. Two sets of
  congruent marks on Right Triangles.

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Examples
HA Hypotenuse Acute Angle
LL Leg Leg
LA - Leg Acute Angle
HL Hypotenuse Leg
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Similarities

• You will notice that SAS looks like LL if the
  sides are the legs.
• ASA looks like LA.
• AAS can look like HA or LA
• HL is the only Right Triangle Congruency Theorem
  that can not have a similar all triangle way to
  prove the triangles are congruent.

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Right Triangle Flow Proof
?
?
?
(HA, LL, LA HL)
CPCTC
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4.6 Isosceles Triangles
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Parts of Isosceles Triangles

• Def A triangle with two or more sides
  congruent.
• The parts have special names.

C
The Congruent Sides are called the Legs
The included angle made by the legs is the
Vertex Angle
The angles opposite the legsare called the base
angles
A
B
The side opposite the vertex is the Base
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Parts of Isosceles Triangles

• The key thing to remember is this
• It doesnt matter which way the triangle is
  oriented, the parts are all in relationship to
  the congruent sides.
• The Base is not on the bottom!
• The Vertex is not on the top!

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Isosceles Triangle Theorem
B
?ABC is Isosceles, with ltB as the Vertex.
Legs AB and BC are Congruent.
Draw Auxiliary Line from B to D (D is MP of
Segment AC.
Segment AD is Congruent to DC (MP Thrm)
C
A
D
Segment BD is Congruent to itself (Ref)
If Two sides of a triangle are congruent,then
the angles opposite those sides are congruent
?ABD is Congruent to ?CBD (SSS)
ltA is Congruent to ltC (CPCTC)
then ?
IF ?
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Converse of Isosceles ? Thrm

• The Converse of the Isosceles Triangle Theorem is
  also true.
• If two angles of a triangle are congruent, then
  the sides opposite them are congruent.

If..
then
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Triangle Corollaries

• A triangle is equilateral if and only if it is
  equiangular.
• Each angle of an equilateral triangle measures 60

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Equilateral Triangles

• Since an Equilateral Triangles are also
  Isosceles, each of the vertices of the triangle
  are Vertex angles.
• Each side is a Leg and a Base.
• All the properties of Isosceles Triangles exist
  for Equilateral Triangles as well.