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Chapter 4
Congruent Triangles
• Identify the
corresponding parts of
congruent figures
• Prove two triangles
are congruent
• Apply the theorems
and corollaries about
isosceles triangles
4.1 Congruent Figures
Objectives
• Identify the corresponding parts of
congruent figures
What we already know…
• Congruent Segments
– Same length
• Congruent Angles
– Same degree measure
Congruent Figures
Exactly the same size and shape. Don’t
ASSume !
C
B
A
F
D
E
Definition of Congruency
Two figures are congruent if corresponding vertices
can be matched up so that:
1. All corresponding sides are congruent
2. All corresponding angles are congruent.
What does corresponding mean
again?
• Matching
• In the same position
Volunteer
• Draw a large scalene triangle (with a ruler)
• Cut out two congruent triangles that are the
same
• Label the Vertices A, B, C and D, E, F
ABC  DEF
You can slide and rotate the triangles around so that
they MATCH up perfectly.
A
E
C
B
F
D
ABC  DEF
The order in which you name the triangles matters !
A
E
C
B
F
D
Based on the definition of
congruency….
• Three pairs of
corresponding angles
• Three pairs of
corresponding sides
1.  A   D
1. AB  DE
2.  B   E
2. BC  EF
3.  C   F
3. CA  FD
It is not practical to cut out and
move the triangles around
 ABC   XYZ
• Means that the letters X and A, which appear
first, name corresponding vertices and that
–  X   A.
• The letters Y and B come next, so
–  Y   B and
–XY  AB
CAUTION !!
• If the diagram doesn’t show the markings
or
• You don’t have a reason
– Shared sides, shared angles, vertical angles,
parallel lines
White Boards
• Suppose  TIM   BER
IM  ___
White Boards
• Suppose  TIM   BER
IM  ER , Why ?
White Boards
• Corresponding Parts of Congruent Triangles
are Congruent
White Boards
• Suppose  TIM   BER
___   R
White Boards
• Suppose  TIM   BER
 M   R, Why?
White Boards
• Corresponding Parts of Congruent Triangles
are Congruent
White Boards
• Suppose  TIM   BER
 MTI   ____
White Boards
• Suppose  TIM   BER
 MTI   RBE
White Boards
• If  ABC   XYZ
m  B = 80
m  C = 50
Name four congruent angles
White Boards
• If  ABC   XYZ
m  B = 80
m  C = 50
 A,  C ,  X,  Z
White Boards
• If  ABC   XYZ
Write six congruences that must be
correct
White Boards
• If  ABC   XYZ
1.  A   X
1. AB  XY
2.  B   Y
2. BC  YZ
3.  C   Z
3. CA  ZX
Remote time
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• An acute triangle is __________ congruent
to an obtuse triangle.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• A polygon is __________ congruent to
itself.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• A right triangle is ___________ congruent
to another right triangle.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• If  ABC   XYZ,  A is ____________
congruent to  Y.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• If  ABC   XYZ,  B is ____________
congruent to  Y.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• If  ABC   XYZ, AB is ____________
congruent to ZY.
4.2 Some Ways to Prove Triangles
Congruent
Objectives
• Learn about ways to prove triangles are
congruent
Don’t ASSume
• Triangles cannot be assumed to be
congruent because they “look” congruent.
and
• It’s not practical to cut them out and match
them up
so,
We must show 6 congruent pairs
• 3 angle pairs and
• 3 pairs of sides
WOW
• That’s a lot of work
Spaghetti Experiment
• Using a small amount of playdough as your
“points” put together a 5 inch, 3 inch and
2.5 inch piece of spaghetti to forma triangle.
• Be careful, IT’S SPAGHETTI, and it will
break.
• Compare your spaghetti triangle to your
neighbors
• Compare your spaghetti triangle to my
spaghetti triangle.
We are lucky…..
• There is a shortcut
– We don’t have to show
• ALL pairs of angles are congruent and
• ALL pairs of sides are congruent
SSS Postulate
If three sides of one triangle are congruent to
the corresponding parts of another triangle,
then the triangles are congruent.
E
B
A
C
F
D
Patty Paper Practice
5 inches
3 inches
2.5 inches
Volunteer
SAS Postulate
If two sides and the included angle are
congruent to the corresponding parts of
another triangle, then the triangles are
congruent.
E
B
C
F
D
ASA Postulate
If two angles and the included side of one
triangle are congruent to the corresponding
parts of another triangle, then the triangles
are congruent.
E
B
A
C
F
D
The order of the letters MEAN
something
• Is SAS the same as SSA or A$$ ?
Construction 2
Given an angle, construct a congruent angle.
Given: ABC
Construct: CDE  ABC
Steps:
Construction 3
Given an angle, construct the bisector of the angle
Given: ABC
Construct: bisector of ABC
Steps:
CAUTION !!
• If the diagram doesn’t show the markings
or
• You don’t have a reason
– Shared sides, shared angles, vertical angles,
parallel lines
Remote Time
Can the two triangles be proved congruent? If so,
what postulate can be used?
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
A.
B.
C.
D.
E.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
I don’t know
White Board
• Decide Whether you can deduce by the
SSS, SAS, or ASA Postulate that the two
triangles are congruent. If so, write the
congruence ( ABC  _ _ _ ). If not write
not congruent.
B
A
D
C
 DBC   ABC
SSS
A
B
C
D
No Congruence
Construction 7
Given a point outside a line, construct a line parallel
to the given line through the point.
Given: line
Construct:
Steps:
l with point A
to l through A
4.3 Using Congruent Triangles
Objectives
• Use congruent triangles to prove other
things
Our Goal
• In the last section, our goal was to prove
that two triangles are congruent.
The Reason
• If we can show two triangle are congruent,
using the SSS, SAS, ASA postulates, then
we can use the definition of Congruent
Triangles to say other parts of the triangles
are congruent.
– Corresponding Parts of Congruent Triangles are
Congruent.
This is an abbreviated way to refer to the definition of
congruency with respect to triangles.
C orresponding
P arts of
C ongruent
T riangles are
C ongruent
Basic Steps
1. Identify two triangles in which the two
segments or angles are corresponding
parts.
2. Prove that those two triangles are
congruent
3. State that the two parts are congruent
using the reason CPCTC.
Given: m  1 = m  2
m3=m4
Prove: M is the midpoint of JK
L
3 4
J
1
2
M
K
L
Given: m  1 = m  2
m3=m4
34
Prove: M is the midpoint of JK
J
LM = LM
1 2
M
Reflexive Property
m  J = m  K If 2 ’s of 1
are  to 2 ’s
of another ,
then the third
’s are .
K
L
Given: m  1 = m  2
m3=m4
34
A A
S
Prove: M is the midpoint of JK
J
LM = LM
A
1A 2 A
M
Reflexive Property
m  J = m  K If 2 ’s of 1
are  to 2 ’s
of another ,
then the third
’s are .
A
K
L
Given: m  1 = m  2
m3=m4
34
A A
S
Prove: M is the midpoint of JK
J
 JLM   KLM ASA
1A 2 A
M
K
L
Given: m  1 = m  2
m3=m4
34
Prove: M is the midpoint of JK
J
Statements
Reasons
1 2
M
K
L
Given: m  1 = m  2
m3=m4
34
Prove: M is the midpoint of JK
J
Statements
Reasons
1. m  1 = m  2
m3=m4
1. Given
1 2
M
K
L
Given: m  1 = m  2
m3=m4
34
Prove: M is the midpoint of JK
J
Statements
Reasons
1. m  1 = m  2
m3=m4
1. Given
5. M is the midpoint of JK
1 2
M
K
L
Given: m  1 = m  2
m3=m4
34
Prove: M is the midpoint of JK
J
Statements
Reasons
1. m  1 = m  2
m3=m4
1. Given
4. JM = KM
5. M is the midpoint of JK
1 2
M
K
L
Given: m  1 = m  2
m3=m4
34
Prove: M is the midpoint of JK
J
Statements
Reasons
1. m  1 = m  2
m3=m4
1. Given
3.  JLM  KLM
4. JM = KM
5. M is the midpoint of JK
4. CPCTC
1 2
M
K
L
Given: m  1 = m  2
m3=m4
34
Prove: M is the midpoint of JK
J
1 2
M
Statements
Reasons
1. m  1 = m  2
m3=m4
2. LM = LM
1. Given
3.  JLM  KLM
3. ASA Postulate
4. JM = KM
4. CPCTC
5. M is the midpoint of JK
2. Reflexive Property
K
L
Given: m  1 = m  2
m3=m4
34
Prove: M is the midpoint of JK
J
1 2
M
Statements
Reasons
1. m  1 = m  2
m3=m4
2. LM = LM
1. Given
3.  JLM  KLM
3. ASA Postulate
4. JM = KM
4. CPCTC
2. Reflexive Property
5. M is the midpoint of JK 5. Definition of midpoint
K
L
Given: m  1 = m  2
m3=m4
34
A A
Prove: M is the midpoint of JK
S
J
A
A
1 2
M
Statements
Reasons
1. m  1 = m  2 A
m3=m4 A
2. LM = LM S
1. Given
3.  JLM  KLM
3. A S A Postulate
4. JM = KM
4. CPCTC
2. Reflexive Property
5. M is the midpoint of JK 5. Definition of midpoint
K
M
Given: MK OK;
KJ bisects  MKO;
Prove: JK bisects  MJO
J
1
2
S
S
O
A 3
A 4
S
34
Definition of  bisector
JK  JK
Reflexive Property
 MKJ  OKJ SAS Postulate
K
M
Given: MK OK;
KJ bisects  MKO;
Prove: JK bisects  MJO
J
1
2
S
S
O
Statements
Reasons
A 3
A 4
S
K
M
Given: MK OK;
KJ bisects  MKO;
Prove: JK bisects  MJO
J
1
2
S
S
O
A 3
A 4
S
Statements
Reasons
1. MK OK;
KJ bisects  MKO
2.  3   4
1. Given
3. JK  JK
3. Reflexive Property
2. Def of  bisector
K
M
Given: MK OK;
KJ bisects  MKO;
Prove: JK bisects  MJO
1
2
J
S
S
O
A 3
A 4
S
Statements
Reasons
1. MK OK;
KJ bisects  MKO
2.  3   4
1. Given
3. JK  JK
3. Reflexive Property
6. JK bisects  MJO
6.
2. Def of  bisector
K
M
Given: MK OK;
KJ bisects  MKO;
Prove: JK bisects  MJO
1
2
J
S
S
O
A 3
A 4
S
Statements
Reasons
1. MK OK;
KJ bisects  MKO
2.  3   4
1. Given
3. JK  JK
3. Reflexive Property
5.  1   2
5. CPCTC
6. JK bisects  MJO
6.
2. Def of  bisector
K
M
Given: MK OK;
KJ bisects  MKO;
Prove: JK bisects  MJO
1
2
J
S
S
O
A 3
A 4
S
Statements
Reasons
1. MK OK;
KJ bisects  MKO
2.  3   4
1. Given
3. JK  JK
3. Reflexive Property
4.  MKJ  OKJ
4. SAS Postulate
5.  1   2
5. CPCTC
6. JK bisects  MJO
6.
2. Def of  bisector
K
M
Given: MK OK;
KJ bisects  MKO;
Prove: JK bisects  MJO
J
1
2
S
S
O
A 3
A 4
S
Statements
Reasons
1. MK OK; S
KJ bisects  MKO
2.  3   4 A
1. Given
3. JK  JK S
3. Reflexive Property
4.  MKJ  OKJ
4. SAS Postulate
5.  1   2
5. CPCTC
6. JK bisects  MJO
6. Def of  bisector
2. Def of  bisector
K
4.4 The Isosceles Triangle Theorem
Objectives
• Apply the theorems and corollaries about
isosceles triangles
Isosceles Triangle
By definition, it is a triangle with two
congruent sides called legs.
X
Legs
Vertex Angle
Base Angles
Y
Z
Base
Experiment - Goal
• Discover Properties of an Isosceles Triangle
Supplies
•
•
•
•
Blank sheet of paper
Ruler
Pencil
Scissors
Procedure
1. Fold a sheet of paper in half.
Procedure
2. Draw a line with the ruler going from the
folded edge (very important) to the corner
of the non folded edge.
Folded edge
Procedure
3. Cut on the red line
Cut here
Procedure
4. Open and lay flat. You will have a triangle
Procedure
5. Label the triangle
P
S
R
Q
Procedure
6. Since  PRQ fits exactly over  PSQ
(because that’s the way we cut it),
P
 PRQ   PSQ
S
R
Q
Procedure
7. What conclusions can you make? Be
careful not to ASSume anything.
P
S
R
Q
Conclusions
1.  PRS   PSR
2. PQ bisects  RPS
P
3. PQ bisects RS
4. PQ  RS at Q
5. PR  PS
S
R
Q
These conclusions are actually
• Theorems and corollaries
Theorem
The base angles of an isosceles triangle are
congruent.
B
A
C
Corollary
•
An equilateral triangle is also equiangular.
Corollary
•
An equilateral triangle has angles that
measure 60.
Corollary
•
The bisector of the vertex angle of an isosceles
triangle is the perpendicular bisector of the base.
Theorem
If two angles of a triangle are congruent, then
it is isosceles.
B
A
C
Corollary
• An equiangular triangle is also equilateral.
White Board Practice
• Find the value of x
30º
xº
x = 75º
White Board Practice
• Find the value of x
2x - 4
2x + 2
x+5
x=9
White Board Practice
• Find the value of x
41
42
56 º
62 º
x
x = 42
4.5 Other Methods of Proving
Triangles Congruent
Objectives
• Learn two new ways to prove triangles are
congruent
Proving Triangles 
We can already prove triangles are congruent
by the ASA, SSS and SAS. There are two
other ways to prove them congruent…
AAS Theorem
If two angles and a non-included side of one
triangle are congruent to the
corresponding parts of another triangle,
then the triangles are congruent.
E
B
A
C
F
D
The Right Triangle
B
leg
A
acute angles
right angle
leg
C
HL Theorem
If the hypotenuse and leg of one right
triangle are congruent to the
corresponding parts of another right
triangle, then the triangles are congruent.
B
A
E
C
F
D
Five Ways to Prove  ’s
All Triangles:
ASA
SSS
SAS
AAS
Right Triangles Only:
HL
White Board Practice
• State which of the congruence methods can
be used to prove the triangles congruent.
You may choose more than one answer.
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
4.6 Using More than One Pair of
Congruent Triangles
Objectives
• Construct a proof using more than one pair
of congruent triangles.
• Sometimes two triangles that you want to
prove congruent have common parts with
two other triangles that you can easily prove
congruent.
More Than One Pair of ’s 
Given: X is the midpt of AF & CD
Prove: X is the midpt of BE
D
A
X
E
B
F
C
Lecture 7 (4-7)
Objectives
• Define altitudes,
medians and
perpendicular
bisectors.
Median of a Triangle
A segment connecting a vertex to the
midpoint of the opposite side.
midpoint
vertex
Median of a Triangle
Each triangle has three Medians
vertex
midpoint
Median of a Triangle
Each triangle has three Medians
vertex
midpoint
Median of a Triangle
• Notice that the three medians will meet
at one point.
If they do not meet,
then you are not
drawing the
segments well.
Altitude of a Triangle
A segment drawn from a vertex
perpendicular to the opposite side.
vertex
perpendicular
Altitude of a Triangle
Each Triangle has three altitudes
perpendicular
vertex
Altitude of a Triangle
Each triangle has three altitudes
perpendicular
vertex
Altitude of a Triangle
Notice that the three altitudes will meet
at one point.
If they do not meet,
then you are not
drawing the
segments well.
Special Cases - Altitudes
Obtuse Triangles: Two of the altitudes
are drawn outside the triangle. Extend
the sides of the triangle
Special Cases - Altitudes
Right Triangles: Two of the altitudes are
already drawn for you.
Perpendicular Bisector
A segment (line or ray) that is
perpendicular to and passes through
the midpoint of another segment.
Must put the
perpendicular
and congruent
markings !
Angle Bisector
A ray that cuts an angle into two
congruent angles.
Theorem
If a point lies on the perpendicular
bisector of a segment of a segment,
then the point is equidistant from the
endpoints.
Theorem
If a point is equidistant from the
endpoints of a segment, then the point
lies on the perpendicular bisector of the
segment.
Remember
• When you measure distance from a point to
a line, you have to make a perpendicular
line.
Theorem
If a point lies on the bisector of an angle
then the point is equidistant from the
sides of the angle.
Construction 10
Given a triangle, circumscribe a circle about the
triangle.
Given: ABC
Construct:
R
Steps:
circumscribed about ABC
Construction 11
Given a triangle, inscribe a circle within the triangle.
Given: ABC
Construct:
Steps:
R inscribed within ABC
Remote Time
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• An altitude is _____________
perpendicular to the opposite side.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• A median is ___________ perpendicular to
the opposite side.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• An altitude is ______________ a
perpendicular bisector.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• An angle bisector is _______________
perpendicular to the opposite side.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• A perpendicular bisector of a segment is
___________ equidistant from the
endpoints of the segment.