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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
– AB  CD
– AB = 4 , CD = 4
• Congruent Angles
– Same degree measure
– ABC   EFG
– mABC = 48 ◦ , m EFG = 48◦
Congruent Figures
Exactly the same size and shape. Don’t
ASSume !
C
B
A
F
D
E
Definition of Congruency
Two polygons 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
Definition of Congruent Triangles
ABC  DEF
You can slide and rotate the triangles around so that
they MATCH up perfectly.
A
E
C
B
F
The order in which you name
the triangles matters !
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
There are 6 pieces of information that we need to have in order
to prove that two triangles are congruent!!
 ABC   XYZ
• Based off this information with or without a
diagram, we can conclude…
• Letters X and A, appear first, naming
corresponding vertices, which means…
–  X   A.
• The letters Y and B come next, so
–  Y   B and
–XY  AB
HINT: If there’s no drawing,
create your own!!!
CAUTION !!
• If the diagram doesn’t show the markings
A
D
C
B
F
F
• Use the information given to you…
– 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 (aka the definition of congruent triangles)
• CPCTC - this is the abbreviated way to say
the statement above
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
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 right triangle is ___________ congruent
to another right triangle.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• If  ABC   XYZ,  B is ____________
congruent to  Y.
4.2 Warm-up (on 4.1 day)
[Use pg. 122 to help fill in the blanks]
FILL IN THE BLANKS….
• AB is opposite _____
• AB is included between L___ and L____
B
• LA is opposite ________
• LA is included between
____ and ____
A
C
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
• WHAT ARE THOSE
6 PAIRS?
–3 angle pairs and
–3 pairs of sides
WOW
• That’s a lot of work
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
• It’s like a lawyer not needing as much
evidence to get a criminal convicted…
Experiment
• Use the 3 connector pieces to create a
triangle
• Compare your groups triangle to your
neighbors
SSS Postulate
Each side matches congruent with the 3
sides of another triangle
E
B
A
C
F
D
**ORDER MATTERS!!!
SAS Postulate
• Two sides match up congruent and…
• The angles between the 2 sides are
congruent.
E
B
A
C
F
D
ASA Postulate
• Two angles match up congruent and…
• The side in between those angles
E
B
A
C
F
D
The order of the letters MEAN
something
• Is SAS the same as SSA or A$$ ? NO!!!!
• SAS – TWO SIDES WITH THE ANGLE
THAT IS IN BETWEEN (INCLUDED  )
• ASA – TWO ANGLES WITH THE SIDE
THAT IS BETWEEN THEM
CAUTION !!
• If the diagram doesn’t show the markings
or
• You don’t have a reason
– Shared sides, shared angles, vertical angles,
parallel lines
A.
B.
C.
D.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
A.
B.
C.
D.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
A.
B.
C.
D.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
A.
B.
C.
D.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
A.
B.
C.
D.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
A.
B.
C.
D.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
A.
B.
C.
D.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
A.
B.
C.
D.
SSS Postulate
SAS Postulate
ASA Postulate
Cannot be proved congruent
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.
 ABC   EDC
SAS
D
A
C
B
E
 CDA   ABC
ASA
A
B
C
D
4.3 Using Congruent Triangles
Objectives
• Use congruent triangles to prove other
things
Pg. 124
• Copy down problem #10
• Talk through with your partner how you would prove the
triangles are congruent
• REMEMEBER… TO SAY THE
TRIANGLES ARE CONGRUENT
MEANS YOU HAVE THE EVIDENCE
FOR 1 OF OUR SHORTCUTS (SSS, ASA,
SAS)
• Write an explanation in paragraph form
Pg. 125
• Copy down problem # 16 in your notes
– Copy down everything!
– Including the diagram
– Complete the proof in your group
Once the triangles are
congruent….
• 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. (CPCTC)
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
Used as a reason in a proof
to say that the rest of the pieces of the
triangles not mentioned are congruent
too. (GETTING A CONVICTION)
CPCTC cannot be used in a proof
until after the triangle is proven
congruent.
STEPS TO FOLLOW…
1. Identify two triangles in which the two
segments or angles are corresponding
parts. (criminals)
2. Prove that those two triangles are
congruent (evidence)
3. State that the two parts are congruent
using the reason CPCTC. ( You convict the
other corresponding pieces as being congruent)
Given: m  1 = m  2
m3=m4
Prove: JM = MK
J
L
34
1 2
M
• Mark the diagram with the given info
• Check for free visual evidence..
• Remember that you must prove the triangles congruent
first, before using CPCTC.
• TALK IT OUT IN YOUR GROUPS WRITE AN
EXPLANATION
K
Given: m  1 = m  2
m3=m4
Prove: JM = MK
J
L
34
1 2
M
K
Chapter 4 Quiz
• Drawing two congruent triangles
– What can you conclude?
– Name the corresponding sides and angles
• Given a diagram…
– Are the two triangles congruent?
– Name the two congruent triangles and the
postulate that proves it (ORDER MATTERS)
• Complete a proof
– Using SSS, SAS, or ASA
– Using CPCTC
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
Does the base
always have to be
at the bottom?
Base Angles
Y
Z
Base
Experiment - Goal
• Discover Properties of an Isosceles Triangle
Procedure
5. Label the triangle
P
Do we have an
isosceles triangle?
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?
–
–
–
Use the 2 smaller congruent
triangles (cpctc)
Every conclusion must be
justified
Be careful not to ASSume
anything.
P
S
R
Q
Conclusions
1. PQ bisects  RPS
P
2. PQ bisects RS
3. PQ  RS at Q
S
R
Q
These conclusions are actually
• Theorems and corollaries
Theorem
If two on
sides
a triangle
arecan
congruent,
then
Based
theof
diagram,
what
we conclude
opposite
those sides?
sides are
ifthe
weangles
have two
congruent
congruent.
B
Always draw the
arrows to show where
the opposite angle is.
A
C
Corollary
equiangular.
1. An equilateral triangle is also __________
2. equilateral triangle = equiangular triangle
60.
3. An equilateral triangle has angles that measure ____
Corollary
If one of the following
occurs, then they all do..
1. Segment bisects vertex
angle
2. Segment bisects base
(segment)
3. Segment is perp. to
base
R
P
Segment coming
from vertex to the
base
S
Q
Theorem
If
two angles
of a triangle
arecan
congruent,
then the
Based
on the diagram,
what
we conclude?
sides
opposite
those
angles
are
congruent.
THINK: Converse of previous theorem
B
A
C
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
X = 42
41
42
56 º
62 º
x
4.5 Other Methods of Proving
Triangles Congruent
Objectives
• Learn two new ways to prove triangles are
congruent
WARM –UP
• CREATE A SEPARATE DRAWING REPRESENTING
EACH OF THE FOLLOWING…
1. SSS congruency
2. SAS congruency
3. ASA congruency
•
Your drawings should use
–
–
Shared sides, vertical angles, parallel lines
Congruency markings and/or written in measurements
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
• Two angles match up congruent and…
• A side not between the angles is congruent.
E
B
A
C
F
How can we
apply one of
the previous
postulates to
help us prove
this theorem?
D
The Right Triangle
Can you label the types of angles this
triangle has?
B
leg
A
What are the specific names for the sides of
a right triangle?
acute angles
right angle
leg
C
HL Theorem
• hypotenuse is congruent in each triangle
• Either leg is congruent
B
E
What is another way I could
have illustrated this theorem?
A
C
F
D
Five Ways to Prove  ’s
All Triangles:
ASA
ORDER MATTERS!!!
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
Section 4-7
Objectives
• Define altitudes,
medians and
perpendicular
bisectors.
Definition: Median of a Triangle
A segment connecting a vertex to the
midpoint of the opposite side.
midpoint
Determine the
definition by looking
at the diagram
vertex
What indicates that the
segment is intersecting the
side at it’s midpoint?
Median of a Triangle
Each triangle has three Medians
vertex
midpoint
Altitude of a Triangle
A segment drawn from a vertex
perpendicular to the opposite side.
vertex
Determine the
definition by
looking at the
diagram
perpendicular
Does the segment
intersect at the midpoint?
Altitude of a Triangle
Each triangle has three altitudes
perpendicular
vertex
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.
**YOU WILL SEE
THIS ON THE
TEST!!
Altitudes
• Acute – all 3 are inside the triangle
• Obtuse – 1 inside – 2 outside
• Right – 1 inside – other 2 are the legs of the
triangle
Perpendicular Bisector
• A segment (line or ray) that is…
– perpendicular to a segment and..
– passes through the midpoint of segment.
Must put the
perpendicular
and congruent
markings !
Perpendicular Bisector in
depth…
• Draw a point on the red line…what can you
say about that point in relation to A and B?
• Any point on the perp. bisector is
equidistant from the endpoints.
A
B
Angle Bisector
A ray that cuts an angle into two
congruent angles.
A
B
C
Remember
• When you measure distance from a point to
a line, you have to make a perpendicular
line.
A
Theorem
Pick a point on the angle bisector… what can
you say about that point using the word
equidistant?
A
B
C
Group Work
Name the following..
1. Median
2. Altitude
3. Angle Bisector
C
D
E
B
F
A
Ch. 4 test
· Take a written statement about 2 congruent
triangles, and deduce information.
o
I.e  XRT congruent BAG
Important definitions: Understanding wording and
based on a diagram
o
Median
o
o
Altitude
Angle bisector
Perpendicular bisector
·Given a diagram…
o Are the two triangles congruent?
o
Name the two triangles and the postulate that
shows they are congruent
o ORDER MATTERS

·
·
Free info
Shared angles / sides
Parallel lines
vert angles
· Solving for x
o Know rules for isosceles and equilateral
triangles
o Set up problem with general rule
o
Then plug in values and solve for x
·
Drawing the following…
o
Perp. Bisect
o
o
Medians
altitudes