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4.0 Students prove basic theorems
involving congruence and similarity.
5.0 Students prove that triangles are
congruent or similar, and they are able
to use the concept of corresponding
parts of congruent triangles.
Agenda for CS 4.0 & 5.0
Although these standards cover
both congruency and similarity,
this presentation will focus on
the congruency aspect and
proving triangles are congruent
of CS 4.0 & CS5.0
The concept of Cloning is helpful
in learning Geo Standards 4 & 5.
Cloning is about making exact copies
Geometry uses the concept of
congruency (exactly the same size)
Both Cloning & Geometry focus on
Corresponding parts (organs) and their
DNA
What we know about CLONES…
• Clones have the
same…DNA
• Clones have the
same physical
characteristics(parts)
• Corresponding Parts
of Clones are the
same (aka Congruent)
Therefore, we proved that these
What
we
know
about
triangles must be congruent
Congruent
Triangles…
because all their
corresponding parts
congruent. (aka CLONED TRIANGLES)
• Congruent
triangles
Can
youare
see
Are
these
two
have
the same
DNA…
Like
Clones,
these
the
connection
(Dimension,
Numbers, Angles)
triangles
triangles
have
between
congruent?
• Congruent Triangles
corresponding
have the
same (parts)
Clones
and
parts
that
are
physical
characteristics
Do
these
Congruent
exactly
thehave
same s3
triangles
Triangles?
•
Corresponding
Parts
size,
thetherefore
same DNA?
of Congruent
they
must be
(Explain…)
Triangles are
Congruent
Congruent (aka CPCTC)
Triangles.
s1
s1
s3
s2
s2
Proving Clones using DNA…
Evidence is analyzed
to compare specific
strands of the DNA
to determine…
1) Inclusion
2) Exclusion
3) Partial Inclusion*
We will use the partial inclusion to
*(new
of tracking
discussmethod
similar triangles
down criminals)
Proving Congruent Triangles…
Euclidean Geometry
provides five
inclusive theorems
and postulates for
students to prove
triangles are
congruent.
I will refer to these as
our DNA Strands
used for matching.
DNA Strand: Hypotenuse-Leg
Using the marking
Only
used
in testing
on the
two
two Rightwe
Triangles
triangles,
for congruency
know…
• The hypotenuse
Hyp
 Hyp
ZX
fromAB
both
triangles
need to be
Leg
CB  Leg YX
congruent.
• Hence,
One pair of legs
need to be
ΔABC
 ΔZXY by
congruent
HL Theorem
DNA Strand: Side-Side-Side
Side(AB)
Side(XY)
• To use this
DNA
Strand,Side(YZ)
you need to
Side(BC)
know the lengths of
Side(CA)
Side(ZX)
all six sides
in the
two triangles
Hence,
• Hence, SSS really
ΔABC
 ΔXYZ
bySS
means
SS, SS,
Side-Side-Side
congruency
DNA Strand: Side-Angle-Side
Hence,
the
The key to
Theorem is
triangles
are angle.
the included
congruent
These are the included Angles,
The
included
because
of angle is
because they are formed by
created
the
sides.
by the pairs
S-A-S
of congruent sides
angle
side
Side
YouSide
have to follow
the physical pattern
of the DNA Strand
angle
side
DNA Strand: Angle-Side-Angle
• A  D
• AC  DF
• C  F
A—A
S—S
A—A
Follows the physical “bonds”
Visualize the
bonds
DNA Strand: Angle-Angle-Side
Use your visual skills to
get
a mental
A
 Y
A—A
ANGLE
picture
what a
C
 Z ofA—A
non-included
AB
 YX
S—S side
means in this
Angle
Theorem…the side
Hence,
Non-Included Side
is notbetween
ΔACB
ΔYZX by the
angles as it-Side
is in ASA.
Angle-Angle
A-A-S
Look at the Physical
Bonds of the DNA.
Once again, look at the physical bonds
required …
Show-Me-That-You-Know
Are these two triangles congruent?
If yes, then explain why…
Yes, Side-Angle-Side Theorem
SMTYK
A  D A—A
1. Why are these two triangles
B  E A—A
congruent?
AC  DF S—S
2. Write an informal proof listing the
Hence,
“DNA
Strand” you used to prove
the triangles
congruent.
ΔABC are
 ΔDEF
by
Angle-Angle-Side
SMTYK
What Theorem “DNA Strand”
is displayed?
SMTYK
If you look at the physical
bonds, then you will notice
Angle-Side-Side…(A-S-S)no bueno
•* * WE CANNOT PROVE THEY ARE CONGRUENT * *
SMTYK
SMTYK
50
Sum of the interior
angles of a triangle
All we can
always equals 180
prove
is that the
Although
AF and thattriangles
is not might
be congruent,
enough information
to
prove a match we cannot
prove it.
To solve some of the triangle
congruency questions you might
Sometimes
things
need to “operate”
and are
pull the
conjoined
triangles
apart…
stuck together
The
operation
was
rated
“R”here
so itto
Meet
little
Joe
and
Jae,
we
are
Dr.was
Blackenstein
is
needed
in
the
removed
from the presentation.
see if they
are congruent…knife
please!
Operating
However,
the math is Room
rated “PG13”.
O
J
E
A
Before
Whatthe
other
operation,
piece ofJOE
information
and JAE do
shared
we a
common
need
side
to know
{JE}.
Therefore,
to prove these
we know
triangles
that are
Side JE 
Given:
OJEAJE
congruent
JE becauseby
of“DNA
the Reflexive
Strand” P.O.C.
A-A-S?
The Operation was a success
O
E
J
J
A
E
Hints when dealing with
conjoined triangle…
• Reflexive Property XYXY, or A A
• Vertical Angles are Congruent
• Parallel lines create Congruent Angles
• Perpendicular lines create Right Angles
• Midpoints of segments are bisectors
• When in doubt, separate the triangles
Where are the conjoined
triangles…A, B or C ?
If you failed to realize it
was B, then please visit
the nurse at lunch time.
Separating Conjoined Triangles
A
S
A
Remember that A-S-A really means
(AA)-(SS)-(AA)
We are matching pairs of angles and sides
More Conjoined Δ’s
Why are the
Since the arrows
that these
The indicates
hearttriangles
angles
are lines
are congruent?
parallel,
these
congruent
because
happy face angles are
they are Vertical
congruent because of
The
DNA
indicates….A-S-A
Angles
Alternate Interior Angle
Theorem (AIA )
Bisectors & Midpoints
R
I
Given: Point A is the midpoint
of segment BI and segment RN
A
Prove: ΔARBΔANI
B
Sides
N
RANA (definition of midpoint—bisect & create  segments)
Hint-1:
Midpoints
create
 segments
Angles
RAB
 NAI (Vertical
Angles)
Sides
BAIA
(def. of midpoint)
Hint-2:
Vertical
Angles
Using CPCTC to prove 
R
I
Given: Point A is the midpoint
of segment BI and segment RN
A
Prove: B  N
B
N
This true because of CPCTC
HINT: We first have to show that the two
RANA (definition of midpoint—bisect & create  segments)
conjoined triangles are congruent, then
Angles RAB  NAI (Vertical Angles)
we know that (like CLONES) the
Sides
Sides
BAIA (def. of midpoint)
corresponding parts of Congruent
triangles must be congruent.
Prove: ΔARBΔANI
Conjoined Δ’s in the
Coordinate Plane
CACA because of Reflex. P.O.C.
(20-4) = 16 side
Yes, there are
Can we
congruent
prove
that
triangles
there
are
congruent of
because
triangles
SSS in
the diagram?
(10-5) = 5
(10-5) = 5
(20-4) = 16 side
Summary and Recap
•
•
•
•
•
•
•
•
•
Hypotenuse-Leg HL
Side-Side-Side
SSS
Side-Angle-Side SAS
Angle-Side-Angle ASA
Angle-Angle-Side AAS
CPCTC (corresponding parts)
Conjoined Δ (split)
Reflex, Vertical ’s, bisectors
// &  Lines, Midpoints
QUIZ: Find & Explain  Δ’s below
ΔDEF  ___
Quiz
Integrity is establishing congruency
with your values and behavior…
Thank your for showing your
integrity by allowing me to
present this information to
you.
Sincerely,
Mr. Blackwood (aka Dr. Blackenstein)