Download Chapter 4.1

DATE:________
Chapter
Goal
Chapter 4: Congruent Triangles
 To determine if two triangles are congruent using SSS,
SAS, ASA, AAS, HL
 To use CPCT (Corresponding Parts of Congruent
Triangles)
 To learn how to write Geometric Proofs
Goal
Congruent
Triangles
To list the corresponding congruent parts of
triangles when they are congruent.
Two polygons are Congruent if they are:
1. exactly the same size
2. exactly the same shape
ex. congruent
polygons
ex.
not congruent
polygons
Corresponding
Congruent When 2 polygons are congruent, then you can list their
Parts Corresponding Congruent Parts
 Corresponding: parts that share the same
position
 Congruent: same length (segment) or same
degree (angle)
List the
B
corresponding
53°
congruent
3 in.
parts of the
congruent
A
polygons
E
5 in.
37°
4 in.
List the corresponding
congruent sides
Congruence
Statement
3 in.
53°
C
D
5 in.
37°
4 in.
F
List the corresponding
congruent angles
1.
1.
2.
2.
3.
3.
Geometry Chapter 4.1
Name__________________
Congruent Triangles
Given each pair of congruent triangles, draw the 2 triangles separately and re-label all
congruencies. Then write a Congruence Statement and list their corresponding sides and angles.
1.
Draw the 2 triangles separately and relabel all congruencies.
Congruence
Statement
Corresponding
Sides
Corresponding
Angles
2.
Draw the 2 triangles separately and relabel all congruencies.
Congruence
Statement
Corresponding
Sides
Corresponding
Angles
3.
Draw the 2 triangles separately and relabel all congruencies.
Congruence
Statement
Corresponding
Sides
Corresponding
Angles
4.
Draw the 2 triangles separately and relabel all congruencies.
Congruence
Statement
Corresponding
Sides
Corresponding
Angles
5.
Draw the 2 triangles separately and relabel all congruencies.
Congruence
Statement
Corresponding
Sides
Corresponding
Angles
6.
Draw the 2 triangles separately and relabel all congruencies.
Congruence
Statement
© 2009 Math Geek
Corresponding
Sides
Corresponding
Angles
DATE:__________
Goal
Theorem 4.1:
Chapter 4: Congruent Triangles
Review copy and angle and copy a segment constructions
If two angles of one triangle are corresponding and
congruent to two angles of a second triangle, then the
third pair of corresponding angle are also congruent.
B
E
If
A
then < C = < F.
C
D
F
A Geometric Investigation
Proofs are used in Geometry as a formal write-up of something that is
believed to be true.
Every statement made in a proof must have a valid reason to support it.
Proofs are written in three different ways:
1. Paragraph Proof: a proof in which the writer of the proof explains
his/her reasoning in complete sentences using statements of what
they believe to be true and the reasons why.
2. 2-Column Proof: a proof which uses a 2 column format. The left
column for statements and the right column for the reasons.
3. Flow Proof: a diagram used to show the logical reasoning of a proof
flowing from one statement to the next with the underlying
reasons.
G
Proofs
Prove the 2 triangles are Congruent:
J
H
I
 Example of a Paragraph Proof:
I am given GJ is congruent to IJ with one pair of tick marks. I am also given that
GH is congruent to IH with two pairs of tick marks. I can conclude that JH is the
same length in both ΔGJH and ΔIJH because of the Reflexive Property. Therefore,
since I have 3 pairs of corresponding and congruent sides, ΔGJH must be congruent
to ΔIJH by Side-Side-Side (SSS).
 Example of a 2 Column Proof:
Statements
1.
2.
3.
4.
Reasons
1. Given (one pair of tick marks)
2. Given (two pairs of tick marks)
3. Reflexive Property
4. Side-Side-Side (SSS)
GJ = IJ
GH = IH
JH = JH
ΔGJH = Δ IJH
 Example of a Flow Proof:
GJ = IJ
GH = IH
JH = JH
given
given
Reflexive Property
ΔGJH = ΔIJH
SSS
Geometry Chapter 4
Name________________
Constructing Congruent Figures
Using Copy a Segment & Angle
Construct a polygon congruent to each given polygon.
DATE:___________
Chapter 4.2: Side-Side-Side
To construct 2 triangles that are congruent using only 3 side
Goal
lengths in the construction.
What’s
The rest of the Chapter deals with one concept:
the point?  to construct congruent triangles, you do not need to use all
6 pieces of info.
 the minimum number of measurements you need to use is
ONLY 3.
 But what three???
Side-Side- If three side lengths of 1 triangle are used to construct a 2nd,
Side
then the two triangles will ALWAYS be congruent.
(SSS)
(ie. you don’t need to measure the angles, they will always
end up the same).
B
A
E
C
D
F
If AB = DE, BC = EF, and AC = DF, then ΔABC = ΔDEF.
DATE:___________
Chapter 4.2: Side-Angle-Side
Goal
To construct congruent triangles only using 2 side lengths
and one angle’s degrees.
Side-Angle- If you use 2 side lengths and the degrees of the angle
Side
between those two sides of one triangle to construct a
(SAS)
second triangle, the 2 triangles will always be congruent.
B
there are 3
possible
combinations
when using
SAS
A
E
C
B
A
E
C
ΔABC = ΔDEF
F
D
B
A
F
D
E
C
D
F
DATE:____________
Chapter 4.3: Angle-Side-Angle
Goal
To construct congruent triangles only using 2 angles and
one side length.
Angle-Side- If you use 2 angle measures and the length of the side
Angle
between the two angles of one triangle to construct a
(ASA)
second triangle, the 2 triangles will always be congruent.
B
there are 3
possible
combinations
when using
ASA
A
E
C
B
A
E
C
ΔABC = ΔDEF
F
D
B
A
F
D
E
C
D
F
DATE:____________
Goal
AngleAngle-Side
Chapter 4.3: Angle-Angle-Side
To construct congruent triangles only using 2 angles and
one side length.
If you use 2 angle measures and the length of the side
between the two angles of one triangle to construct a
second triangle, the 2 triangles will always be congruent.
E
E
B
B
there are 6
possible
combinations
A
when using
AAS
B
C
B
A
B
E
A
C D
F A
C D
F A
D
C D
ΔABC = ΔDEF
F
A
E
C
B
E
F
F
D
E
C D
F
DATE:___________
Shortcuts That Don’t Work-Angle-Side-Side and Angle-Angle-Angle
Goal
To identify the two shortcuts that don’t work:
Angle-Side- Angle-Side-Side is not a valid way to prove 2 triangles are
Side
congruent. Why? There are 2 different triangles that can
(ASS)
be drawn using 2 side lengths and the angle not between
them. Therefore, they are not always congruent.
These 2 triangles are different sized and shaped and
therefore not congruent, even though the same angle and
2 sides were used to construct them.
Angle-AngleAngle
These 2 triangles share the same 3 angle degrees, but
they are not the same size (they are the same shape), but
still they are not congruent.
DATE:___________
Chapter 4.4: CPCTC=Corresponding Parts of Congruent Triangles are
Congruent
Goal
Once two Δ’s are proven congruent (using SSS, SAS, ASA, or
AAS), we can make assumptions about the other three pieces
of information that we didn’t know (sides and angles).
Recall-we
B
E
started
53°
53°
5 in.
the
5 in.
3
in.
3
in.
chapter
with
37°
37°
congruent
A
4 in.
4 in.
F
C D
Δ’s
List three pairs of sides
List 3 pairs of angles
ΔABC = ΔDEF
SSS
B
3 in.
A
E
5 in.
4 in.
3 in.
C
D
5 in.
4 in.
F
S: AB = DE
S: BC = EF
S: AC = DF
ΔABC = ΔDEF by SSS.
What can we conclude about the corresponding angles (the 3 missing
pieces of information)? If we use SSS to prove the two triangles are
congruent (identical in all 6 measures), then we can conclude:
< A = < D, < B = < E, < C = < F.
This reasoning is called “Corresponding Parts of Congruent
Triangles” are congruent. We abbreviate this “CPCT”.
DATE:_________
Chapter 4.6: Hypotenuse-Leg Congruence Theorem
Goal
To use the Hypotenuse-Leg Congruence Theorem to prove
Right Triangles are congruent.
Hypotenuse- If the hypotenuse and leg of one right triangle are
Leg
corresponding and congruent to the hyp. and leg of a 2nd
Congruence triangle, then the two triangles will be congruent.
Theorem
E
B
(HL)
A
C
AB = DE
BC = EF
D
F
leg
hypotenuse
ΔABC = ΔDEF
HL Congruence
Why? But why is this enough info. for right triangles?
3
x
x 12
12
4
x
x
3
4
5
5
DATE:___________
Chapter 4.7: Congruence in Overlapping Triangles
Goal
Given two triangles that are overlapping, re-draw them
marking all known congruencies then determining if they
are congruent using SSS, SAS, ASA, AAS, or HL.
Overlapping
triangles
You will see only the second picture. It is the goal for you
to separate the two picture back into the original 2
triangles, re-marking any congruencies that are given.
Then determining if the two triangles are congruent by
SSS, SAS, ASA, AAS, or HL.