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904
L
Geometry
904
Date:
Name:
Geometry 904
Table of Contents
Objectives ................................................................................................................................. 2
Vocabulary: ............................................................................................................................... 2
Character Trait: ......................................................................................................................... 2
Chapter 1: Proving Triangles are Congruent ............................................................................ 3
Section 1: Corresponding Parts of Triangles. ....................................................................... 3
Section 2: Congruent Triangles ............................................................................................ 5
Section 3: Postulates about Angles and Sides of Congruent Triangles ................................ 6
Chapter 1 Review ................................................................................................................ 12
Chapter 2: More Methods of Proving Congruence ................................................................. 14
Section 1: Overlapping Triangles ....................................................................................... 14
Section 2: AAS and HY-LEG ............................................................................................. 16
Chapter 2 Review .................................................................Error! Bookmark not defined.
Character Lesson:.....................................................................Error! Bookmark not defined.
Chapter 3: Applications of Congruent Triangles .....................Error! Bookmark not defined.
Section 1: Proving Segments and Angles are Congruent ....Error! Bookmark not defined.
Section 2: Proving Bisectors and Perpendicular Lines with Congruent Triangles. ..... Error!
Bookmark not defined.
Chapter 3 Review .................................................................Error! Bookmark not defined.
Chapter 4: Area .......................................................................Error! Bookmark not defined.
Section 1: Areas of Polygons ...............................................Error! Bookmark not defined.
Chapter 4 Review .................................................................Error! Bookmark not defined.
Unit Review .............................................................................Error! Bookmark not defined.
Appendix A: Theorem reference list.......................................Error! Bookmark not defined.
Appendix B: .............................................................................Error! Bookmark not defined.
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Page 1
Objectives
Upon the successful completion of this unit you will be able to:

Explain what is required for triangles to be congruent.

Use shortcut methods for proving triangles are congruent.

Prove overlapping triangles are congruent

Use AAS and Hy-Leg methods of proving congruence of triangles

Use congruent triangles to prove segments and angles are congruent.

Use congruent triangles to prove special properties of lines.

Find the areas of triangles, trapezoids, polygons.

Define and give examples of TACTFUL
Vocabulary:








Correspondence Agreement between particular things.
SSS Side-Side-Side method of proving congruence when three sides of one triangle
are congruent to the corresponding sides of another triangle.
SAS Side-Angle-Side method of proving congruence when two sides and the
included angle of one triangle are congruent to the corresponding parts of another
triangle.
ASA Angle-Side-Angle method of proving congruence when two angles and the
included side are congruent to the corresponding parts of another triangle.
AAS Angle-Angle-Side method of proving congruence when two angles and an
opposite side are congruent to the corresponding parts of another triangle.
Hy-Leg Hypotenuse-Leg method of proving the congruence of right triangles when
the hypotenuse and a leg of a right triangle are congruent to the corresponding parts
of another right triangle.
Included Side The side of a triangle between two of the vertices.
Included Angle An angle of a triangle that is between two of the sides.
Character Trait:
Tactful: Choosing to be kind with my words.
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Chapter 1: Proving Triangles are Congruent
Section 1: Corresponding Parts of Triangles.
Correspondence is visible in many of the everyday things in the world about us. You can see
correspondence between two cars on the road. Each car may be a different make, model,
color, and age. Yet, each will have
four wheels, a windshield and an
engine. The agreement between the
parts of cars indicates
correspondence. Examine a typical
human hand, and you will see five
fingers. Examine another, and you will see five fingers. The fingers on one typical human
hand correspond to the fingers on another.
Y
X
Corresponding parts can be used for comparison.
B
The parts of two triangles can be ‘paired’ to
establish corresponding parts. Paired of
corresponding vertices are seen below.
A
C
The diagram shows the following correspondence:
vertex X corresponds to vertex A
vertex Y corresponds to vertex B
vertex Z corresponds to vertex C
Z
Pairing Corresponding Vertices
Relationships of correspondence can be written with mathematical symbols. The symbol 
is read “corresponds to.”
AX
BY
CZ
The order of the vertices in the triangle name is important in establishing a correspondence
between triangles. We can note the correspondence between triangles as follows:
ΔABC  ΔXYZ (triangle ABC corresponds to triangle XYZ)
The first vertex listed in the first triangle corresponds to the first vertex listed in the second
triangle.
Examine this statement.
ΔDEF ΔLMN (triangle DEF corresponds to triangle LMN)
There are three pairs of corresponding vertices. You must list the corresponding vertices in
the same order that they are listed in the statement of triangle correspondence.
DL
ΔDEF
EM
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FN
Example 1: Determine the corresponding vertices when ΔGHI ΔRST
You know that both triangles correspond and that the corresponding vertices are written in
the same order. From the statement of triangle correspondence you can identify three pairs
of corresponding vertices. You can write each pair as a corresponding vertices statement.
Your answer has three statements:
GR
HS
I T
Corresponding vertices determine the corresponding sides of a triangle. You can use logic to
deduce these relationships.
AX
BY
It follows that
and
are corresponding line segments. Notice how this is shown in the
diagram below.
We have used capital letters to indicate points,
lines, and line segments while lowercase letters
have been used to indicate interior angles. We
will now use capital letters to indicate vertices as
well as the angle that occurs inside the vertex of
a closed figure. Lowercase letters will still be
used to indicate angles that occur with
intersecting lines and line segments.
Y
X
B
A
C
Z
Pairing Corresponding Sides
Examine the corresponding vertices C and Z. Because the vertices correspond, we can say
that C  Z. Notice that the sides opposite the corresponding angles are corresponding
sides.
Given that ΔABC  ΔXYZ , let’s review the corresponding parts between these two
triangles.
Angle Correspondence
A X
B Y
C Z
Side Correspondence
 Y
B


A
X
Y
C
Z
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Establishing angle and side correspondence does not establish that sides and angles
congruency. Correspondence is used to compare two figures and their parts. We will
examine the conditions necessary for congruency in the next section.
Given, the conditions below, identify the
corresponding parts of ΔABC ΔUVW:
W
A
U
1.
AU
BC  _______
2.
ΔABC  ΔUVW
AB  _______
3.
ΔABC  ΔUVW
BC  _______
4.
T OR F
B
V
C
ΔABC  ΔUVW and ΔBAC
 ΔVUW
5.
T OR F
ΔABC  ΔUVW and ΔCAB  ΔWUV
Score the above questions.
Correct wrong answers.
 Rescore.
Teacher’s
Initials:_____
Section 2: Congruent Triangles
Congruent Parts:
To establish that triangles are congruent requires more than correspondence. Congruence
requires that the triangles meet the following definition:
Two triangles are congruent when all corresponding parts are congruent.
Congruence of corresponding sides and angles establish congruent triangles.
 Every triangle has three sides. In order for two triangles to be congruent, all three
pairs of corresponding sides must be congruent.
 Every triangle has three angles. In order for two triangles to be congruent, all three
pairs of corresponding angles must be congruent.
 Six points of agreement (3 sides + 3 angles) must be congruent in order to know that
the triangles are congruent.
C
If:
XA
≅
X ≅ A
≅
Y ≅ B
≅
Z ≅ C
54°
X
51°
75°
51°
75°
≅
Y
B
54°
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Z
Page 5
A
Then: ΔABC ≅ ΔXYZ
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Included Parts:
In any triangle such as ΔQRS at the right, the terms ‘included
side’ and ‘included angle’ are commonly used to describe the
parts. An angle is included between two sides. Between R and
S is the included side
. You can also have an angle included
between two sides. R is an included angle between
and .
R
Q
Circle the correct answer using Figure 1:
1.
True False
M
ΔDEF ≅ ΔKLM
≅
D ≅ K
≅
E ≅ L
≅
F ≅ M
S
F
D
L
Figure 1
K
E
Using Figure 2, answer the following questions:
A
2. The included side of A and 
Figure 2
3. The included angle of side AC
Score the above questions.
4.
Side BC is the included
5.
Side AC is the included
Correct wrong answers.
C
 Rescore.
B
Teacher’s
Initials:_____
Section 3: Postulates about Angles and
We will begin examining relationships between
Investigation 1: Follow the steps below.
1. On a separate sheet of paper use a ruler, to draw a triangle. Note the measurements of
each side.
2. Draw another triangle, making the sides the same
length as the sides on the first triangle.
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Cut out the second triangle with scissors.
3. Lay the triangle you cut out on top of the first triangle you drew. (You may need to
rotate it.) The two triangles should match perfectly both in the lengths of the sides
and the measurements of the angles.

1
2
Teacher’s
Show your work to your teacher.
Initials:_____
You did not need to measure the angles in order to create congruent triangles. The angles are
congruent because all of the sides of the triangles are congruent. This is the basis of the
Side-Side-Side postulate: This is postulate is often referred to by the letters SSS Postulate.
Side-Side-Side Postulate
If the vertices of one triangle can be paired with the vertices of another triangle so that the
sides of the first triangle are congruent to the corresponding sides of the second triangle,
then the two triangles are congruent.
This postulate is helpful in proving that triangles
Using this postulate only the side congruency


M

F
D
L
Figure 1
E
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K
Page 8
Other relationships between corresponding parts of triangles can be used to prove that two
triangles are congruent. Examine the following postulate.
This is known as the Angle-Side-Angle (ASA) Postulate.
Angle-Side-Angle Postulate
If the vertices of one triangle can be paired with the vertices of another triangle so that two
angles and the included side of the first triangle are congruent to the corresponding parts
of the second triangle, then the two triangles are congruent.
The arc marks on figure 2 show:
DK
EL
F
D
The hash marks show:
 .
M
L
Figure 2
E
K
Using this information it can be determined  DEF   KLM
Side-Angle-Side Postulate
If the vertices of one triangle can be paired with the vertices of another triangle such that
two sides and the included angle of one triangle are congruent to the corresponding parts
of a second triangle, then the two triangles are congruent.
This is known as the Side-Angle-Side (SAS) Postulate
Having two sides and the included angle of
the first triangle congruent to the
corresponding parts of the second triangle
also proves that the triangles are
congruent.
M
F
D
L
E
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K
Page 9
Example 1:
Construct a proof.
R
Given: Figure 3
Figure 3
Prove: ΔRST ≅ ΔRPT
S
P
SOLUTION: These two triangles share a common
side, . The hash marks indicate that
 .
There are two sides and an included angle that are
congruent.
T
This allows us to use the SAS Postulate to prove the triangles are congruent.
PROOF
Statements
Reasons
1. ΔRTS ≅ ΔRTP
1.
GIVEN
≅
≅
2.
GIVEN
3.
Reflexive Property
4. ΔRST ≅ ΔRPT
4.
SAS Postulate
2.
3.
Figure 4
Example 2: Construct a Proof
Given:
A
Figure 4
AB ∥ DC
ADX  BCY
X

Prove:
A
B
T
D
U
X
B
Y
D
Figure 5
Y
C
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C
First, create two triangles by drawing line
segments perpendicular to the parallel
lines and through the vertices D and C
forming right triangles (as shown in
figure 5).
Label the points where the perpendicular
line segments cross
as T and U.
Page 10
The segments you added create two triangles Δ ADT and ΔBCU that have corresponding
parts for comparison. Each triangle includes a segment that you want to prove is congruent
to the other. ΔADT contains AD, and ΔBCU contains BC.
This proof will use the definition of complementary angles and the definition of parallel lines
to show that two angles and an included side are congruent. The ASA Postulate can then be
used to prove the triangles are congruent. When the triangles are proven to be congruent,
then the corresponding sides, AD and BC will be proven as congruent.
PROOF
Statements
Reasons
1.
GIVEN
2.
Definition of complementary angles
3.
Transitive Property
4.
Substitution
5.
Subtraction Property of Equality
6.
Definition of Parallel Lines
7. m ATD = 90° = m∠BUC
7.
Definition of Perpendicular Lines
8. ATD ≅ BUC
8.
Transitive Property
9. Δ ATD ≅ Δ BUC
9.
ASA Postulate
1. ADX ≅ ∠BCY
2. mADX + mADT= 90°
mBCY + mBCU = 90°
3. mADX + mADT = mBCY +
mBCU
4. mBCY + mADT= m∠BCY +
m∠BCU
5. ADT ≅ BCU
6.
10.


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10.
Corresponding parts of congruent triangles
Page 11
Answer the following questions:
1.
Is ΔA  ΔB?
Y
N
B
Explain Your answer:
2.
Is ΔA  ΔB?
A
Y
N
2.5
Explain Your answer:
2.5
1.5
A
1.5
B
2
2
3.
Is ΔX  ΔY?
Y
N
Explain Your answer:
X
55°
55°
55°
55°
Y
4.
Is Δ1  Δ2 ?
Y
N
(Given: This is a parallelogram)
2
Explain Your answer:
1
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5.
Is Δ4  Δ3 ?
Y
N
4
(Given: This is a regular pentagon)
Explain Your answer:
6.
3
5
Are Δ5, Δ6, Δ7, & Δ8 congruent triangles?
Y
6
N
(Given: This is a regular Octagon)
Explain Your answer:
8
7
7.
What is the shape in the middle of the octagon in problem 6? ___________
How do you know? ____________________________________________
Score the above questions.
Correct wrong answers.
 Rescore.
Teacher’s
Initials:_____
Chapter 1 Review
Identify the corresponding parts using figure 1:
1.
∠A  ∠U
 _______
2.
ΔABC  ΔUVW
 _______
3.
ΔABC  ΔUVW
 _______
4.
T or F
ΔABC  ΔUYW and ΔBAC  ΔYUW
5.
T or F
ΔABC  ΔUYW and ΔCAB  ΔWUY
Y
U
B
A
C
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W
Figure 1
Page 13
Circle the correct answer:
6.
True False
M
ΔDEF ≅ ΔKLM
≅
D ≅ K
≅
E ≅  L
≅
F ≅  M
F
L
E
K
E
Using this triangle answer the following questions:
A
B
C
7.
Side BC is the included side of B and C.
True
False
8.
Side AC is the included side of A and  B.
True
False
Fill in the missing statements in the following proof:
9.
Complete the proof:
R
Given: ST  PT
Prove: ΔRST ≅ ΔRPT
S
P
SOLUTION: Because both these
T
triangles share a common side, , then
there are two sides and an included angle that are congruent. This allows us to use
the SAS Postulate to prove the triangles are congruent.
PROOF
Statements
Reasons
1.
______________________
2.
GIVEN
3. RT  RT
3.
Reflexive Property
4. ΔRST ≅ ΔRPT
4.
_______________________
1. RTS ≅ RTP
2. ST  PT
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Is ΔABC  ΔADE ?
10.
Y
N
B
(Given: This is a regular pentagon)
A
C
Explain Your answer:
2
D
E
Score the above questions.
Correct wrong answers.
 Rescore.
Teacher’s
Initials:_____
Chapter 2: More Methods of Proving Congruence
Section 1: Overlapping Triangles
It is not always easy to distinguish between triangles in a complex figure. Overlapping
triangles make it tricky to identifying corresponding sides and angles. Examine figure 1
below. Can you see the five triangles? How about the twelve angles?
Remember relationships: These sides
are the same side in this example!
You can use different strategies to identify corresponding triangles, sides, and angles.


Separate and reposition the triangles by redrawing them to help keep them
organized:
Use colored pencils or pens to outline triangles.
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Marking Triangles with colors:
Redraw Triangles
REMEMBER
Same Angles:
Mark one pair of corresponding triangles in each figure using colored pens or pencils:
1.
2.
3.
4.
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5.
6.
Score the above questions.
Correct wrong answers.
 Rescore.
Teacher’s
Initials:_____
Section 2: AAS and HY-LEG
AAS Theorem:
In the following triangles, ΔFGH & ΔCDE, there are two pairs of corresponding angles and
one pair of corresponding sides that are congruent. The corresponding sides are not included
sides. Can we determine if the triangles are congruent?
The postulates we’ve learned about in
this unit do not fit this example.
We cannot use the SSS Postulate,
ASA Postulate, or the SAS Postulate
to establish that the triangles are
congruent.
G
D
Figure 1
F
H
C
E
We can use a corollary.
Corresponding and Congruent Angles Corollary
If two angles of a triangle are congruent to two angles of another triangle then the
remaining angles are congruent.
For Figure 1 we are can examine the arc marks and hash marks to state the givens:

F C
GD
By applying the corollary we determine that E and  H are congruent.
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Angle-Angle-Side (AAS) Theorem
If two triangles have two corresponding angles that are congruent and the side opposite
one of the angles in the first triangle is congruent to a corresponding side in the second
triangle then the two triangles are congruent.
Example 1: Construct a Proof
Given:
A
D
•
 ABE ≅  DEB
•
≅
 EDF ≅  CAB
•F
Prove: ∆DEF ≅ ∆ABC
E
•C
B
SOLUTION: Since ABC and DEF are supplementary to congruent angles, they are
congruent. With two angles congruent and an opposite side congruent, the AAS Theorem
can be applied to prove the triangles are congruent.
PROOF
Statements
1. ABE ≅ DEB
Reasons
1.
Given
2.
Given
3. EDF ≅ CAB
3.
Given
4. ABC & ABE are supplementary
4.
Definition of supplementary angles
5. DEF & DEB are supplementary
5.
Definition of supplementary angles
6. ABC ≅ DEF
6.
7. ∆DEF ≅ ∆ABC
7.
Supplementary angles of congruent
angles are congruent.
AAS Theorem
2.
≅
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