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Section:
2-4
Name:
Topic: Reasoning in
Algebra
Class: Geometry 1A
Standard: 2
Date:
Proof
A proof is a convincing argument that uses ________________
___________________.
Period:
In a geometric proof, every ________________________ that is made
must be _________________________.
Ways to write a proof
A proof can be written in many forms.
1. ______________________
2. ______________________
3. ______________________
4. ______________________
5. ______________________
Two-column proof
For now, we’ll focus on one type of proof.
In a two-column proof:

The left column lists the ________________; these are the
deductive steps that connect what you are given to what you are
trying to prove.

The right column provides the _________________ behind the
statements. The reasons can be ____________________ ,
___________________, ____________________, or
__________________that we’ve learned.
1
Some Familiar
Properties from
Algebra
Algebra Example
If a = b,
then a + __ = b + ___
If a = b,
then a – ___ = b – __
If a = b,
then a  ___ = b  ___
________________
Property of Equality
________________
Property of Equality
________________
Property of Equality
If a = b and c  0 ,
a
b

then
________________
Property of Equality
a(b+c) = ____ + ____
________________
Property
Given: 2  m  3  2m  5m
Prove: m  6
Statements
Additional Properties
Reasons
If a = b,
then b can replace a in any expression.
_____________
Property
a=a
_____________
Property
If a = b,
then _____ = _____
_____________
Property
If a = b, and b = c,
then _____ = _____.
_____________
Property
Summary: Why do you have to show each step in a proof?
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
2
Section:
4-1 A
Topic: Introduction to
Proofs
Standard: 2
Assumptions from
Diagrams
Name:
Class: Geometry 1A
Period:
Date:
There are certain things that you can assume when you look at a
diagram; however, you must ________________ your statements.
Reflexive Property
1. Reflexive
 Segment
Statements
Reasons
B
A
C
Be sure to mark
the ___
___________on
the diagram.
D
Reflexive
 Angle
B
D
Reasons
Statements
Reasons
E
A
C
2. Vertical Angles
Vertical Angles
You need ____
___________ to
name an angle if it
isn’t clear which
angle you want.
Statements
B
A
E
D
C
3
Linear Pair
3. Linear Pair (supplementary)
Statements
Reasons
A
x
B
Conclusions based on
given information.
3x+20
D
C
Generally speaking, given information will lead you to a conclusion.
Think of it as a conditional statement. All you have to do is justify your
conclusion using a ____________________, __________________,
__________________________, or _________________________.
Midpoint
4. Midpoint
 If you know a point is the midpoint of a segment, then
____________________________________________.
Given: E is the midpoint of AC .
B
A
Statements
Reasons
E
C
D
Segment Bisector
5. Bisector  Segment

If a segment, ray, or line bisects a segment, then
____________________________________________.
Given: CD bisects AB .
A
Statements
Reasons
D
B
C
4
Angle Bisector

Angle

If a segment, ray, or line bisects an angle, then
______________________________________________
.
Given: CD bisects ACB
Statements
Reasons
A
D
B
C
Parallel Lines
Corresponding Angles
6. Parallel lines cut by a transversal
 Corresponding Angles

If parallel lines are cut by a transversal, then
______________________________________________
.
Given: DE AC
Parallel Lines
Alternate Interior
Angles

Alternate Interior Angles

If parallel lines are cut by a transversal, then
______________________________________________
Given: AD BC
5
Parallel Lines
Same-Side Interior
Angles

Same-Side Interior

If parallel lines are cut by a transversal, then
______________________________________________
.
Given: AD BC
A
B
D
Perpendicular Lines
C
7. Perpendicular Lines
 A two step process
1. If two lines are perpendicular, then ______________.
2. If ____________________, then ________________
.
Given: BD  AC .
B
A
D
C
6
Perpendicular
Bisector
8. Perpendicular Bisector
 Two conclusions can be drawn.

If a segment, ray or line is the perpendicular bisector of a
segment, then ___________________________ and

___________________________________________.
Given: BD is the  bisector of AC .
B
A
Isosceles Triangle
D
C
9. Isosceles Triangle
 If two sides of a triangle are congruent, then
________________________________________________
Given: AB  BC
B
A
D
C
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
Isosceles Triangle
If two angles of a triangle are congruent, then _________
________________________________________________
Given: A  C
B
A
D
C
Summary: How is a perpendicular bisector different from an angle bisector? Draw pictures of
each to help you explain your answer.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
8
Section:
4-1 B
Name:
Topic: Congruent Figures Class: Geometry 1A
Period:
Standard: 2
Date:
Congruent Figures
Congruent figures have the same ______________ and
______________.
Match the congruent
polygons.
B
A
C
What would you have to
do to the numbered
polygon to make it fit on
top of the lettered
polygon?
1
Congruent Polygons
2
3
Congruent Polygons have congruent __________________ parts.
Z
Given: ABC  ZXY
B
This is called the
__________ statement.
X
A
C
Name the corresponding
parts in the congruent
triangles.
Angles
Y
Sides
9
Example
Use the congruency statement to draw two congruent triangles.
ABC  MNO
Mark the corresponding
parts congruent.
List the congruent corresponding parts.
Angles
Segments
Look at the congruency statement and your list of congruent
corresponding parts. Could you have made the list without
drawing the picture? Explain.
____________________________________________________
____________________________________________________
____________________________________________________
3rd Angle Theorem
If two angles of one triangle are congruent to two angles of another
triangle, then _______________________________________.
Summary: List the congruent corresponding parts for the quadrilaterals. ABCD  WXYZ
Angles
Segments
10
Section: 4-2 & 4-3
Name:
Topic: Congruent
Triangles
Class: Geometry 1A
Period:
Standard: 2 & 5
Date:
Congruent Triangles
We found that congruent polygons have congruent corresponding
____________ and ____________. From the “What’s in Your
Bag” activity you found that you only need ____ pieces of
information to create congruent triangles.
___ - ___ - ____
_________ - _________ - _________
 If ______ sides of one triangle are ______ to ______ sides of
another triangle, then the triangles are ______.
___ - ___ - ____
_________ - _________ - _________
 If ______ sides of one triangle are ______ to the corresponding
______ sides of another triangle and the ___________ angle is
___ in each triangle, then the triangles are ______.
11
___ - ___ - ____
_________ - _________ - _________
 If ______ angles of one triangle are ______ to the corresponding
______ angles of another triangle and the ___________ side is
___ in each triangle, then the triangles are ______.
___ - ___ - ____
_________ - _________ - _________
 If ______ angles of one triangle are ______ to the corresponding
______ angles of another triangle and the corresponding
____-______________ sides are ___, then the triangles are ____.
Note: You can NOT use _________ - __________ - _________ to
prove triangles are congruent.
Summary: Determine which of the following pairs of triangles are congruent. Then state the
reason why (SSS, SAS, ASA or AAS). Be sure to mark the pictures with information you know.
A.
B.
C.
D.
12
Section: 4-6
Name:
Topic: Congruence in
Right Triangles
Class: Geometry 1A
Standard: 2 & 5
Date:
Parts of Right Triangles
In a right triangle, the right angle is formed by the ________.
Period:
The side opposite the right angle is the _____________________ the _______________ side of the triangle.
Congruent Right Triangles
We saw that we weren’t able to use _______ to prove triangles
congruent. But since right triangles have special properties
between their sides, you can use a unique theorem for right
triangles only.
___ - ___
_____________________ - ________
 If the _______________ and _______ of one ___________
triangle are congruent to the ________________ and _______ of
Remember they MUST
be ______________
triangles
another ___________ triangle, then the triangles are _____.
Remember they MUST
be ______________
triangles
13
Examples What additional information is needed to prove these triangles are
congruent by HL?
A.
B.
C.
Summary: For what values of x and y are the triangle congruent by HL?
y+1
x+3
x=
x
6
y=
14
Section: 4-4
Name:
Topic: Corresponding
Parts of Congruent s
Class: Geometry 1A
Standard: 2 & 5
Date:
What does it mean for two
triangles to be congruent?
Are the triangles below congruent? How can you tell?
Period:
B
A
E
C
D
What do you also know?
Angles
Segments
CPCTC
Once we have two congruent triangles we can say that all of their
corresponding parts (____________ and ___________) are also
congruent.
C________________________________
P________________________________
You must
__________ this
first, before you
can use this rule!
C________________________________
T________________________________
C________________________________
15
Example
Given: AB is an altitude to BE , DE is an altitude to BE
A  D, C is the midpoint of BE A
Prove: AC  DC
Statements
AB is an alt. to BE ,
1.
2.
3.
4.
5.
B
D
C
E
Reasons
1. Given
DE is an alt. to BE ,
A  D,
C is the midpt of BE
QUESTION 1
B and E are right angles .
B   E
QUESTION 3
6. ABC  DEC
7. AC  DC
1. A. BC  EC
B. AB  BE , DE  BE
C. AC  BE , DC  BE
D. AB  DE
3. A. BC  EC
B. AC  DC
C. AB  DE
D. ACB  DCE
2. An alt. forms perpendicular lines.
3. QUESTION 2
4. All right angles are congruent.
5. A midpoint divides a segment
into 2 congruent segments.
6. QUESTION 4
7. QUESTION 5
2. A. Right angles measure 90
B. Right angles are
perpendicular
C. B and E measure 90
each
D. Perpendicular lines form
right angles
4. A. SAS
B. HL
C. AAS
D. ASA
5. A.
B.
C.
D.
Summary: Draw a picture to represent CEO  HDF.
Name all the pairs of corresponding congruent parts.
Prove
CPCTC
AAS
Given
Angles
Segments
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