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Geometry
Lesson Notes 2.6
Date ________________
Objective: Write algebraic proofs. Use properties of equality to write geometric proofs.
Properties of Equality for Real Numbers
For all numbers a, b, and c,
Reflexive Property
a=a
Symmetric Property
If a = b then b = a
Transitive Property
Addition and
If a = b and b = c, then a = c
Subtraction Properties
If a = b, then a + c = b + c
Multiplication and
Division Properties
If a = b, then ac = bc and a/c = b/c if c  0
Substitution Property
If a = b, then a may be replaced by b in any equation
or expression
Distributive Property
a(b + c) = ab + ac
NOTE: We will be assuming the Commutative and Associative Properties of addition and
multiplication. No need to state them in a proof.
You must be able to recognize and use these properties!
You can use these properties to justify every step as you solve an equation. The group of
algebraic steps used to solve problems is called a deductive argument.
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Example 1 (p 94): Verify Algebraic Relationships
Solve ½ (x + 16) = 5x − 1 for x and give a reason for each step.
Statement
Reason
½ (x + 16) = 5x − 1
Given
x + 16 = 2(5x − 1)
Multiplication / Substitution properties
x + 16 = 10x − 2
Distributive / Substitution properties
16 = 9x − 2
Subtraction / Substitution properties
18 = 9x
Addition / Substitution properties
2=x
Division / Substitution properties
x=2
Symmetric property
This deductive argument is an example of an algebraic proof of a conditional statement.
The conditional statement would be: If ½ (x + 16) = 5x − 1, then x = 2.
The hypothesis is the starting point (the given) of the proof.
The conclusion is the end of the proof, what we need to prove.
Listing the reasons (properties) for each step makes this a proof.
Two-column, or formal, proof: contains statements (the steps) and reasons
(the properties that justify each step) organized in two columns.
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Example 2 (p 94): Write a Two-Column (Algebraic) Proof
Write a two-column proof of the following conditional statement.
If
7
1
 n  4  n , then n  2 .
2
2
Given:
7
2
 n  4  1n
Prove:
n2
2
Statements:
1.
Reasons
7
1
 n  4 n
2
2
1. Given
2. 2   n   2  4  n 
2 
2


2. Multiplication Property
3. 7  2n  8  n
3. Distributive Property
4. 7  2n  n  8  n  n
4. Addition Property
5. 7  n  8
5. Substitution Property (Combining Like Terms)
6. 7  n  7  8  7
6. Subtraction Property
7.  n  1
7. Substitution Property (Combining Like Terms)
7
8.
n
1
1

1
1
9. n  1
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8. Division Property
9. Substitution Property
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Proofs in geometry are presented in the same manner. Algebra properties as well as definitions,
postulates, and other true statements can be used as reasons in a geometric proof.
Since geometry also uses variables, numbers, and operations, we are able to use many of the
properties of equality to prove geometric properties.
Segment measures and angle measures are real numbers, so we can use the properties of
equality to describe relationships between segments and between angles.
Examples:
Property
Segments
Angles
Reflexive
AB = AB
mC = mC
Symmetric
If XY = YZ, then YZ = XY
If m1 = m2, then m2 = m1
Transitive
If MN = NO and NO = OP,
then MN = OP
If mK = mL and mL = mM,
then mK = mM
Practice: Name the property of equality that justifies each statement.
Statement
Property
If 5 = x, then x = 5
_______________________________
If ½ x = 9, then x = 18
_______________________________
If AB = 2x and AB = CD, then CD = 2x
_______________________________
If 2AB = 2CD, then AB = CD
_______________________________
Example 3 (p 96): Justify Geometric Relationships
If GH + JK = ST and ST  RP , then which of the following conclusions is true?
I.
GH + JK = RP
II.
PR = TS
III. GH + JK = ST + RP
A. I only
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B. I and II
C. I and III D. I, II, and III
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E
Example 4 (p 96): Geometric Proof
A starfish has five arms. If the length of arm 1 is 22 cm, and arm 1 is congruent to arm
2, and arm 2 is congruent to arm 3, prove that arm 3 has length of 22 cm.
Given:
A
_____________________________
22 cm
_____________________________
E
Prove:
_____________________________
Proof:
B
S
C
D
C
Statements
Reasons
1. ________________________
1. ___________________________________________
2. ________________________
2. ___________________________________________
3. ________________________
3. ___________________________________________
4. ________________________
4. ___________________________________________
 HW: A7a pp 97-100 #14-25, 29-31, 37-38
A7b 2-6 Skills Practice / Practice
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