Properties and Proofs
Name: _________________________________ HOUR: ________
3(x – 10)
Distributive
5x – 6 – 10x
Simplify
3x – 8 = 14
Addition
4x + 9 = 2
Subtraction
𝒙
=𝟗
𝟒
Multiplication
-5x = 20
Division
4r + 8 when r = 8
Substitution
HB  HB
Reflexive
If A  B then _________
Symmetric
If LM  OP and OP  QR, then ___________
Transitive
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Properties Review: Name the properties and answer the questions below:
____________________1) If1  2 and 2  3, then 1   3.
____________________2) If a + 5 = 26 and a = mR, then mR + 5 = 26.
____________________3) Explain the difference between questions 1 and 2
____________________4) Y  Y
____________________5) If ST 
QR ,
then
QR
 ST
____________________6) What do questions #1, 4, and 5 have in common?
____________________7) If m1 + 70 = 100, then the m1 = 30.
____________________8) What happened on both sides of problem #7
____________________9) If EF – CD = 12, then EF = 12 + CD
____________________10) What happened on both sides of problem #9
____________________11) If 9x = 72, then x = 8.
____________________12) If AB = CD and RS = ST, then AB + RS = CD + ST
____________________13) In problem #12, if RS = 5, then ST = 5, so what
happened on both sides of the equal sign?
____________________14) If 2(mA) = 120°, then mA = 60°.
____________________15) Complementary angles add up to ___ degrees
____________________16) Supplementary angles add up to ___ degrees
____________________17) A linear pairs angles do what?
____________________18) Vertical angles have the _____ measure.
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Definitions, Theorems, and Postulates that you may have to use besides the properties:
Statement
Drawing or Given Item
Reason
B
A
C
Definition of Midpoint
F
Segment Addition Postulate
“B is the midpoint of AC”
E
D
“E is between DF”
L
N
Triangle Sum Theorem
M
Given:  1 and  2 are
complementary
Given:  1 and  2
are supplementary
Definition of Complementary
Definition of Supplementary
Linear Pair
4
3
5
Vertical Angles
6
1=5
5=4
51
Proof: Complementary Angles
Given: Angle 1 and Angle 2 are complementary
Angle 2 = 47
1
Prove: Angle 1 = 43
STATEMENT
2
REASON
Given
Given
Def of __________
Angle 1 = 43
L
Proof: Triangle Sum Theorem
N
Given: Angle N = 42
Angle L= 116
Prove: Angle M= 22
STATEMENT
M
REASON
52
Proof: Supplementary Angles
Given: Angle 1= 145
Angle 2 = 35
1
2
Prove: Angle 1 and 2 are supplementary
STATEMENT
REASON
Angle 1 = 145
Angle 2 = 35
Angle 1 = Angle 1
Angle 1 + Angle 2 = Angle 1 + Angle 2
Angle 1 + Angle 2 = 145 + 35
Angle 1 + Angle 2 = 180
Angle 1 and Angle 2 are supplementary
Proof Puzzle #1: Algebraic Proof
Given:
Prove:
2x  6  2
5
x=2
STATEMENTS
REASONS
53
d
54
Proof Puzzle #2: Parallel Lines
Cut out the reasons and statements from
the next page to complete the proof
Given:
1
3
7
a.
b.
b.
1=4
8
d.
e.
e.
 5 +  6 = 180
 5 = 35
Given
c.
d.
g.
b
REASON
a.
f.
6
a
Angle 6 = 145
STATEMENT
c.
4
5
𝑎∥𝑏
Angle 1 = 35
Prove:
2
Alternate Interior Angles
f.
g.
h.
h.
i.
i.
USE THESE STATEMENTS AND REASONS TO FILL IN YOUR PROOF ABOVE:
𝑎∥𝑏
Substitution
Angle 6 = 145
Alternate Interior
Angle 1 = Angle 4
Linear Pair
Angle 4 = Angle 5
Given
Angle 5 + Angle 6 = 180
Given
Angle 1 = 35
Substitution
Angle 4 = 35
Vertical Angles
Angle 5 = 35
Substitution
35 + Angle 6 = 180
Subtraction
55
Statements
Reasons
Given: B is the midpoint of AC
AB = 5
A
Prove: AC = 10
Statements
B
C
Reasons
Definition of Midpoint
Angle Segment Postulate
56
Given: 15y + 7 = 12 – 20y
Prove: 𝑦
=
1
7
Statements
Reasons
15y + 7 = 12 – 20 y
35y + 7 = 12
35y = 5
𝑦=
1
7
Given: m1 + m2 = 100
m1 = 80
Prove: m2=20
Statements
Reasons
Given: B is the midpoint of AC
AB = 5
A
B
C
Prove: BC = 5
Statements
Reasons
57
Given: 1 = 30
2= 30
Prove: 1+2=60
Statements
Reasons
1 = 30
2= 30
1=1
1+2=1+2
1+2=30+30
1+2=60
Given: B is between A and C
AB = 5
AC = 12
A
B
C
Prove: BC = 7
Statements
Reasons
58