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Properties from Algebra
Name ____________ Per ___
Section 2-2 – the beginning of Proofs
Properties of Equality
Wording
Addition prop. of =
Subtraction prop. of =
Multiplication prop of =
Division prop. of =
Substitution prop. of =
Reflexive prop. of =
Symmetric prop. of =
Transitive prop. of =
(same as substitution, >2 items)
Distributive Prop.
Algebra Example
3x – 4 = 20
3x = 24
3x + 4 = 20
3x = 16
½x=5
x = 10
2x = 10
x=5
2x + y = 70
y = 3x
2x + 3x = 70
a=a
a=b
b=a
a = b, b = c, c = d
a=d
2( x  3)  2 x  6
Example: State the properties of equality to justify each step of solving the following equation: 3 x  6 
Given: 3 x  6 
1
x
2
Steps
12
7
Reason
1
x
2
2.
6x  12  x
3.
7 x  12
12
x
4.
7
Properties of Congruence
1.
Prove: x 
3x  6 
Wording
Reflexive. Prop. of 
Symmetric. Prop. of 
Transitive Prop. of 
1. Given
2.
3.
4.
Example
1
x
2
Segment Reasons
Wording
Example
Seg. Add. Post.
Def. of midpoint
Def. of seg. bisec.
Angle Reasons
Wording
Example
Def. of  bisector
 bisector thrm
Practice with Proof Statements:
Justify each statement with a property from algebra or a property of congruence.
1. P  P
2. If AB  CD and CD  EF , then AB  EF .
3. If RS=TW, then TW=RS.
5. If 5 y  20 , then y  4 .
4. If x+5=16, then x=11.
6. If
z
 10 , then z = 50.
5
7. 2 (a+b) = 2a+2b
8. If 2z  5  3 , then 2 z  2
9. If 2 x  y  70 and y= 3x, then 2x+3x=70.
10. If AB=CD, CD=EF, and EF=23, then AB=23.
2
b  8  2b
3
2b  3(8  2b)
11. Justify each step. 2b  24  6b
8b  24
b3
12. Complete each proof by supplying missing reasons and statements.
Given: m1  m3; m2  m4
Prove: mABC  mDEF
H
C
G
F
2
4
1
A
S
1. m1  m3; m2  m4
2. m1  m2  m3  m4
m1  m2  mABC
3.
m3  m4  mDEF
4. mABC  mDEF
R
1. ?
2. ?
3. ?
4. ?
13. Complete each proof by supplying missing reasons and statements.
Given: ST=RN; IT=RU
S
Prove: SI=UN
R
I
T
U
S
1. ST = RN
2.
_______ = SI+IT; ______ = RU+UN
3. SI + IT = RU + UN
4. IT = RU
5. ?
E
B
N
R
1. ?
2. ?
3. ?
4. ?
5. ?
3
D
Proving Theorems:
Section 2-3 – using deductive reasoning
Setting up a proof
Reasons Used in Proof
1. Copy Diagram
1. Given information
2. State Given: and Prove:
2. Definitions
3. Set up Statement and Reason headings of two columns
3. Postulates (Incl. Properties of Algebra
4. Theorems that have been proven
Midpoint Theorem (Theorem 2-1) If M is the midpoint of AB , then AM 
1
1
AB and BM  AB .
2
2
Angle Bisector Theorem (Theorem 2-2) If BX is the bisector of ABC , then mABX 
mCBX 
1
mABC .
2
1
mABC and
2
Practice:
What postulate, definition, or theorem justifies the statement about the diagram.
2. mAEB  mBEC  mAEC
B
3. AE+EF=AF
C
4. mAEB  mBEF  180
A
5. If E is the midpoint of AF , then AE  EF .
6. If E is the midpoint of AF , then AE 
E
F
1
AF .
2
7. If E is the midpoint of AF , then EC bisects AF .
8. If EB bisects AF , then E is the midpoint of AF .
9. If EB is the bisector of AEC , then mAEB 
1
mAEC .
2
10. If BEC  CEF , then EC is the bisector of BEF .
11. Deduce and prove.
A
B
Given: AC bisects BD ; BD bisects AC ; CE=BE.
Prove: ??
E
C
D
S
REMINDER: There is no “substitution” property of congruence.
R