Download Properties of Equality for Real Numbers

Reasoning in Algebra and Geometry
Geometry
Date
Name
Properties of Equality
If a = b, then a +c = b + c
Addition Property
If a = b, then a – c = b - c
Subtraction Property
If a = b, then ac = bc
Multiplication Property
If a = b, then a/c = b/c
Division Property
a=a
Reflexive Property
If a = b, then b = a
Symmetric Property
If a = b and b = c, then a = c
Transitive Property
Substitution Property
If a = b and a = c, then b = c AND
if x = 2a + 4, then x = 2b + 4
a(b + c) = ab + ac
Distributive Property
*Combine like terms – Combine terms on one side of an equation.
*Reflexive, Symmetric, and Transitive Properties have corresponding
properties of congruence.
1. Name the property that justifies each statement.
a. If AB + BC = DE + BC, then AB = DE.
subtraction
b. mABC = mABC
reflexive
c. If XY = PQ and XY = RS, then PQ = RS.
substitution
x
d. If
= 5, then x = 15.
multiplication
3
9
e. If 2x = 9, then x = .
division
2
2. Name the property that justifies each statement.
a. If 3x = 120, then x = 40.
division
b. If 12 = AB, then AB = 12.
symmetric
c. If AB = BC, and BC = CD, then AB = CD.
transitive
d. If y = 75 and y = mA, then mA = 75.
substitution
Deductive reasoning can be used when solving an equation. Each step can be
justified with a postulate, property, or a definition.
M
3. Find the value of x? Justify each step.
(2x + 30)°
A
1.
2. AOM and MOC are
supplementary
3. Two angles that form a line
are supplementary.
2. mAOM + mMOC = 180°
O
x°
C
Supplementary angles add up to 180
degrees.
(2x + 30) + x = 180°
substitution
3x + 30 = 180°
Combine like terms
3x = 150°
subtraction
x = 50°
division
4. What is the value of x? Justify each step.
R
B
x°
Given: AB bisects RAN
(2x – 75)°
A
N
4. AB bisects RAN
Given
2. mRAB = mBAN
A bisector divides an angle into 2 congruent
angles.
x = 2x – 75
substitution
-x = -75
subtraction
x = 75
division
A proof is a convincing argument that uses deductive reasoning. A proof
logically shows why a conjecture is true. A two-column proof lists each
statement on the left, and the justification, or the reason for each
statement on the right.
Review the Segment Addition Postulate and Angle Addition Postulate.
5. Write a two-column proof.
Given: m1 = m3
Prove: mAEC = mDEB
A
B
1
Proof:
Statements
1. m1 = m3
Reasons
1. Given
2. m2 = m2
2. Reflexive
3. m1 + m2 = m3 + m2
3. Addition
4. m1 + m2 = mAEC,
m3 + m2 = mDEB
4. Angle Addition
5. mAEC = mDEB
5. Substitution
E
2
C
3
D
6. Write a two-column proof.
Given: AB = CD
Prove: AC = BD
A
B
C
Proof:
Statements
1. AB = CD
Reasons
1. Given
2. BC = BC
2. Reflexive
3. AB + BC = CD + BC
3. Addition
4. AB + BC = AC,
CD + BC = BD
4. Segment Addition
5. AC = BD
5. Substitution
7. Write a two-column proof.
Given: mABD = mCBE
Prove: m1 = m3
Proof:
Statements
1. mABD = mCBE
2.
3. 2. m1 + m2 = mABD,
4.
m2 + m3 = mCBE
1
B
Reasons
1. 1. Given
2. Angle Addition
3. m1 + m2 = m2 + m3
3. Substitution
4. m2 = m2
4. Reflexive
5. 5. m1 = m3
C
A
5. Subtraction
D
2
3
E
D
8. Write a two-column proof.
Given: AC = BD
Prove: AB = CD
Proof:
Statements
5. 1. AC = BD
D
C
A
Reasons
1. Given
2. AB + BC = AC,
BC + CD = BD
2. Segment Addition
3. AB + BC = BC + CD
3. Substitution
4. BC = BC
4. Reflexive
5. AB = CD
5. Subtraction
B