Prentice Hall Geometry

Table 4 Properties of Real Numbers

Unless otherwise stated, a, b, c, and d represent real numbers.

Identity Properties
Addition eh plus . 0 equals eh and 0 plus . eh equals eh
Multiplication eh middle dot . 1 equals eh and 1 middle dot . eh equals eh
Commutative Properties
Addition eh plus , b equals . b plus eh
Multiplication eh middle dot , b equals . b middle dot eh
Associative Properties
Multiplication open eh plus b close . plus , c equals , eh plus . open b plus c close
Addition open eh middle dot b close . middle dot , c equals , eh middle dot . open b middle dot c close
Inverse Properties
Addition The sum of a number and its opposite, or additive inverse, is zero.
eh plus . open negative eh close . equals 0 and negative , eh plus . eh equals 0
Multiplication The reciprocal, or multiplicative inverse, of a rational number eh over b is b over eh , open eh comma b , not equal to 0 close .
eh middle dot , 1 over eh , equals 1 and 1 over eh , middle dot , eh equals . 1 open eh not equal to 0 close
Distributive Properties
eh open b plus c close . equals eh b . plus eh c open b plus c . close eh equals b eh . plus c eh
eh open b minus c close . equals eh b . negative eh c open b minus c . close eh equals b eh . negative c eh
Properties of Equality
Addition If eh equals b comma then eh plus , c equals . b plus c .
Subtraction If eh equals b comma then eh minus , c equals . b minus c .
Multiplication If eh equals b comma then eh middle dot , c equals . b middle dot c .
Division If eh equals b and c not equal to 0 comma then eh over c , equals , b over c . .
Substitution If eh equals b comma then b can replace a in any expression.
Reflexive eh equals eh
Symmetric If eh equals b comma then b equals eh , .
Transitive If eh equals b and b equals c comma then eh equals c .
Properties of Proportions
eh over b , equals , c over d . open eh comma b comma c comma d , not equal to 0 close is equivalent to
  1. eh d equals b c
  2. b over eh , equals , d over c
  3. eh over c , equals , b over d
  4. fraction eh plus b , over b end fraction . equals . fraction c plus d , over d end fraction
Zero-Product Property
If eh b equals 0 comma then eh equals 0 or b equals 0 , .
Properties of Inequality
Addition If eh greater than b and c greater than or equal to d comma then eh plus , c greater than . b plus d .
Multiplication If eh greater than b and c greater than 0 comma then eh c greater than b c . .
  If eh greater than b and c less than 0 comma then eh c less than b c . .
Transitive If eh greater than b and b greater than c comma then eh greater than c , .
Comparison If eh equals . b plus c comma and c greater than 0 comma then eh greater than b , .
Properties of Exponents
For any nonzero numbers a and b, any positive number c, and any integers m and n,
Zero Exponent eh to the , equals 1
Negative Exponent eh super negative n end super , equals , fraction 1 , over eh to the n end fraction
Product of Powers eh to the m , middle dot , eh to the n , equals . eh super m plus n end super
Quotient of Powers fraction eh to the m , over eh to the n end fraction . equals . eh super m minus n end super
Power to a Power open , c to the m . close to the n , equals , c super m n end super
Product to a Power open eh b , close to the n . equals , eh to the n , b to the n
Quotient to a Power open , eh over b , close to the n . equals . fraction eh to the n , over b to the n end fraction
Properties of Square Roots
For any nonnegative numbers a and b, and any positive number c,
Product of Square Roots square root of eh dot square root of b equals , square root of eh b end root
Quotient of Square Roots fraction square root of eh , over square root of c end fraction , equals , square root of eh over c end root

End ofPage 841

Table of Contents

Prentice Hall Geometry
Contents in Brief
Series Authors
Consulting Authors
Program Authors
Reviewers National
From the Authors
PowerGeometry.com
Big Ideas
Exploring Concepts
Thinking Mathematically
Active Learning
Practice Makes Perfect
Pearson Video Challenge
Acing the Test
BIG ideas
Entry-Level Assessment
Chapter 1 Tools of Geometry
Get Ready!
1-1 Nets and Drawings for Visualizing Geometry
1-2 Points, Lines, and Planes
1-3 Measuring Segments
1-4 Measuring Angles
1-5 Exploring Angle Pairs
1 Mid-Chapter Quiz
Concept Byte: Compass Designs
1-6 Basic Constructions
Concept Byte: Exploring Constructions
1-7 Midpoint and Distance in the Coordinate Plane
Review: Classifying Polygons
1-8 Perimeter, Circumference, and Area
Concept Byte: Comparing Perimeters and Areas
1 Pull It All Together
1 Chapter Review
1 Chapter Test
1 Cumulative Test Prep
Chapter 2 Reasoning and Proof
Get Ready!
2-1 Patterns and Inductive Reasoning
2-2 Conditional Statements
Concept Byte: Logic and Truth Tables
2-3 Biconditionals and Definitions
2 Mid-Chapter Quiz
2-4 Deductive Reasoning
2-5 Reasoning in Algebra and Geometry
2-6 Proving Angles Congruent
2 Pull It All Together
2 Chapter Review
2 Chapter Test
2 Cumulative Test Prep
Chapter 3 Parallel and Perpendicular Lines
Get Ready!
3-1 Lines and Angles
Concept Byte: Parallel Lines and Related Angles
3-2 Properties of Parallel Lines
3-3 Proving Lines Parallel
3-4 Parallel and Perpendicular Lines
Concept Byte: Perpendicular Lines and Planes
3-5 Parallel Lines and Triangles
Concept Byte: Exploring Spherical Geometry
3 Mid-Chapter Quiz
3-6 Constructing Parallel and Perpendicular Lines
3-7 Equations of Lines in the Coordinate Plane
3-8 Slopes of Parallel and Perpendicular Lines
3 Pull It All Together
3 Chapter Review
3 Chapter Test
3 Cumulative Test Prep
Chapter 4 Congruent Triangles
Get Ready!
4-1 Congruent Figures
Concept Byte: Building Congruent Triangles
4-2 Triangle Congruence by SSS and SAS
4-3 Triangle Congruence by ASA and AAS
Concept Byte: Exploring AAA and SSA
4 Mid-Chapter Quiz
4-4 Using Corresponding Parts of Congruent Triangles
Concept Byte: Paper-Folding Conjectures
4-5 Isosceles and Equilateral Triangles
Algebra Review: Systems of Linear Equations
4-6 Congruence in Right Triangles
4-7 Congruence in Overlapping Triangles
4 Pull It All Together
4 Chapter Review
4 Chapter Test
4 Cumulative Test Prep
Chapter 5 Relationships Within Triangles
Get Ready!
Concept Byte: Investigating Midsegments
5-1 Midsegments of Triangles
5-2 Perpendicular and Angle Bisectors
Concept Byte: Paper Folding Bisectors
5-3 Bisectors in Triangles
Concept Byte: Special Segments in Triangles
5-4 Medians and Altitudes
5 Mid-Chapter Quiz
5-5 Indirect Proof
Algebra Review: Solving Inequalities
5-6 Inequalities in One Triangle
5-7 Inequalities in Two Triangles
5 Pull It All Together
5 Chapter Review
5 Chapter Test
5 Cumulative Test Prep
Chapter 6 Polygons and Quadrilaterals
Get Ready!
Concept Byte: Exterior Angles of Polygons
6-1 The Polygon Angle-Sum Theorems
6-2 Properties of Parallelograms
6-3 Proving That a Quadrilateral Is a Parallelogram
6-4 Properties of Rhombuses, Rectangles, and Squares
6-5 Conditions for Rhombuses, Rectangles, and Squares
6-6 Trapezoids and Kites
6 Mid-Chapter Quiz
Algebra Review: Simplifying Radicals
6-7 Polygons in the Coordinate Plane
6-8 Applying Coordinate Geometry
Concept Byte: Quadrilaterals in Quadrilaterals
6-9 Proofs Using Coordinate Geometry
6 Pull It All Together
6 Chapter Review
6 Chapter Test
6 Cumulative Test Prep
Chapter 7 Similarity
Get Ready!
7-1 Ratios and Proportions
Algebra Review: Solving Quadratic Equations
7-2 Similar Polygons
Concept Byte: Fractals
7-3 Proving Triangles Similar
7 Mid-Chapter Quiz
7-4 Similarity in Right Triangles
Concept Byte: The Golden Ratio
Concept Byte: Exploring Proportions in Triangles
7-5 Proportions in Triangles
7 Pull It All Together
7 Chapter Review
7 Chapter Test
7 Cumulative Test Prep
Chapter 8 Right Triangles and Trigonometry
Get Ready!
Concept Byte: The Pythagorean Theorem
8-1 The Pythagorean Theorem and Its Converse
8-2 Special Right Triangles
Concept Byte: Exploring Trigonometric Ratios
8-3 Trigonometry
8 Mid-Chapter Quiz
Concept Byte: Measuring From Afar
8-4 Angles of Elevation and Depression
Concept Byte: Law of Sines and Law of Cosines
8-5 Vectors
8 Pull It All Together
8 Chapter Review
8 Chapter Test
8 Cumulative Test Prep
Chapter 9 Transformations
Get Ready!
9-1 Translations
Concept Byte: Paper Folding and Reflections
9-2 Reflections
9-3 Rotations
Concept Byte: Tracing Paper Transformations
9-4 Symmetry
9 Mid-Chapter Quiz
9-5 Dilations
Concept Byte: Transformations Using Vectors and Matrices
9-6 Compositions of Reflections
Concept Byte: Frieze Patterns
Concept Byte: Creating Tessellations
9-7 Tessellations
9 Pull It All Together
9 Chapter Review
9 Chapter Test
9 Cumulative Test Prep
Chapter 10 Area
Get Ready!
Concept Byte: Transforming to Find Area
10-1 Areas of Parallelograms and Triangles
10-2 Areas of Trapezoids, Rhombuses, and Kites
10-3 Areas of Regular Polygons
10-4 Perimeters and Areas of Similar Figures
10 Mid-Chapter Quiz
10-5 Trigonometry and Area
10-6 Circles and Arcs
Concept Byte: Circle Graphs
Concept Byte: Exploring the Area of a Circle
10-7 Areas of Circles and Sectors
Concept Byte: Exploring Area and Circumference
10-8 Geometric Probability
10 Pull It All Together
10 Chapter Review
10 Chapter Test
10 Cumulative Test Prep
Chapter 11 Surface Area and Volume
Get Ready!
11-1 Space Figures and Cross Sections
Concept Byte: Perspective Drawing
Algebra Review: Literal Equations
11-2 Surface Areas of Prisms and Cylinders
11-3 Surface Areas of Pyramids and Cones
11 Mid-Chapter Quiz
11-4 Volumes of Prisms and Cylinders
Concept Byte: Finding Volume
11-5 Volumes of Pyramids and Cones
11-6 Surface Areas and Volumes of Spheres
Concept Byte: Exploring Similar Solids
11-7 Areas and Volumes of Similar Solids
11 Pull It All Together
11 Chapter Review
11 Chapter Test
11 Cumulative Test Prep
Chapter 12 Circles
Get Ready!
12-1 Tangent Lines
Concept Byte: Paper Folding With Circles
12-2 Chords and Arcs
12-3 Inscribed Angles
12 Mid-Chapter Quiz
Concept Byte: Exploring Chords and Secants
12-4 Angle Measures and Segment Lengths
12-5 Circles in the Coordinate Plane
12-6 Locus: A Set of Points
12 Pull It All Together
12 Chapter Review
12 Chapter Test
End-of-Course Assessment
Skills Handbook
Using a Ruler and Protractor
Classifying Triangles
Measurement Conversions
Measurement, Rounding Error, and Reasonableness
The Effect of Measurement Errors on Calculations
Squaring Numbers and Finding Square Roots
Evaluating and Simplifying Expressions
Simplifying Ratios
Absolute Value
The Coordinate Plane
Solving and Writing Linear Equations
Percents
Probability
Reference
Table 1 Measures
Table 2 Formulas
Table 3 Reading Math Symbols
Table 4 Properties of Real Numbers
Postulates, Theorems, and Constructions
Visual Glossary
Selected Answers
Index
Acknowledgments
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