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1 Understanding Geometry Table of Contents TABLE OF CONTENTS Why Use This Book...ii Teaching Suggestions...vi About the Author...vi Student Introduction...vii Dedication...viii Chapter 1 Fundamentals of Geometry Reasoning... Pages 1-7 Geometry Notation Geometry Notation Practice... 3 All About Lines... 4 All About Planes... 5 Build It!... 6 Three Planes in Space... 7 Chapter 2 Uncovering All the Angles... Pages 8-20 Types of Angles... 8 Angles Activity... 9 Triangle Activity Triangle Practice Quadrilateral Activity Complementary and Supplementary Angles Vertical Angles Angles Puzzle Angles Puzzle Parallel Lines and Transversals I Parallel Lines and Transversals II Puzzle 3 - The 20-Angle Problem Chapter 3 Triangle Properties... Pages Sum of Angle Measures Exterior Angle Exploration Sum of Two Sides Triangle Properties Practice Triangle Opposite Property Properties of Equilateral and Isosceles Triangles Algebra and Geometry Chapter 4 The Pythagorean Theorem... Pages Exploration - What Is a Theorem? Pythagorean Triples Pythagorean Triples Practice Using the Pythagorean Theorem Pythagorean Theorem Applications Solving Multi-Step Right Triangle Problems Special Right Triangles - An Introduction The Critical Thinking Co iii

2 Understanding Geometry Table of Contents TABLE OF CONTENTS (Cont.) Chapter 5 Uncovering All Polygons... Pages Polygons Investigation of Angles and Sides Polygons Angle Exploration Summary of Polygon Properties Diagonal Exploration The Handshake Problem Chapter 6 Quadrilateral and Parallelogram Properties... Pages The Quadrilateral Family Venn Diagram Activity Parallelogram Discovery Parallelogram Properties Working With Parallelograms Experimenting With Parallelograms A Look at Trapezoids Kite! Kite! Kite! Critical Thinking About Quadrilaterals Midsegment Investigation Using Algebra to Solve Quadrilateral Problems Quadrilateral Matching Chapter 7 Metric Geometry... Pages Perimeter and Circumference Pi (π) Investigation Perimeter and Circumference Activity Archimedes Idea for Approximating Pi Area of Parallelograms Parallelogram Area Activity Area of Triangles and Trapezoids Discovering the Area of a Trapezoid Area of a Trapezoid Area of Circles The Arbelos Problem Chapter 8 Geometric Constructions...Pages A Geometric Construction How to Bisect an Angle How to Copy an Angle How to Construct a Perpendicular Bisector to a Line Finding the Median in a Triangle How To Construct a Line Parallel To Another Line Geometric Construction Review Problem-Solving With Geometric Constructions Problem-Solving With Impossible Constructions iv 2012 The Critical Thinking Co

3 Understanding Geometry Table of Contents Chapter 9 The Geometry of Three Dimensional Shapes... Pages D Shapes - Prisms The Volume of Cylinders Volumes of Pyramids and Cones Turn Up the Volume! Volume of a Sphere Surface Area of Prisms Surface Area of a Cylinder Activity Finding Surface Area Euler s Formula Chapter 10 Symmetry and Transformations... Pages What is Vertical, Horizontal, and Point Symmetry? Transformations - Reflections Transformations - Translations Transformations - Rotations Transformations - Dilations Glide Reflections and Compositions Tessellations Chapter 11 Proving Triangles Congruent... Pages Introduction to Proofs - Congruency SSS Activity SAS Activity ASA and AAS Activities SSA Activity The Essence of a Good Geometric Proof Picture, Statement, and Reason Finding Congruent Triangles Two Column Proofs Investigate Hypotenuse - Leg Theorem Similar Figures and Introduction to Similarity Proofs Proving Triangles Similar Test Your Reasoning Skills Chapter 12 Coordinate Geometry... Pages What is the Slope of a Line Slope Formula How to Write an Equation of a Line The Midpoint Formula The Distance Formula Review Your Formulas Introduction to Coordinate Proofs Glossary... Pages Answers... Pages The Critical Thinking Co v

4 UndersTanding geometry Uncovering All the Angles Angles Puzzle 1 Use your thinking skills to find the missing angles and record in degrees below Figures are not to scale, so do not measure e 46 f g h i d 83 c b a m l j k 60 a b c d h i j k BBP Math Reasoning Geometry Book.indb 16 e f g l m 2012 The Critical Thinking Co /29/2012 3:20:48 PM

5 UndersTanding geometry Quadrilateral and Parallelogram Properties Quadrilateral Matching Match the quadrilateral with its properties You may use an answer more than once 1 Rectangle 2 Rhombus 3 Parallelogram 4 Kite 5 Trapezoid 6 Isosceles Trapezoid a b c d e f g Diagonals Opposite sides Opposite angles Consecutive angles are supplementary Diagonals are perpendicular All sides All angles h Sum of the angle measures equals Square The Critical Thinking Co

6 Geometric Constructions UndersTanding geometry Chapter 8 - Geometric Constructions A Geometric Construction A geometric construction is a construction done with only a compass and a straightedge Of course, a good pencil is important In a classic geometric construction, you are not allowed to measure or erase If you make a mistake, it s best you start over Before we problem solve and do some fun thinking problems with constructions, it is a good idea to review how to do these constructions Bisecting an angle Note Copying an angle Constructing a perpendicular bisector to a line Constructing a perpendicular bisector to a line from a point on the not on the line Constructing a line parallel to another line through a given point Constructing some popular polygons My constructions are actual drawings since they were done on the computer, but if you follow the directions for each of these, you will make perfect constructions Geometric constructions are used by engineers and designers They will help you see the beauty of geometry For fun, create your own design with a compass and a straightedge Try to fill up a page of paper with your own design 2012 The Critical Thinking Co BBP Math Reasoning Geometry Book.indb /29/2012 3:22:13 PM

7 Geometric Constructions UndersTanding geometry Problem-Solving With Geometric Constructions (Cont.) Plato ( BC), a famous Greek philosopher and 7 Using only a compass, given segment AB, construct segments that are 2, 3, and 4 times bigger than AB A 8 B Ancient Greek philosopher drawing his theorem in the sand Remember You cannot use a straightedge so you can only show the points that mark such distances I lost the center of My Circle construction below If you draw a circle and forget where the center is, draw two non-parallel chords Construct the perpendicular bisectors of each chord and where the perpendicular bisectors meet is your missing center Try it! Hint Once you find the center, make two triangles Use the center as one point, and the points where the chord and the circle intersect to help you see what is happening mathematician, discovered that many constructions can be made with only a compass and no straightedge Those constructions are now called Mascheroni constructions in honor of Italian mathematician Lorenzo Mascheroni ( ) who wrote a book called The Geometry of Compasses Why does this work? Explain your thinking 2012 The Critical Thinking Co BBP Math Reasoning Geometry Book.indb /29/2012 3:22:37 PM

8 UndersTanding geometry Proving Triangles Congruent Picture, Statement, and Reason Refer to the picture and write a reason for each statement Statement Reason 1 B a AB CB a Given (The fact is stated in the picture or in the information ) A 2 R 3 D T V C W b A C a TV TV a 1 2 b _ a _ a 2 _ The Critical Thinking Co

9 UndersTanding geometry Proving Triangles Congruent Two Column Proofs (Cont.) Refer to the picture and fill in the missing reasons or statements for each proof Statement Reason Proof 3 1 _ 1 Given A R C Given: RC AB and AC BC Prove: 1 2 Proof 4 A B B 2 _ 3 ACR and BCR are right angles 4 _ 5 RC RC 6 ACR BCR _ 2 _ 3 _ 2 Given 3 _ 4 All right angles are congruent 5 _ 6 _ 7 _ 8 _ 1 Given 2 Given 3 Given W R L H 4 RL RL 4 _ Given: WR HL, W H, ALW BRH 5 WL HR 5 _ Prove: AWL BHR and A B 6 AWL BHR 6 _ 7 A B 7 _ 2012 The Critical Thinking Co

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