Table of Contents Table of Contents Introduction to the Math Practice Series...1 Common Mathematics/Geometry Symbols and Terms...2 Chapter 1: Introduction To Geometry...13 Shapes, Congruence, Similarity, and Symmetry...13 Practice...16 Challenge Problems...18 Checking Progress...19 Chapter 2: Coordinate and Non-coordinate Geometry...21 Midpoint Formulas, Slope, Equations, Distance...21 Practice...27 Challenge Problems...31 Checking Progress...32 Chapter 3: Angles...33 Measuring and Classifying Angles...33 Practice...38 Challenge Problems...40 Checking Progress...43 Chapter 5: Triangles...51 Parts of a Triangle, Similar and Congruent Triangles, Properties of Triangles, Perimeter and Area...51 Practice...57 Challenge Problems...60 Checking Progress...62 Chapter 6: Polygons and Quadrilaterals...63 Polygons and Quadrilaterals...63 Practice...71 Challenge Problems...73 Checking Progress...75 Chapter 7: Circles...76 Radius, Diameter, Circumference and Area, Chords, Tangents, Secants, Arcs, Inscribed Angles, Finding the Equation of a Circle...76 Practice...81 Challenge Problems...87 Geometry Check-Up...88 Practice Answer Keys...93 Chapter 4: Patterns and Reasoning...44 Patterns, Structure of Geometry, Conditional Statements, and Deductive Reasoning...44 Practice...46 Challenge Problems...48 Checking Progress...50 Check-Up Answer Keys...118 References...122 Mark Twain Media, Inc., Publishers iii
Introduction to the Math Practice Series Introduction to the Math Practice Series The Math Practice series of books will introduce students in middle school and high school to the course topics of Pre-algebra, Algebra, Algebra II, and Geometry. The content of all of the practice books are aligned with the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics. (NCTM 2000) This series is written for classroom teachers, parents, families, and students. The practice books in this series can be used as a full unit of study or as individual lessons to supplement textbooks or curriculum programs. Parents and students can use this series as an enhancement to what is being done in the classroom or as a tutorial at home. Students will be given a basic overview of the concepts, examples, practice problems, and challenge problems using the concepts introduced in the section. At the end of each section, there will be a set of problems to check progress on the concepts and a challenge set of problems over the whole section. At the end of the book, there will be problems for each section that can be used for assessment. According to the Mathematics Education Trust and NCTM, new technologies require the fundamentals of algebra and algebraic thinking be a part of the background for all citizens. These technologies also provide opportunities to generate numerical examples, graph data, analyze patterns, and make generalizations. An understanding of algebra is also important because business and industry require higher levels of thinking and problem solving. NCTM also suggests that understanding geometry, including the characteristics and properties of two- and three-dimensional shapes, spatial relationships, symmetry, the use of visualization, and spatial reasoning, can also be used in solving problems. The NCTM Standards suggest content and vocabulary are necessary, but of equal importance are the processes of mathematics. The process skills described in the Standards include: problem solving, reasoning, communication, and connections. The practice books in this series will address both the content and processes of algebra and algebraic thinking and geometry. This practice book, Geometry Practice, will help students practice geometric concepts. Mark Twain Media, Inc., Publishers 1
Basic Overview: Shapes, Congruence, Similarity, and Symmetry Polygons have names determined by the number of sides: triangles have three sides, quadrilaterals have four sides, pentagons have five sides, hexagons have six sides, and octagons have eight sides. The polygons above are in one plane or surface. Objects can also be solid and three-dimensional and be in more than one plane. Two figures are congruent if they have the same shape and the same size. If two figures are similar, their sides are proportional and corresponding angles are congruent. Geometric figures are similar if figures have the same shape. They may or may not have the same size. If a figure can be divided into two parts, each of which is a mirror image of the other, it has line symmetry, or reflection symmetry. A figure in a plane has a line of symmetry if it can be mapped onto itself by a reflection. The lines of symmetry can, but need not be, vertical or horizontal. Figures can also have rotational symmetry if the figure can be mapped on itself by rotating the figure. Translational symmetry is created when an object is moved without rotating or reflecting it. Glide reflection symmetry is a transformation that consists of a translation followed by a reflection with the line of translation also essentially serving as the line of symmetry. Examples of Shapes, Congruence, Similarity, and Symmetry Examples of Polygons in One Plane: Examples of Three-dimensional Shapes: Mark Twain Media, Inc., Publishers 13
(cont.) Examples of Congruence: Examples of Similarity: Examples of Symmetry: Line Symmetry: Each dashed line can be a line of symmetry. Mark Twain Media, Inc., Publishers 14
R R Geometry Practice (cont.) Rotational Symmetry: Rotating 180 Rotating 120 TOP BOTTOM A C BOTTOM TOP C B B A Translational Symmetry: B B Glide Reflection Symmetry: R Mark Twain Media, Inc., Publishers 15
(cont.) Practice: Shapes, Congruence, Similarity, and Symmetry Directions: Draw an example of the polygons below. Explain why your drawing shows an example of the polygon. 1. Octagon Explanation: Drawing: 2. Hexagon Explanation: Drawing: 3. Triangle Explanation: Drawing: 4. Quadrilateral Explanation: Drawing: Mark Twain Media, Inc., Publishers 16
(cont.) 5. Which figures are similar and which are congruent? Similar: Congruent: A. B. C. D. Directions: Divide the shaded regions into two congruent parts. Show a different possible solution in each of the regions below. 6. 7. 8. 9. Mark Twain Media, Inc., Publishers 17
Practice Answer Keys Practice Answer Keys Chapter 1: Practice: Shapes, Congruence, Similarity, and Symmetry (pages 16 17) 1. Octagon Drawing: Drawings will vary, but the polygon must have eight sides. Explanation: The polygon has eight sides. 2. Hexagon Drawing: Drawings will vary, but the polygon must have six sides. Explanation: The polygon has six sides. 3. Triangle Drawing: Drawings will vary, but the polygon must have three sides. Explanation: The polygon has three sides. 4. Quadrilateral Drawing: Drawings will vary, but the polygon must have four sides. Explanation: The polygon has four sides. 5. A and C are congruent; B and D are similar; A and C are also similar. Possible solutions include: 6. 7. 8. 9. Mark Twain Media, Inc., Publishers 93