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Coach is the leader in standards-based, state-customized instruction for grades K–12 in English language arts, mathematics, science, and social studies. Our student texts deliver everything you need to meet your state standards and prepare your class for grade-level success! Coach lessons have just what you’re looking for: ✔✔ Easy-to-follow, predictable lesson plans Florida Coach, Standards-Based Instruction, Geometry Standards-Based Curriculum Support! ✔✔ Focused instruction with modeled examples ✔✔ Guided practice with hints and support ✔✔ Higher-level thinking activities PLUS Chapter Reviews that target assessed skills Used by more students in the U.S. than any other state-customized series, Coach books are proven effective. Triumph Learning has been a trusted name in educational publishing for more than 40 years, and we continue to work with teachers and administrators to keep our books up to date— improving test scores and maximizing student learning. Please visit our Web site for detailed product descriptions of all our instructional materials, including sample pages and more. www.triumphlearning.com Phone: (800) 221-9372 • Fax: (866) 805-5723 • E-mail: [email protected] 188FL_Geo_HS_SE_Cvr.indd 1 188FL This book is printed on paper containing a minimum of 10% post-consumer waste. Developed in Consultation with Florida Educators 12/3/09 11:02:04 AM 188FL_Mth_Geo_SE_FM_PDF.indd Page 3 9/26/09 9:34:58 AM u-s008 /Volumes/109/TL00110/work%0/indd%0/SE/FM Table of Contents Benchmarks Letter to the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Test-Taking Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Florida Mathematics Standards Correlation Chart . . . . . . . . . . . . . . . . . 8 Chapter 1 Mathematical Reasoning and Logic . . . . . . . . . . . . . . . . 13 Lesson 1 Introduction to Euclidean Geometry . . . . . . . . . 14 MA.912.G.8.1 Lesson 2 Converses, Inverses, and Contrapositives . . . . . . . . . . . . . . . . . . . . . . . . . 18 MA.912.D.6.2 Geometric Proofs . . . . . . . . . . . . . . . . . . . . . . . 24 MA.912.D.6.4, MA.912.G.8.2, Lesson 3 MA.912.G.8.4, MA.912.G.8.5 Lesson 4 Logical Equivalence and Validity . . . . . . . . . . . 29 MA.912.D.6.3, MA.912.D.6.4, MA.912.G.8.3 Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 2 Points, Lines, Angles, and Planes . . . . . . . . . . . . . . . . . . 39 Lesson 5 Distance and Midpoint . . . . . . . . . . . . . . . . . . . 40 MA.912.G.1.1 Lesson 6 Constructing Congruent Segments and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 MA.912.G.1.2, MA.912.G.8.6 Constructing Bisectors and Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . 50 MA.912.G.1.2, MA.912.G.4.2, Lesson 7 MA.912.G.8.6 Lesson 8 Parallel Lines and Transversals. . . . . . . . . . . . . 56 MA.912.G.1.3 Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter 3 Polygons and Quadrilaterals. . . . . . . . . . . . . . . . . . . . . . . 69 Lesson 9 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 MA.912.G.2.1, MA.912.G.2.2 Lesson 10 Solving Problems with Congruent and Similar Polygons . . . . . . . . . . . . . . . . . . . . 75 MA.912.G.2.3, MA.912.G.8.2 Lesson 11 Transformations, Congruence, Similarity, and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 81 MA.912.G.2.4 Lesson 12 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 MA.912.G.2.4 Lesson 13 Perimeter and Area of Polygons . . . . . . . . . . . . 93 MA.912.G.2.5 Duplicating any part of this book is prohibited by law. 3 P DF P ass 188FL_Mth_Geo_SE_FM_PDF.indd Page 4 9/26/09 9:34:59 AM u-s008 /Volumes/109/TL00110/work%0/indd%0/SE/FM Lesson 14 Changes in the Dimensions of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 MA.912.G.2.7 Lesson 15 Special Quadrilaterals . . . . . . . . . . . . . . . . . . . 102 MA.912.G.3.1, MA.912.G.3.2 Lesson 16 Proving Properties of Quadrilaterals . . . . . . . . 109 MA.912.G.3.3 Lesson 17 Proving Quadrilateral Theorems . . . . . . . . . . . 115 MA.912.G.3.4 Chapter 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Chapter 4 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Lesson 18 Classifying and Constructing Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 MA.912.G.4.1, MA.912.G.8.6 Lesson 19 Points of Concurrency in Triangles . . . . . . . . . 132 MA.912.G.4.2, MA.912.G.8.6 Lesson 20 Constructing Congruent Triangles . . . . . . . . . 139 MA.912.G.4.3, MA.912.G.8.6 Lesson 21 Solving Problems with Congruent and Similar Triangles. . . . . . . . . . . . . . . . . . . . 144 MA.912.G.4.4, MA.912.G.4.5, MA.912.G.8.2 Lesson 22 Proving Congruence and Similarity . . . . . . . . 149 MA.912.G.4.5, MA.912.G.4.6 Lesson 23 Triangle Inequality Theorems . . . . . . . . . . . . . 156 MA.912.G.4.7 Lesson 24 Right Triangle Theorems . . . . . . . . . . . . . . . . . 161 MA.912.G.5.1, MA.912.G.5.2 Lesson 25 Pythagorean Theorem . . . . . . . . . . . . . . . . . . 165 MA.912.G.5.1, MA.912.G.5.4 Lesson 26 Trigonometric Ratios and Special Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . 169 MA.912.G.5.3, MA.912.G.5.4, MA.912.T.2.1 Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Chapter 5 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Lesson 27 Parts of a Circle . . . . . . . . . . . . . . . . . . . . . . . 182 MA.912.G.6.2 Lesson 28 Measures of Arcs and Related Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 MA.912.G.6.4 Lesson 29 Solving Real-World Problems with Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 MA.912.G.6.5, MA.912.G.8.2 Lesson 30 Equations of Circles . . . . . . . . . . . . . . . . . . . . 197 MA.912.G.6.6, MA.912.G.6.7 Chapter 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4 Duplicating any part of this book is prohibited by law. P DF P ass 188FL_Mth_Geo_SE_FM_PDF.indd Page 5 9/26/09 9:34:59 AM u-s008 /Volumes/109/TL00110/work%0/indd%0/SE/FM Table of Contents Chapter 6 Polyhedra and Other Solids . . . . . . . . . . . . . . . . . . . . . . 207 Lesson 31 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 MA.912.G.7.1, MA.912.G.7.2 Lesson 32 Nets of Polyhedra . . . . . . . . . . . . . . . . . . . . . . 215 MA.912.G.7.1 Lesson 33 Spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 MA.912.G.7.4 Lesson 34 Surface Area and Volume . . . . . . . . . . . . . . . . 226 MA.912.G.7.5 Lesson 35 Changes in the Dimensions of Solids . . . . . . . 232 MA.912.G.7.7 Lesson 36 Congruent and Similar Solids . . . . . . . . . . . . . 237 MA.912.G.7.6 Chapter 6 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Duplicating any part of this book is prohibited by law. 5 P DF P ass 188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:132 19 9/26/09 1:27:30 PM u-s008 /Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04 Points of Concurrency in Triangles MA.912.G.4.2, MA.912.G.8.6 If three or more lines are concurrent, then they intersect at the same point, which is called the point of concurrency. Knowing about four points of concurrency in a triangle can help you solve many different types of problems in mathematics and in the real world. The incenter is the point at which the angle bisectors of a triangle intersect. An angle bisector is a line or ray that divides an angle in half. The incenter, as its name suggests, is also the center of the circle that can be inscribed in the triangle. angle bisector incenter angle bisector angle bisector The orthocenter is the point at which the three altitudes of a triangle intersect. An altitude is the shortest distance between a vertex of a triangle and the opposite side (even if the opposite side needs to be extended to find that shortest distance). The shortest distance is always perpendicular to the opposite side. altitude altitude orthocenter altitude 132 Duplicating any part of this book is prohibited by law. P DF P ass 188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:133 9/26/09 1:27:31 PM u-s008 /Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04 Lesson 19: Points of Concurrency in Triangles The circumcenter is the point at which the three perpendicular bisectors of the sides of a triangle intersect. A perpendicular bisector is a line that is perpendicular to a line segment and passes through the midpoint of the line segment. The circumcenter is also the center of the circle that could be circumscribed around the triangle such that all three vertices of the triangle are on the circle. Also, the circumcenter is equidistant (the same distance) from each of the triangle’s vertices. perpendicular bisector perpendicular bisector circumcenter perpendicular bisector The centroid is the point at which the medians of a triangle intersect. The median of a triangle is a line segment drawn from a vertex of a triangle to the midpoint of the opposite side. The centroid divides each median into two segments, and the longer segment is twice as long as the shorter segment. The centroid is also the center of gravity of the triangle. midpoint midpoint centroid midpoint You can construct each of these points of concurrency, using tools such as a compass. Duplicating any part of this book is prohibited by law. 133 P DF P ass 188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:134 9/26/09 1:27:31 PM u-s008 /Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04 EXAMPLE 1 Construct the incenter of XYZ shown below. X Z STRATEGY STEP 1 Y Use a compass to draw an angle bisector for each vertex of the angle. Then find the point at which those angle bisectors intersect. Construct an angle bisector for vertex X. Draw an arc from vertex X so that the arc crosses both sides of the angle. Plot points where the arc crosses each side. Draw an arc from each of the points you plotted into the triangle’s interior, using the same compass opening each time. Plot a point where the two arcs meet. Finally, draw a line segment from vertex X through the intersection of the two arcs to the opposite side. This is an angle bisector. X X X Y Z Z Y Z Y STEP 2 Using the steps described above, construct angle bisectors for vertices Y and Z. STEP 3 Label the point where the angle bisectors intersect as the incenter. Note: You can inscribe a circle in the triangle, using the incenter as its center. X X incenter Z SOLUTION 134 incenter Y Z Y The construction in Step 3 shows the incenter of XYZ. Duplicating any part of this book is prohibited by law. P DF P ass 188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:135 9/26/09 1:27:31 PM u-s008 /Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04 Lesson 19: Points of Concurrency in Triangles EXAMPLE 2 Evansville A county wants to build a new public library. They want to build it at a location that is equidistant from the centers of three towns—Derry, Evansville, and Franklinton—shown on the map below. Which point of concurrency could you draw on this map to show the ideal location for the library? STRATEGY STEP 1 Franklinton Derry Consider which point of concurrency is the same distance from all the vertices of a triangle. Construct this point. Determine which point of concurrency is equidistant from the vertices. The circumcenter of a triangle is the center of a circle which has all three vertices of the triangle on the circle. The distance from each vertex to the circumcenter is equal to a radius of the circle. Each radius is the same length, so each vertex will be equidistant from the circumcenter. STEP 2 Construct the circumcenter of the triangle. Construct the perpendicular bisector of the segment joining Derry and Evansville. Evansville Franklinton Derry Repeat this process for the other two sides. radii Evansville radius Franklinton Derry SOLUTION circumcenter The circumcenter of the triangle is the ideal location for the library. Duplicating any part of this book is prohibited by law. 135 P DF P ass 188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:136 9/26/09 1:27:32 PM u-s008 /Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04 EXAMPLE 3 ___ Point T is the centroid of MNP. If TR 6 cm, what is the length of MT? N Q M STRATEGY STEP 1 T R P S Use what you know about medians and centroids. What do you know about medians and centroids? The centroid of a triangle divides each median into two segments. The longer segment is twice the length of the shorter segment. ___ STEP 2 Find the length of MT. ____ Centroid T divides the median MR into two segments. ___ ___ The shorter segment, TR, is 6 cm long, so the longer segment, MT, will be twice that length. ___ MT 6 2 12 cm SOLUTION ____ The length of MT is 12 cm. COACHED EXAMPLE ___ ___ ___ ___ ___ ___ In the diagram below, AH BC, BJ AC, and CK AB. What point of concurrency is point M ? B K H M A J C THINKING IT THROUGH The line segment from the vertex of a triangle that is perpendicular to the opposite side is a(n) of the triangle. intersect at the point of concurrency called the The three ___ ___ Since point M shows the intersection of . ___ AH, BJ, and CK, it is the of ABC. 136 Duplicating any part of this book is prohibited by law. P DF P ass 188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:137 9/26/09 1:27:32 PM u-s008 /Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04 Lesson 19: Points of Concurrency in Triangles Lesson Practice Choose the correct answer. 1. In the diagram below, lines DE, FG, and HJ are perpendicular bisectors of the sides of ABC. 3. ____ If MD 20 units, what is the length of DA? A. 10 units B. 20 units B C. 30 units F D D. 40 units H K A 4. G J Point Z is the orthocenter of TUV. C E U X Which describes point K ? Y Z A. centroid B. circumcenter C. incenter T D. orthocenter Point D is the centroid of KLM below. K D M 2. V ____ Which term accurately describes TY ? Use this diagram for questions 2 and 3. C W A altitude B angle bisector C median of triangle D perpendicular bisector A B L _____ If KC 22 units, what is the length of CM ? A. 11 units B. 20 units C. 22 units D. 44 units Duplicating any part of this book is prohibited by law. 137 P DF P ass 188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:138 9/26/09 1:27:33 PM u-s008 Use the map below for questions 5 and 6. Briarville Lo vie Ma in Roa d ng Avon 5. Peach Road w 6. /Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04 A school is equidistant from each of the three towns shown on the map. Which could be used to find the school’s location? A. centroid Ro ad B. circumcenter Claytown D. orthocenter C. incenter A museum is equidistant from each of the three roads shown on the map. Which could be used to find the museum’s location? A. centroid B. circumcenter C. incenter D. orthocenter 138 Duplicating any part of this book is prohibited by law. P DF P ass