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DG4GP_905_13.qxd 12/27/06 10:28 AM Page 53 CHAPTER 13 Geometry as a Mathematical System Content Summary Having experienced all the concepts of a standard geometry course, students are ready to examine the framework of the geometry knowledge they have built. Students now review and deepen their understanding of those concepts by proving some of the most important conjectures in the context of a logical system, starting with the premises of geometry. Premises and Theorems A complete deductive system must begin with some assumptions that are clearly stated and, ideally, so obvious that they need no defense. Chapter 13 begins by laying out its assumptions: properties of arithmetic and equality, postulates of geometry, and a definition of congruence for angles and line segments. These basic assumptions are called premises. Everything else builds on these premises. Next, students develop proofs of their conjectures concerning triangles, quadrilaterals, circles, similarity, and coordinate geometry. Once a conjecture has been proved, it is called a theorem. Each step of a proof must be supported by a premise or a previously proved theorem. Developing a Proof Developing a proof is more art than science. Mathematicians don’t sit down and write a proof from beginning to end, so encourage your student not to expect to do so. Proofs require thought and creativity. Generally, the student will start by writing down what’s given and what’s to be shown—the beginning and end of the proof. Then, perhaps using diagrams, the student will restate these first and last statements in several ways, looking for an idea of how to get from one to the other logically. You might remind your student about the reasoning strategies that can help in planning a proof: ● ● ● ● ● ● Draw a labeled diagram and mark what you know Represent a situation algebraically Apply previous conjectures and definitions Break a problem into parts Add an auxiliary line Think backward Questions you may ask your student include “What can you conclude from the given statements?” and “What’s needed to prove the last statement?” Seeing connections, the student can develop a plan, perhaps expressed using flowcharts. There are several ways to express the plan, and your student may use more than one. Then he or she can write the proof, being careful to check the reasoning. A good way to be careful about details is to write a two-column proof, with statements in the first column and reasons in the second. Your student might find gaps in his or her reasoning and have to go back to the planning stage. If there doesn’t seem to be any way to prove the statement, you might suggest indirect reasoning, in which the student proves that the negation of the theorem is false. It then follows that the theorem must be true. ©2008 Kendall Hunt Publishing (continued) Discovering Geometry: A Guide for Parents 53 DG4GP_905_13.qxd 12/27/06 10:28 AM Page 54 Chapter 13 • Geometry as a Mathematical System (continued) Summary Problem Draw a diagram that gives a family tree for all triangle theorems that appear in the exercises for Lesson 13.3. Include postulates and theorems, but not definitions or properties. Questions you might ask in your role as student to your student: ● ● Can you build on the tree on page 707? What do the arrows represent in words? Sample Answers A family tree shows how theorems support each other. A theorem may rely on several theorems, each of which relies on other theorems, and so on, all the way up to the postulates of geometry. The top of the family tree should be all postulates, and all the arrows should flow down from there. The diagrams can get quite complex. Don’t worry too much about neatness or completeness; the goal is to see how the structure can be built while reviewing the theorems. Here is the complete tree. SAS Postulate ASA Postulate Parallel Postulate CA Postulate Linear Pair Postulate Angle Addition Postulate SSS Postulate VA Theorem AIA Theorem Perpendicular Bisector Theorem Isosceles Triangle Theorem Triangle Sum Theorem Third Angle Theorem SAA Congruence Theorem Converse of Isosceles Triangle Theorem Angle Bisector Theorem Converse of Angle Bisector Theorem Angle Bisector Concurrency Theorem Angle Bisectors to Congruent Sides Theorem Altitudes to Congruent Sides Theorem Medians to Congruent Sides Theorem Perpendicular Bisector Concurrency Theorem 54 Discovering Geometry: A Guide for Parents Converse of Perpendicular Bisector Theorem ©2008 Kendall Hunt Publishing DG4GP_905_13.qxd 12/27/06 10:28 AM Page 55 Chapter 13 • Review Exercises Name Period Date 1. (Lesson 13.1) Name the property that supports each statement: CD and CD EF , then AB EF . a. If AB CD , then AB CD. b. If AB 2. (Lessons 13.2, 13.3) In Lesson 13.2, Example B, the Triangle Sum Theorem is proved with a flowchart proof. Rewrite this proof using a two-column proof. Given: 1, 2, and 3 are the three angles of ABC Show: m1 + m2 + m3 180° 3. (Lessons 13.2, 13.4) Answer the following questions for the statement, “The diagonals of an isosceles trapezoid are congruent.” a. Task 1: Identify what is given and what you must show. b. Task 2: Draw and label a diagram to illustrate the given information. c. Task 3: Restate what is given and what you must show in terms of your diagram. 4. (Lesson 13.6) Write a proof for the Parallel Secants Congruent Arcs Theorem: Parallel lines intercept congruent arcs on a circle. 5. (Lesson 13.7) Write a proof for the Corresponding Altitudes Theorem: If two triangles are similar, then corresponding altitudes are proportional to the corresponding sides. ©2008 Kendall Hunt Publishing Discovering Geometry: A Guide for Parents 55 DG4GP_905_13.qxd 12/27/06 10:28 AM Page 56 SOLUTIONS TO CHAPTER 13 REVIEW EXERCISES 1. a. Transitive Property b. Definition of Congruence 2. K 4 C 2 5 1 A 3 B Statement Reason 1, 2, and 3 of ABC Given AB Construct KC Parallel Postulate 1 4; 3 5 Alternate Interior Angles Theorem m1 m4; m3 m5 Definition of Congruence m4 m2 mKCB Angle Addition Postulate KCB and 5 are supplementary Linear Pair Postulate mKCB m5 180° Definition of Supplementary m4 m2 + m5 180° Substitution Property of Equality m1 m2 m3 180° Substitution Property of Equality 3. a. Given: Isosceles trapezoid Show: Diagonals are congruent b. A D B C DC ; AD BC c. Given: AB Show: AC DB 4. B C A D DC Given: AB Show: AD BC 56 Discovering Geometry: A Guide for Parents Statement Reason DC AB Given Construct AC Line Postulate DCA BAC Alternate Interior Angles Theorem mDCA mBAC Definition of Congruence 1 mBAC 2 Multiplication Property of Equality 1 mDCA 2 1mDCA mAD 2 Inscribed Angle Theorem 1mBAC mBC 2 Inscribed Angle Theorem mA A D mBC Substitution Property of Equality BC AD Definition of Congruence ©2008 Kendall Hunt Publishing DG4GP_905_13.qxd 12/27/06 10:28 AM 5. Page 57 J C K B L R P D and JR Given: CBD JKL; Altitudes CP C P CB Show: JR JK Statement Reason and JR CBD JKL; Altitudes CP Given BD ; JR KL CP Definition of Altitude CPB and JRK are right angles Definition of Perpendicular CPB JRK Right Angles are Congruent Theorem CBD JKL Corresponding Angles of Similar Triangles are Congruent CBP JKR AA Similarity Postulate C P C B JR JK Corresponding Sides of Similar Triangles are Proportional ©2008 Kendall Hunt Publishing Discovering Geometry: A Guide for Parents 57