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The Premises of Geometry

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The Premises of Geometry DG4CL_895_13.qxd 12/4/06 1:39 PM Page 169 CONDENSED LESSON 13.1 The Premises of Geometry In this lesson you will ● Learn about Euclid’s deductive system for organizing geometry properties ● Become familiar with the four types of premises for geometry Beginning in about 600 B.C.E., mathematicians began to use logical reasoning to deduce mathematical ideas. The Greek mathematician Euclid (ca. 325–265 B.C.E.) created a deductive system to organize geometry properties. He started with a simple collection of statements called postulates. He considered these postulates to be obvious truths that did not need to be proved. Euclid then systematically demonstrated how each geometry discovery followed logically from his postulates and his previously proved conjectures, or theorems. Up to now, you have used informal proofs to explain why certain conjectures are true. However, you often relied on unproved conjectures in your proofs. A conclusion in a proof is true if and only if your premises are true and all of your arguments are valid. In this chapter you will look at geometry as Euclid did. You will start with premises and systematically prove your earlier conjectures. Once you have proved a conjecture, it becomes a theorem that you can use to prove other conjectures. Read the four types of premises on page 693 of your book. You are already familiar with the first type of premise. You have learned the undefined terms—point, line, and plane—and you have a list of definitions in your notebook. The second type of premise is the properties of arithmetic, equality, and congruence. Read through the properties of arithmetic and equality in your book. You have used these properties many times to solve algebraic equations. The example in your book shows the solution to an algebraic equation, along with the reason for each step. This type of step-by-step solution is actually an algebraic proof. Here is another example. EXAMPLE Prove that if (a ϩ b)(a Ϫ b) ϭ (a ϩ b)(a ϩ b) and a ϩ b 0, then b ϭ 0. ᮣ Solution (a ϩ b)(a Ϫ b) ϭ (a ϩ b)(a ϩ b) Given a Ϫ b ϭ a ϩ b Division property of equality (a ϩ b 0) a ϭ a ϩ 2b Addition property of equality 0 ϭ 2b Subtraction property of equality 0 ϭ b Division property of equality Just as you use equality to express a relationship between numbers, you use congruence to express a relationship between geometric figures. Read the definition of congruence on page 695 of your book. Here are the properties of congruence. (continued) Discovering Geometry Condensed Lessons CHAPTER 13 169 ©2008 Kendall Hunt Publishing DG4CL_895_13.qxd 12/4/06 1:39 PM Page 170 Lesson 13.1 • The Premises of Geometry (continued) Properties of Congruence In the statements below, “figure” refers to a segment, an angle, or a geometric shape. Reflexive Property of Congruence Any figure is congruent to itself. Transitive Property of Congruence If Figure A Х Figure B and Figure B Х Figure C, then Figure A Х Figure C. Symmetric Property of Congruence If Figure A Х Figure B, then Figure B Х Figure A. The third type of premise is the postulates of geometry. Postulates are very basic statements that are useful and easy for everyone to agree on. Read the postulates of geometry on pages 696–697 of your book. Some of the postulates allow you to add auxiliary lines, segments, and points to a diagram. For example, you can use the Line Postulate to construct a diagonal of a polygon, and you can use the Perpendicular Postulate to construct an altitude in a triangle. Notice that the Corresponding Angles Conjecture and its converse are stated as a postulate, but the Alternate Interior Angles Conjecture is not. This means that you will need to prove the Alternate Interior Angles Conjecture before you can use it to prove other conjectures. Similarly, the SSS, SAS, and ASA Congruence Conjectures are stated as postulates, but SAA is not, so you will need to prove it. The fourth type of premise is previously proved geometry conjectures, or theorems. Each time you prove a conjecture, you may rename it as a theorem and add it to your theorem list. You can use the theorems on your list to prove conjectures. 170 CHAPTER 13 Discovering Geometry Condensed Lessons ©2008 Kendall Hunt Publishing DG4CL_895_13.qxd 12/4/06 1:39 PM Page 171 CONDENSED LESSON 13.2 Planning a Geometry Proof In this lesson you will ● Learn the five tasks involved in writing a proof ● Prove several conjectures about angles ● Learn how to create a logical family tree for a theorem A proof in geometry is a sequence of statements, starting with a given set of premises and leading to a valid conclusion. Each statement must follow from previous statements and must be supported by a reason. The reason must come from the set of premises you learned about in Lesson 13.1. Writing a proof involves five tasks. Task 1 is to identify what is given and what is to be proved. This is easiest if the conjecture is a conditional, or “if-then,” statement. The “if” part is what you are given, and the “then” part is what you must show. If a conjecture is not given this way, you can often restate it. For example, the conjecture “Vertical angles are congruent” can be rewritten as “If two angles are vertical angles, then they are congruent.” Task 2 is to make a diagram and to mark it so it illustrates the given information. Task 3 is to restate the “given” and “show” information in terms of the diagram. Task 4 is to make a plan for your proof, organizing your reasoning either mentally or on paper. Task 5 is to use your plan to write the proof. Page 703 of your book summarizes the five tasks involved in writing a proof. A flowchart proof of the Vertical Angles Conjecture is given on page 704 of your book. Notice that the proof uses only postulates, definitions, and properties of equality. Thus, it is a valid proof. You can now call the conjecture the Vertical Angles (VA) Theorem and add it to your theorem list. In Lesson 13.1, the Corresponding Angles (CA) Conjecture was stated as a postulate, but the Alternate Interior Angles (AIA) Conjecture was not. Example A in your book goes through the five-task process for proving the AIA Conjecture. Read the example carefully, and then add the AIA Theorem to your theorem list. Example B proves the Triangle Sum Conjecture. The proof requires using the Parallel Postulate to construct a line parallel to one side of the triangle. After you read and understand the proof, add the Triangle Sum Theorem to your theorem list. Pages 706–707 of your book give a proof of the Third Angle Conjecture. Read the proof, and then add the Third Angle Theorem to your theorem list. A logical family tree for a theorem traces the theorem back to all the postulates on which the theorem relies. On page 707 of your book, you’ll see how to construct the logical family tree for the Third Angle Theorem. Make sure you understand how this theorem is rooted in four postulates: the Parallel Postulate, the CA Postulate, the Linear Pair Postulate, and the Angle Addition Postulate. On page 707 you can see that the Third Angle Theorem follows from the Triangle Sum Theorem without relying on any other postulate or theorem. A theorem that is an immediate consequence of another proven theorem is called a corollary. So, the Third Angle Theorem is a corollary of the Triangle Sum Theorem. (continued) Discovering Geometry Condensed Lessons CHAPTER 13 171 ©2008 Kendall Hunt Publishing DG4CL_895_13.qxd 12/4/06 1:39 PM Page 172 Lesson 13.2 • Planning a Geometry Proof (continued) The next example is Exercise 8 in your book. It illustrates the five-task process of proving the Converse of the Alternate Interior Angles Conjecture. EXAMPLE Prove the Converse of the AIA Conjecture: If two lines are cut by a transversal forming congruent alternate interior angles, then the lines are parallel. Then create a family tree for the Converse of the AIA Theorem. ᮣ Solution Task 1: Identify what is given and what you must show. Given: Two lines cut by a transversal to form congruent alternate interior angles Show: The lines are parallel ᐉ Task 2: Draw and label a diagram. 3 (Note: You may not realize that labeling Є3 is useful until you make your plan.) ᐉ 1 1 Task 3: Restate the given and show information in 2 ᐉ terms of your diagram. 3 2 ᐉ ᐉ ᐉ Given: 1 and 2 cut by transversal 3; Є1 Х Є2 ᐉ ʈ ᐉ Show: 1 2 Task 4: Make a plan. ᐉ ʈ ᐉ Plan: I need to prove that 1 2. The only theorem or postulate I have for proving lines are parallel is the CA Postulate. If I can show that Є1 Х Є3, ᐉ ʈ ᐉ Є Х Є I can use the CA Postulate to conclude that 1 2. I know that 1 2. By the VA Theorem, Є2 Х Є3. So, by the transitive property of congruence, Є1 Х Є3. Task 5: Create a proof. Flowchart Proof Є Є Є Є Є Є ᐉ ʈ 1 2 2 3 1 3 1 2 Given VA Theorem Transitive CA Postulate property Here is a logical family tree for the Converse of the AIA Theorem.
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