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To construct a logical argument using algebraic properties
2.5 Reason Using Algebra Properties 2.6 Prove Statements about Segments and Angles Objectives: To construct a logical argument using algebraic properties To understand the role of proof in a deductive system To write proofs using geometric theorems
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Example 1 Solve 2x +5 = 20 – 3x. Write a reason for each step.
Given 5x +5 = 20 Addition Prop of = 5x = 15 Subtraction Prop of = x = 3 Division Prop of =
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An Algebraic Flashback
Algebraic Properties of Equality Let a, b, and c be real numbers. Addition Property Subtraction Property Multiplication Property Division Property Substitution Property
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Oh, Here’s One More Distributive Property
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Example 2 Solve −4(11x + 2) = 80. Write a reason for each step.
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Example 3 Solve the formula below for b1. Write a reason for each step.
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Even More Properties!
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Even More Properties! Any of these properties can be used as reasons in an algebraic or geometric proof.
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Example 4 Complete the conditional statement using the property indicated. Transitive: If JD = SB and SB = RA, then ___________________. Symmetric: If m<4 = m<2, then ___________________. Substitution: If RA = SB and SB + JN = 7, then ___________________.
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A Brief History of Math Over thousands of years the Babylonians and Egyptians discovered many geometric principles and developed a collection of “rule-of-thumb” procedures for doing practical geometry.
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A Brief History of Math The result of trial and error, these procedures were used to compute simple areas and volumes. The procedures were used in surveying to reestablish land boundaries after floods, and they were practical instructions for building canals and tombs.
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A Brief History of Math By 600 B.C. a prosperous new civilization had begun to grow in the trading towns along the coast of Asia Minor and later in Greece, Sicily, and Italy. People had free time to discuss and debate issues of government and law.
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A Brief History of Math This led to an insistence on reasons to support statements made in debate. Mathematicians began to use logical reasoning to deduce mathematical ideas.
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A Brief History of Math Greek mathematician Thales of Miletus made a number of valuable geometric conjectures. Unlike most other mathematicians before him, Thales supported his discoveries with logical reasoning.
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A Brief History of Math Over the next 300 years, the process of supporting mathematical conjectures with logical arguments became more and more refined. Other Greek mathematicians, including Thales’ most famous student, Pythagoras, began linking together chains of logical reasoning.
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A Brief History of Math Later students of mathematics at Plato’s Academy linked even longer chains of geometric properties together by deductive reasoning. Euclid, in his famous work about geometry and number theory, Elements, established a single chain of deductive arguments for most of the geometry then known.
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A Brief History of Math Euclid started from a collection of statements that he regarded as obviously true (postulates). He then systematically demonstrated that one after another geometric discovery followed logically from his postulates and his previously verified conjectures (theorems). In doing this, Euclid created a deductive system. Text by Michael Serra
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Thanks a lot, Euclid! So it’s the development of civilization in general and specifically a series of clever ancient Greeks who are to be thanked (or blamed) for the insistence on reason and proof in mathematics.
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When in Greece… Recall that inductive reasoning leads to conjectures in mathematics which must be proven with deductive reasoning. In a mathematical proof, every statement must be the consequence of other previously accepted or proven statements.
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Premises in Geometric Arguments
The following is a list of premises that can be used in geometric proofs: Definitions and undefined terms Properties of algebra, equality, and congruence Postulates of geometry Previously accepted or proven geometric conjectures (theorems)
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Amazing Usually we have to prove a conditional statement. Think of this proof as a maze, where the hypothesis is the starting point and the conclusion is the ending. p q
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Amazing Your job in constructing the proof is to link p to q using definitions, properties, postulates, and previously proven theorems. p q
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Un-Amazing With proofs, sometimes this is the case: And so is this:
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Example 1 Construct a two-column proof of:
If m1 = m3, then mDBC = mEBA.
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Example 5 Given: m1 = m3 Prove: mDBC = mEBA
Statements Reasons 1. m1 = m3 1.Given 2. m1 + m2 = m3 + m2 2.Addition Property 3. m1 + m2 = mDBC 3.Angle Addition Postulate 4. m3 + m2 = mEBA 4.Angle Addition Postulate 5. mDBC = mEBA 5.Substitution Property
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Two-Column Proof Notice in a two-column proof, you first list what you are given (hypothesis) and what you are to prove (conclusion). The proof itself resembles a T-chart with numbered statements on the left and numbered reasons for those statements on the right. Before you begin your proof, it is wise to try to map out the maze from p to q.
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Generic Two-Column Proof
Given: ____________ Prove: ____________ Insert illustration here Statements Reasons 1. 2. 3.
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Proof Activity In this activity, you and your group members will have a couple of two-column proofs to assemble. For the first one, you only have to put the reasons in the correct order. For the second one, you will have to put both the statements and reasons in the correct order.
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Assignment P : 6-20 even, 21-26, 39 P : 1-4, 16-19, 22-24 Challenge Problems
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