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You can use these properties to justify every step as you solve an equation. The group of algebraic steps used to solve problems is called a deductive argument. Example 1 (p 94): Verify Algebraic Relationships Solve (x + 16) = 5x " 1 for x and give a reason for each step. Statement Reason (x + 16) = 5x " 1 Given x + 16 = 2(5x " 1) Multiplication / Substitution properties x + 16 = 10x " 2 Distributive / Substitution properties 16 = 9x " 2 Subtraction / Substitution properties 18 = 9x Addition / Substitution properties 2 = x Division / Substitution properties x = 2 Symmetric property This deductive argument is an example of an algebraic proof of a conditional statement. The conditional statement would be: If (x + 16) = 5x " 1, then x = 2. The hypothesis is the starting point (the given) of the proof. The conclusion is the end of the proof, what we need to prove. Listing the reasons (properties) for each step makes this a proof. Two-column, or formal, proof: contains statements (the steps) and reasons (the properties that justify each step) organized in two columns. Example 2 (p 94): Write a Two-Column (Algebraic) Proof Write a two-column proof of the following conditional statement. If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . Given:  EMBED Equation.DSMT4  Prove:  EMBED Equation.DSMT4   Statements: Reasons 1.  EMBED Equation.DSMT4  1. Given 2.  EMBED Equation.DSMT4  2. Multiplication Property 3.  EMBED Equation.DSMT4  3. Distributive Property 4.  EMBED Equation.DSMT4  4. Addition Property 5.  EMBED Equation.DSMT4  5. Substitution Property (Combining Like Terms) 6.  EMBED Equation.DSMT4  6. Subtraction Property 7.  EMBED Equation.DSMT4  7. Substitution Property (Combining Like Terms) 8.  EMBED Equation.DSMT4  8. Division Property 9.  EMBED Equation.DSMT4  9. Substitution Property Proofs in geometry are presented in the same manner. Algebra properties as well as definitions, postulates, and other true statements can be used as reasons in a geometric proof. Since geometry also uses variables, numbers, and operations, we are able to use many of the properties of equality to prove geometric properties. Segment measures and angle measures are real numbers, so we can use the properties of equality to describe relationships between segments and between angles. Examples: Property Segments Angles Reflexive AB = AB m(C = m(C Symmetric If XY = YZ, then YZ = XY If m(1 = m(2, then m(2 = m(1 Transitive If MN = NO and NO = OP, If m(K = m(L and m(L = m(M, then MN = OP then m(K = m(M Practice: Name the property of equality that justifies each statement. Statement Property If 5 = x, then x = 5 _______________________________ If x = 9, then x = 18 _______________________________ If AB = 2x and AB = CD, then CD = 2x _______________________________ If 2AB = 2CD, then AB = CD _______________________________ Example 3 (p 96): Justify Geometric Relationships If GH + JK = ST and  EMBED Equation.DSMT4 , then which of the following conclusions is true? I. GH + JK = RP II. PR = TS III. GH + JK = ST + RP A. I only B. I and II C. I and III D. I, II, and III Example 4 (p 96): Geometric Proof A starfish has five arms. If the length of arm 1 is 22 cm, and arm 1 is congruent to arm 2, and arm 2 is congruent to arm 3, prove that arm 3 has length of 22 cm.  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