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Geometry Lesson Notes 2.6 Date ________________ Objective: Write algebraic proofs. Use properties of equality to write geometric proofs. Properties of Equality for Real Numbers For all numbers a, b, and c, Reflexive Property a=a Symmetric Property If a = b then b = a Transitive Property Addition and If a = b and b = c, then a = c Subtraction Properties If a = b, then a + c = b + c Multiplication and Division Properties If a = b, then ac = bc and a/c = b/c if c 0 Substitution Property If a = b, then a may be replaced by b in any equation or expression Distributive Property a(b + c) = ab + ac NOTE: We will be assuming the Commutative and Associative Properties of addition and multiplication. No need to state them in a proof. You must be able to recognize and use these properties! 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. 493717193 Page 1 of 5 Example 1 (p 94): Verify Algebraic Relationships Solve ½ (x + 16) = 5x − 1 for x and give a reason for each step. Statement Reason ½ (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 ½ (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. 493717193 Page 2 of 5 Example 2 (p 94): Write a Two-Column (Algebraic) Proof Write a two-column proof of the following conditional statement. If 7 1 n 4 n , then n 2 . 2 2 Given: 7 2 n 4 1n Prove: n2 2 Statements: 1. Reasons 7 1 n 4 n 2 2 1. Given 2. 2 n 2 4 n 2 2 2. Multiplication Property 3. 7 2n 8 n 3. Distributive Property 4. 7 2n n 8 n n 4. Addition Property 5. 7 n 8 5. Substitution Property (Combining Like Terms) 6. 7 n 7 8 7 6. Subtraction Property 7. n 1 7. Substitution Property (Combining Like Terms) 7 8. n 1 1 1 1 9. n 1 493717193 8. Division Property 9. Substitution Property Page 3 of 5 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 mC = mC Symmetric If XY = YZ, then YZ = XY If m1 = m2, then m2 = m1 Transitive If MN = NO and NO = OP, then MN = OP If mK = mL and mL = mM, then mK = mM Practice: Name the property of equality that justifies each statement. Statement Property If 5 = x, then x = 5 _______________________________ If ½ 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 ST RP , then which of the following conclusions is true? I. GH + JK = RP II. PR = TS III. GH + JK = ST + RP A. I only 493717193 B. I and II C. I and III D. I, II, and III Page 4 of 5 E 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. Given: A _____________________________ 22 cm _____________________________ E Prove: _____________________________ Proof: B S C D C Statements Reasons 1. ________________________ 1. ___________________________________________ 2. ________________________ 2. ___________________________________________ 3. ________________________ 3. ___________________________________________ 4. ________________________ 4. ___________________________________________ HW: A7a pp 97-100 #14-25, 29-31, 37-38 A7b 2-6 Skills Practice / Practice 493717193 Page 5 of 5