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2.5 Reasoning with Properties of Algebra Mrs. Padilla Geometry September 2011 Standards/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: Use properties from Algebra • Use properties of length and measure to justify segment and angle relationships, such as the angles at the turns of a racetrack. Algebraic properties – – – – – – – – Addition property Subtraction property Multiplication property Division property Reflexive property Symmetric property Transitive property Substitution property Distributive Property • a (b + c) = ab + ac • Can be used to solve equations • Example: x + 3 = 7 By subtracting 3 from each side of the equation, you obtain 4. Example 1: Writing Reasons Solve 5x – 18 = 3x +2 1. 5x – 18 = 3x + 2 2. 2x – 18 = 2 3. 2x = 20 4. x = 10 1. Given 2. 3. 4. Subtraction property Addition property Division property Example 2: Writing Reasons • Solve 55z – 3(9z + 12)= -64 1. 2. 3. 4. 5. 55z – 3(9z + 12)= -64 55z – 27z – 36 = -64 28z – 36 = -64 28z = -28 z = -1 1. 2. 3. 4. 5. Given Distributiveproperty Simplify Addition property Division property Example 3: Using properties in Real Life • Before exercising, you should find your target heart rate. This is the rate at which you achieve an effective workout while not placing too much strain on your heart. Your target heart rate (r) –in beats per minute can be determined from your age (a)—in years using the equation a = 22 – 10/7 4 Find the following: a. Solve the equation for r and write a reason for each step. b. Use the result to find the target heart rate for a 16-year old. c. Find the target rate for the following ages: 20, 30, 40, 50, and 60. What happens to the target heart rate as a person gets older? a. a = 220 – 10/7 r 1. 2. 3. 4. a = 220 – 10/7r a + 10/7 r = 220 10/7r = 220 – a r = 7/10(220 – a) 1. 2. 3. 4. Given Addition property Subtraction property Multiplication property b. Using a = 16, the target rate is: 1. 2. 3. r = 7/10(220 – a) r = 7/10(220 – 16) r = 142.8 The target rate for a 16 year old is about 143 beats per minute 1. 2. 3. Given Substitute 16 for a Simplify c. Age Rate 20 140 30 133 40 126 50 119 60 112 • From the table, the target heart rate appears to decrease as the person gets older. Example 4: Using properties of length 1. AB = CD 1. Given 2. 3. 4. 5. 2. 3. AB + BC = BC + CD AC = AB + BC BD = BC + CD AC = BD 4. 5. Addition property Segment addition postulate Segment addition postulate Substitution property Example 5: Using properties of measure 1. 2. 3. 4. 5. 6. 7. m1 + m 2 = 66 m 1 + m 2+m 3 = 99 66 + m 3 = 99 m 3 = 33 m 3 = m 1, m 1 = m 4 m3=m4 m 4 = 33 1. 2. 3. 4. 5. 6. 7. Given Given Substitution Subtraction Given Transitive Substitution Assignment: • 2.5 N ; • Worksheets