2.4 Reasoning with Properties of Algebra

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2.4 Reasoning with Properties of Algebra
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Objectives/Assignment

• 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.
• Assignment 2-30 even, 36-50 even

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Goal 1 Using Properties from Algebra

• Pg. 96
• Addition property If a b, then ac bc
• Subtraction property If ab, then a-cb-c
• Multiplication property If ab, then acbc
• Division property If ab, then a?cb?c
• Reflexive property aa
• Symmetric property If ab, then ba
• Transitive property If ab and bc, then ac
• Substitution property If ab, then a can be
• substituted for b in any
• expression or equation.

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Distributive Property

• a (b c) ab ac
• Algebra Properties can be used to solve equations
• Example
• x 3 7 By subtracting 3 from each side of
  the equation, you obtain 4.

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Example 1 Writing Reasons

• Solve 5x 18 3x 2
• 5x 18 3x 2
• 2x 18 2
• 2x 20
• x 10

• Given
• Subtraction property
• Addition property
• Division property

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Example 2 Writing Reasons

• Solve 55z 3(9z 12) -64
• 55z 3(9z 12) -64
• 55z 27z 36 -64
• 28z 36 -64
• 28z -28
• z -1

• Given
• Distributiveproperty
• Simplify
• Addition property
• Division property

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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
  220 10/7 r.

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Find the following

1. Solve the equation for r and write a reason for
   each step.
2. Use the result to find the target heart rate for
   a 16-year old.
3. 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?

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a. a 220 10/7 r

1. a 220 10/7r
2. a 10/7 r 220
3. 10/7r 220 a
4. r 7/10(220 a)

1. Given
2. Addition property
3. Subtraction property
4. Multiplication property

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b. Using a 16, the target rate is

• 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. Given
2. Substitute 16 for a
3. Simplify

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c.

• From the table, the target heart rate appears to
  decrease as the person gets older.

Age Rate
20 140
30 133
40 126
50 119
60 112
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Goal 2 Using Properties of Length and Measure

• Segment Length
• Reflexive AB AB
• Symmetric If ABCD, then CDBA
• Transitive If ABCD and CDEF, then ABEF.

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Angle Measure Properties

• Reflexive m?Am?A
• Symmetric If m?A m?B, then m?B m?A.
• Transitive If m?A m?B and m?B m?C, then m?A
  m?C.

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Example 4 Using properties of length

• AB CD
• AB BC BC CD
• AC AB BC
• BD BC CD
• AC BD

• Given
• Addition property
• Segment addition postulate
• Segment addition postulate
• Substitution property

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Example 5 Using properties of measure

1. m?1 m ? 2 66?
2. m ? 1 m ? 2m ? 3 99 ?
3. 66 ? m ? 3 99?
4. m ? 3 33 ?
5. m ? 3 m ? 1, m ? 1 m ? 4
6. m ? 3 m ? 4
7. m ? 4 33 ?

1. Given
2. Given
3. Substitution
4. Subtraction
5. Given
6. Transitive
7. Substitution