2.4 reasoning with properties of algebra mrs. spitz geometry september 2004

2.4 Reasoning with Properties of Algebra

Mrs. SpitzGeometry

September 2004

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.

Assignment:

• pp. 99-100 #1-31

Algebraic properties

• Pg. 96– 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. Given2. Subtraction property

3. Addition property

4. Division property

Example 2: Writing Reasons

• Solve 55z – 3(9z + 12)= -64

1. 55z – 3(9z + 12)= -64

2. 55z – 27z – 36 = -64

3. 28z – 36 = -64

4. 28z = -28

5. z = -1

1. Given

2. Distributiveproperty

3. Simplify

4. Addition property

5. 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. 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

b. Using a = 16, the target rate is:

1. r = 7/10(220 – a)

2. r = 7/10(220 – 16)

3. 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

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

Example 4: Using properties of length

1. AB = CD2. AB + BC = BC + CD

3. AC = AB + BC

4. BD = BC + CD

5. AC = BD

1. Given2. Addition property

3. Segment addition postulate

4. Segment addition postulate

5. Substitution property

Example 5: Using properties of measure

1. m1 + m 2 = 662. m 1 + m 2+m 3 =

99 3. 66 + m 3 = 99 4. m 3 = 33 5. m 3 = m 1, m 1 =

m 46. m 3 = m 47. m 4 = 33

1. Given

2. Given

3. Substitution

4. Subtraction

5. Given

6. Transitive

7. Substitution