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 +21. 5x – 18 = 3x + 22. 2x – 18 = 23. 2x = 204. x = 10

1. Given2. Subtraction property3. Addition property4. Division property

Example 2: Writing Reasons• Solve 55z – 3(9z + 12)= -64

1. 55z – 3(9z + 12)= -642. 55z – 27z – 36 = -643. 28z – 36 = -644. 28z = -285. z = -1

1. Given2. Distributiveproperty3. Simplify4. Addition property5. 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 – 10/7r2. a + 10/7 r = 2203. 10/7r = 220 – a4. r = 7/10(220 – a)

1. Given2. Addition property3. Subtraction property4. 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. Given2. Substitute 16 for a3. Simplify

c.

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

Age Rate20 14030 13340 12650 11960 112

Example 4: Using properties of length

1. AB = CD2. AB + BC = BC + CD3. AC = AB + BC4. BD = BC + CD5. AC = BD

1. Given2. Addition property3. Segment addition postulate4. Segment addition postulate5. 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. Given2. Given3. Substitution4. Subtraction5. Given6. Transitive7. Substitution