Transcript 2.4Algebraic Reasoni..

2.4 Reasoning with
Properties of Algebra
Mrs. Spitz
Geometry
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. 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.
m1 + 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
m3=m4
m  4 = 33 
1.
2.
3.
4.
5.
6.
7.
Given
Given
Substitution
Subtraction
Given
Transitive
Substitution