Download Practice B 2-5

Name
LESSON
2-5
Date
Class
Practice B
Algebraic Proof
Solve each equation. Show
1 (a 10) 3
1. __
5
1 (a 10) 5(3)
5 __
5
a 10 15
a 10 10 15 10
a –25
[
]
all your steps and write a justification for each step.
2. t 6.5 3t 1.3
t 6.5 t 3t 1.3 t
(Mult. Prop. of )
6.5 2t 1.3
(Simplify.)
6.5 1.3 2t 1.3 + 1.3
(Subtr. Prop. of ) 7.8 2t
(Simplify.)
2t
7.8 __
___
2
2
3.9 t
t 3.9
(Subtr. Prop. of )
(Simplify.)
(Add. Prop. of )
(Simplify.)
(Div. Prop. of )
(Simplify.)
(Symmetric Prop. of )
3. The formula for the perimeter P of a rectangle with length ᐉ and width w is
1 feet.
P = 2(ᐉ w). Find the length of the rectangle shown here if the perimeter is 9 __
2
Solve the equation for ᐉ and justify each step. Possible answer:
P 2( w)
1)
1 2( 1 __
9 __
4
2
1 2 2 __
1
9 __
2
2
(Given)
7 2
(Simplify.)
(Subst. Prop. of )
2
7 __
__
(Div. Prop. of )
1
3 __
2
1
3 __
2
(Simplify.)
2
(Distrib. Prop.)
1 2 __
1 2 2 __
1 2 __
1 (Subtr. Prop. of )
9 __
2
2
2
2
2
11–4 ft
(Symmetric Prop. of )
Write a justification for each step.
7X 3
4.
2X 6
(
) 3X 3 *
HJ HI IJ
7x 3 (2x 6) (3x 3)
7x 3 5x 3
7x 5x 6
2x 6
x3
Seg. Add. Post.
Subst. Prop. of Simplify.
Add. Prop. of Subtr. Prop. of Div. Prop. of Identify the property that justifies each statement.
5. m n, so n m.
6. ABC ABC
Symmetric Prop. of _
Reflexive Prop. of _
7. KL LK
8. p q and q 1, so p 1.
Reflexive Prop. of Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Transitive Prop. of or Subst.
36
Holt Geometry
Name
LESSON
2-5
Date
Class
Name
Practice A
2-5
For Exercises 1–12, write the letter of each property next to its definition.
The letters a, b, and c represent real numbers.
F
C
2. If a � b, then ac � bc.
_
_
3. AB � AB
4. a � a
A. Addition Property of Equality
J
C. Multiplication Property of Equality
E
D. Division Property of Equality
5. If a � b, then a � c � b � c.
A
F. Symmetric Property of Equality
G
7. If a � b and b � c, then a � c.
G. Transitive Property of Equality
8. If �P � �Q, then
�Q � �P.
K
I. Distributive Property
12. If a � b, then a � c � b � c.
K. Symmetric Property of
Congruence
L. Transitive Property of Congruence
_
35
Date
Class
Holt Geometry
LESSON
2-5
Solve Exercises 1 and 2. Write justifications for each step in your solutions.
2. Solve for m�ZYX in terms of m�CBD.
Given: __
�XYZ
� �ABC.
›
BD is the angle bisector of �ABC.
�XYZ � �ABC
(Given)
�ZYX � �XYZ
(Reflexive Prop. of �)
�ZYX � �ABC
(Trans. Prop. of �)
m�ZYX � m�ABC
(Def. of �)
�ABD � �CBD
(Def. of � bisector)
m�ABD � m�CBD
(Def. of �)
m�ABC � m�ABD
� m�CBD
(� Add. Post.)
m�ABC � m�CBD
� m�CBD
(Subst. Prop. of �)
m�ABC � 2m�CBD
(Simplify.)
m�ZYX � 2m�CBD
(Subst. Prop. of �)
Algebraic Proof
Symbols
Examples
Addition
If a � b, then a � c � b � c.
If x � �4, then x � 4 � �4 � 4.
Subtraction
If a � b, then a � c � b � c.
If r � 1 � 7, then r � 1 � 1 � 7 � 1.
Multiplication
If a � b, then ac � bc.
Division
If a � 2 and c � 0, then _a_ � _b_ .
Reflexive
a�a
If _k_ � 8, then _k_ (2) � 8(2).
2
2
3t .
If 6 � 3t, then _6_ � __
3
3
15 � 15
c
c
Symmetric
If a � b, then b � a.
If n � 2, then 2 � n.
Transitive
If a � b and b � c, then a � c.
2
2
If y � 3 and 3 � 9, then y � 9.
Substitution
If a � b, then b can be substituted
for a in any expression.
If x � 7, then 2x � 2(7).
2(a � 1) � �6
Given equation
2a � 2 � �6
�2
cx � cy � dx � dy
Distributive Property
�2
Subtraction Property of Equality
2a � �8
2a � ___
�8
___
2
2
a � �4
4. Explain logically how the Transitive Property of Equality can be derived from the
Substitution Property of Equality and the Symmetric Property of Equality.
Possible answer: The Substitution Property states that if a � b, then b can
be substituted for a in any expression. Applying the Symmetric Property to
Simplify.
Division Property of Equality
Simplify.
Solve each equation. Write a justification for each step.
1. _n_ � 3 � 10
6
the Substitution Property shows that if b � a, then a can be substituted for
�3 �3
_n_ � 13
6
_n_ (6) � 13(6)
6
n � 78
b in any expression. So if a � b and b � c, then a � c by the Substitution
Property, and this is also the Transitive Property.
5. Explain why there is no Substitution Property of Congruence.
y�4
3. _____ � 3
7
Possible answer: Consider the points A(0, 1), B (1, 0), C (0, �1), and
_
Given equation
Add. Prop. of �
Simplify.
_
AB � AD, but you cannot conclude that the image of AD is CB for
reflection across the x-axis.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
67
2. 5 � x � 2x
Given equation
� x � x Subtr. Prop. of �
Simplify.
5�x
x�5
Sym. Prop. of �
Mult. Prop. of �
Simplify.
Given equation
y_____
�4
(7) � 3(7)
7
y � 4 � 21
�4 �4
y � 17
D (�1, 0). For reflection across the x-axis, the image of AB is CB.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Holt Geometry
When solving an algebraic equation, justify each step by using a definition,
property, or piece of given information.
c (x � y) � d (x � y ) � cx � cy � dx � dy ; (x � y )(c � d) �
37
Class
Reteach
Properties of
Equality
(x � y) � a ; a(c � d) � ac � ad; ac � ad � c(x � y) � d (x � y );
_
Date
A proof is a logical argument that shows a conclusion is true. An algebraic proof uses
algebraic properties, including the Distributive Property and the properties of equality.
3. Use the Distributive Property to find (x � y )(c � d ). Write out all the steps.
(Hint: Let (x � y ) � a.)
_
36
Name
Algebraic Proof
_
Transitive Prop. of � or Subst.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Practice C
1. Solve for m�3 in terms of m�1.
Given: �1 and �2 are complementary.
�2 and �3 are supplementary.
m�1 � m�2 � 90�
(Given)
m�2 � m�3 � 180�
(Given)
m�2 � m�3 � (m�1 � m�2)
� 180� � 90�
(Subtr. Prop. of �)
m�3 � m�1 � 90�
(Simplify.)
m�3 � m�1 � 90�
(Add. Prop. of �)
8. p � q and q � �1, so p � �1.
Reflexive Prop. of �
Mult. Prop. of �
x�9
Reflexive Prop. of �
_
7. KL � LK
Subtr. Prop. of �
3
(Symmetric Prop. of �)
6. �ABC � �ABC
Symmetric Prop. of �
Simplify.
_1_ x � 3
(Simplify.)
Seg. Add. Post.
Subst. Prop. of �
Simplify.
Add. Prop. of �
Subtr. Prop. of �
Div. Prop. of �
5. m � n, so n � m.
Subst.
3
2
11–4 ft
�
Identify the property that justifies each statement.
Seg. Add. Post.
_1_ x � 8 � 11
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
� 3� � 3 �
x�3
� _13_ x � 1 � � 7 � 11
_
(Simplify.)
2x � 6
�
DE � EF � DF
2-5
3 _1_ � �
2
� � 3 _1_
2
7x � 5x � 6
11
LESSON
(Div. Prop. of �)
(Distrib. Prop.)
7x � 3 � 5x � 3
Symmetric Property of Equality
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
2�
_7_ � __
7x � 3 � (2x � 6) � (3x � 3)
14. Write a justification for each step.
Name
7 � 2�
2
2� � 6
�
Simplify.
7
(Simplify.)
(Symmetric Prop. of �)
(Subst. Prop. of �)
HJ � HI � IJ
Substitution Property of Equality
�1 �
(Div. Prop. of �)
Write a justification for each step.
7� � 3
4.
Given equation
8 )�c
20.32 � c
c
� 20.32
(Subtr. Prop. of �)
(Simplify.)
(Add. Prop. of �)
(Simplify.)
(Given)
9 _1_ � 2 _1_ � 2� � 2 _1_ � 2 _1_ (Subtr. Prop. of �)
2
2
2
2
B
2.54i � c
1
–�
3
2. t � 6.5 � 3t � 1.3
t � 6.5 � t � 3t � 1.3 � t
6.5 � 2t � 1.3
(Simplify.)
6.5 � 1.3 � 2t � 1.3 + 1.3
(Subtr. Prop. of �) 7.8 � 2t
(Simplify.)
2t
7.8 � __
___
2
2
3.9 � t
t � 3.9
(Mult. Prop. of �)
P � 2(� � w)
9 _1_ � 2(� � 1 _1_)
4
2
9 _1_ � 2� � 2 _1_
2
2
J. Reflexive Property of Congruence
13. Cali measures her textbook and finds that it is 8 inches wide. She wants to know
how many centimeters wide her textbook is. The formula to convert inches to
centimeters is 2.54i � c, where i is the length in inches and c is the length in
centimeters. Fill in the blanks to find the answer. The justifications will guide you.
�
all your steps and write a justification for each step.
3. The formula for the perimeter P of a rectangle with length � and width w is
P = 2(� � w). Find the length of the rectangle shown here if the perimeter is 9 _1_ feet.
2
Solve the equation for � and justify each step. Possible answer:
H. Substitution Property of Equality
9. If �A � �B and �B � �C,
then �A � �C.
L
_
_b_
D
10. If a � b and c � 0, then _a
c � c.
11. If a � b, then b can be substituted for a
in any expression.
H
2.54(
]
E. Reflexive Property of Equality
I
6. a (b � c) � ab � ac
Algebraic Proof
Solve each equation. Show
1. _1_(a � 10) � �3
5
5 _1_ (a � 10) � 5(�3)
5
a � 10 � �15
a � 10 � 10 � �15 � 10
a � –25
[
B. Subtraction Property of Equality
Class
Practice B
LESSON
Algebraic Proof
1. If a � b, then b � a.
Date
4. 4(t � 3) � �20
Mult. Prop. of �
Simplify.
Subtr. Prop. of �
Simplify.
38
4t � 12 � �20
� 12 � 12
4t � �8
4t � ___
�8
__
4
4
t � �2
Given equation
Distr. Prop.
Add Prop. of �
Simplify.
Div. Prop. of �
Simplify.
Holt Geometry
Holt Geometry