LESSON Reteach Algebraic Proof - CRHS Mathematics · PDF fileCopyright © by Holt, Rinehart and Winston. 68 Holt Geometry All rights reserved. Copyright © by Holt, Rinehart and Winston

Copyright © by Holt, Rinehart and Winston. 38 Holt GeometryAll rights reserved.

Name Date Class

LESSON

� x � x Subtr. Prop. of � 5 � x Simplify. x � 5 Sym. Prop. of �

y � 4 _____ 7 (7) � 3(7) Mult. Prop. of �

y � 4 � 21 Simplify. �4 �4 Subtr. Prop. of � y � 17 Simplify.

4t � 12 � �20 Distr. Prop. � 12 � 12 Add Prop. of � 4t � �8 Simplify.

4t __ 4 � �8 ___

4 Div. Prop. of �

t � �2 Simplify.

ReteachAlgebraic Proof

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.

Properties of Equality

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. If k __ 2 � 8, then k __

2 (2) � 8(2).

Division If a � 2 and c � 0, then a __ c � b __ c . If 6 � 3t, then 6 __ 3 � 3t __

3 .

Reflexive a � a 15 � 15

Symmetric If a � b, then b � a. If n � 2, then 2 � n.

Transitive If a � b and b � c, then a � c. If y � 32 and 32 � 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).

When solving an algebraic equation, justify each step by using a definition, property, or piece of given information.

2(a � 1) � �6 Given equation

2a � 2 � �6 Distributive Property

� 2 � 2 Subtraction Property of Equality

2a � �8 Simplify.

2a ___ 2 � �8 ___

2 Division Property of Equality

a � �4 Simplify.

Solve each equation. Write a justification for each step.

1. n __ 6

� 3 � 10 Given equation 2. 5 � x � 2x Given equation � 3 � 3 Add. Prop. of � n __

6 � 13 Simplify.

n __ 6 (6) � 13(6) Mult. Prop. of �

n � 78 Simplify. 3.

y � 4 _____

7 � 3 Given equation 4. 4(t � 3) � �20 Given equation

2-5


Name Date Class

LESSON ReteachAlgebraic Proof continued2-5

When writing algebraic proofs in geometry, you can also use definitions, postulates, properties, and pieces of given information to justify the steps.

m�JKM � m�MKL Definition of congruent angles

(5x � 12)� � 4x � Substitution Property of Equality

x � 12 � 0 Subtraction Property of Equality

x � 12 Addition Property of Equality

Properties of Congruence Symbols Examples

Reflexive figure A � figure A �CDE � �CDE

SymmetricIf figure A � figure B, then figure B � figure A.

If _

JK � _

LM , then _

LM � _

JK .

TransitiveIf figure A � figure B and figure B � figure C, then figure A � figure C.

If �N � �P and �P � �Q, then �N � �Q.

Write a justification for each step.

5. CE � CD � DE Segment Addition Postulate 3 78

6

6x � 8 � (3x � 7) Substitution Property of Equality

6x � 15 � 3x Simplify.

3x � 15 Subtraction Property of Equality

x � 5 Division Property of Equality

6. m�PQR � m�PQS � m�SQR Angle Addition Postulate

90� � 2x � � (4x � 12)� Substitution Property of Equality

90 � 6x � 12 Simplify.

102 � 6x Addition Property of Equality

17 � x Division Property of Equality

Identify the property that justifies each statement.

7. If �ABC � �DEF, then �DEF � �ABC. 8. �1 � �2 and �2 � �3, so �1 � �3.

Symmetric Property of Congruence Transitive Property of Congruence

9. If FG � HJ, then HJ � FG. 10. _

WX � _

WX

Symmetric Property of Equality Reflexive Property of Congruence



Name Date Class

LESSON

For Exercises 1–12, write the letter of each property next to its definition. The letters a, b, and c represent real numbers.

1. If a � b, then b � a. F

2. If a � b, then ac � bc. C

3._AB �

_AB J

4. a � a E

5. If a � b, then a � c � b � c. A

6. a (b � c) � ab � ac I

7. If a � b and b � c, then a � c. G

8. If �P � �Q, then�Q � �P. K

9. If �A � �B and �B � �C,then �A � �C. L

10. If a � b and c � 0, then a__c � b__

c . D

11. If a � b, then b can be substituted for ain any expression. H

12. If a � b, then a � c � b � c. B

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.

2.54i � c Given equation

2.54( 8 ) � c Substitution Property of Equality

20.32 � c Simplify.

c � 20.32 Symmetric Property of Equality

14. Write a justification for each step.

71–3� � 1� ��

11

DE � EF � DF Seg. Add. Post.

� 1__3

x � 1 � � 7 � 11 Subst.

1__3

x � 8 � 11 Simplify.

1__3

x � 3 Subtr. Prop. of �

x � 9 Mult. Prop. of �

2-5Practice AAlgebraic Proof

A. Addition Property of Equality

B. Subtraction Property of Equality

C. Multiplication Property of Equality

D. Division Property of Equality

E. Reflexive Property of Equality

F. Symmetric Property of Equality

G. Transitive Property of Equality

H. Substitution Property of Equality

I. Distributive Property

J. Reflexive Property of Congruence

K. Symmetric Property of Congruence

L. Transitive Property of Congruence


Name Date Class

LESSON

t � 6.5 � t � 3t � 1.3 � t (Subtr. Prop. of �)6.5 � 2t � 1.3 (Simplify.)6.5 � 1.3 � 2t � 1.3 + 1.3 (Add. Prop. of �)7.8 � 2t (Simplify.)7.8___2

� 2t__2

(Div. Prop. of �)

3.9 � t (Simplify.)t � 3.9 (Symmetric Prop. of �)

�

11–4 ft

7 � 2� (Simplify.)

7__2

� 2� __2

(Div. Prop. of �)

31__2

� � (Simplify.)

� � 3 1__2

(Symmetric Prop. of �)

Solve each equation. Show all your steps and write a justification for each step.

1. 1__5

(a � 10) � �3 2. t � 6.5 � 3t � 1.3

5[1__5

(a � 10) ] � 5(�3) (Mult. Prop. of �)

a � 10 � �15 (Simplify.)a � 10 � 10 � �15 � 10 (Subtr. Prop. of �)a � –25 (Simplify.)

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__

2 feet.

Solve the equation for � and justify each step. Possible answer:

P � 2(� � w) (Given)

9 1__2

� 2(� � 1 1__4

) (Subst. Prop. of �)

9 1__2

� 2� � 2 1__2

(Distrib. Prop.)

9 1__2

� 2 1__2

� 2� � 2 1__2

� 2 1__2

(Subtr. Prop. of �)

Write a justification for each step. 4.

2� � 6 3� � 3� ��

7� � 3

HJ � HI � IJ Seg. Add. Post.

7x � 3 � (2x � 6) � (3x � 3) Subst. Prop. of �

7x � 3 � 5x � 3 Simplify.

7x � 5x � 6 Add. Prop. of �

2x � 6 Subtr. Prop. of �

x � 3 Div. Prop. of �


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 � Transitive Prop. of � or Subst.

Practice BAlgebraic Proof2-5


Name Date Class

LESSON

�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 �)

Practice CAlgebraic Proof

Solve Exercises 1 and 2. Write justifications for each step in your solutions.

1. Solve for m�3 in terms of m�1. 2. Solve for m�ZYX in terms of m�CBD.Given: �1 and �2 are complementary. Given: �XYZ � �ABC.

�2 and �3 are supplementary. __›BD is the angle bisector of �ABC.

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 �)

3. Use the Distributive Property to find (x � y )(c � d ). Write out all the steps. (Hint: Let (x � y ) � a.)

(x � y ) � a ; a (c � d ) � ac � ad ; ac � ad � c (x � y ) � d (x � y );

c (x � y ) � d (x � y ) � cx � cy � dx � dy ; (x � y )(c � d ) �

cx � cy � dx � dy

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

the Substitution Property shows that if b � a, then a can be substituted for

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.

Possible answer: Consider the points A(0, 1), B (1, 0), C (0, �1), and

D (�1, 0). For reflection across the x-axis, the image of_AB is

_CB.

_AB �

_AD, but you cannot conclude that the image of

_AD is

_CB for

reflection across the x-axis.

2-5


Name Date Class

LESSON

� x � x Subtr. Prop. of � 5 � x Simplify.

x � 5 Sym. Prop. of �

y � 4_____7

(7) � 3(7) Mult. Prop. of �y � 4 � 21 Simplify.

�4 �4 Subtr. Prop. of �y � 17 Simplify.

4t � 12 � �20 Distr. Prop.� 12 � 12 Add Prop. of �

4t � �8 Simplify.

4t__4

� �8___4

Div. Prop. of �

t � �2 Simplify.

ReteachAlgebraic Proof

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.

Properties of Equality

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. If k__2

� 8, then k__2

(2) � 8(2).

Division If a � 2 and c � 0, then a__c � b__

c . If 6 � 3t, then 6__3

� 3t__3

.

Reflexive a � a 15 � 15

Symmetric If a � b, then b � a. If n � 2, then 2 � n.

Transitive If a � b and b � c, then a � c. If y � 32 and 32� 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).

When solving an algebraic equation, justify each step by using a definition, property, or piece of given information.

2(a � 1) � �6 Given equation

2a � 2 � �6 Distributive Property

� 2 � 2 Subtraction Property of Equality

2a � �8 Simplify.2a___2

� �8___2

Division Property of Equality

a � �4 Simplify.

Solve each equation. Write a justification for each step.

1. n__6

� 3 � 10 Given equation 2. 5 � x � 2x Given equation� 3 � 3 Add. Prop. of �

n__6

� 13 Simplify.

n__6

(6) � 13(6) Mult. Prop. of �n � 78 Simplify.

3.y � 4_____

7� 3 Given equation 4. 4(t � 3) � �20 Given equation

2-5



Name Date Class

LESSON ReteachAlgebraic Proof continued2-5

When writing algebraic proofs in geometry, you can also use definitions, postulates, properties, and pieces of given information to justify the steps.

m�JKM � m�MKL Definition of congruent angles �

�

��

��

��

(5x � 12)� � 4x � Substitution Property of Equality

x � 12 � 0 Subtraction Property of Equality

x � 12 Addition Property of Equality

Properties of Congruence Symbols Examples

Reflexive figure A � figure A �CDE � �CDE

SymmetricIf figure A � figure B, then figure B �figure A.

If_JK �

_LM , then

_LM �

_JK .

TransitiveIf figure A � figure B and figure B �figure C, then figure A � figure C.

If �N � �P and �P � �Q,then �N � �Q.

Write a justification for each step.

5. CE � CD � DE Segment Addition Postulate3� � 78� ��

6�

6x � 8 � (3x � 7) Substitution Property of Equality

6x � 15 � 3x Simplify.

3x � 15 Subtraction Property of Equality

x � 5 Division Property of Equality

6. m�PQR � m�PQS � m�SQR Angle Addition Postulate

��

��

�

�

�

�

90� � 2x � � (4x � 12)� Substitution Property of Equality

90 � 6x � 12 Simplify.

102 � 6x Addition Property of Equality

17 � x Division Property of Equality


7. If �ABC � �DEF, then �DEF � �ABC. 8. �1 � �2 and �2 � �3, so �1 � �3.

Symmetric Property of Congruence Transitive Property of Congruence

9. If FG � HJ, then HJ � FG. 10._WX �

_WX

Symmetric Property of Equality Reflexive Property of Congruence


Name Date Class

LESSON ChallengeHave a Good Reason2-5

Mathematical induction is a type of proof that uses deductive reasoning to prove statements about positive integers. To prove that 13n

� 11 is divisible by 12, for example, you assume that there is a whole number r such that 13 k

� 11 � 12r.It is important to be able to justify each step of these types of proofs.

1. Write a justification for each step of the mathematical induction proof.

13 k� 11 � 12r Inductive hypothesis

13 k� 12r � 11 a. Subtr. Prop. of �

13(13 k) � 13(12r � 11) b. Mult. Prop. of �

13 k � 1� 13(12r � 11) c. Simplify.

13 k � 1� 156r � 143 d. Distributive Property

13 k � 1� 11 � 156r � 132 e. Add. Prop. of �

13 k � 1� 11 � 12(13r � 11) f. Distributive Property

2. Write a justification for each step of the mathematical induction proof, which proves that 10n

� 1 is divisible by 9.

10 k� 1 � 9r Inductive hypothesis

10 k� 9r � 1 a. Add. Prop. of �

10(10 k) � 10(9r � 1) b. Mult. Prop. of �

10 k � 1� 10(9r � 1) c. Simplify.

10 k � 1� 90r � 10 d. Distributive Property

10 k � 1� 1 � 90r � 9 e. Subtr. Prop. of �

10 k � 1� 1 � 9(10r � 1) f. Distributive Property

3. In the figure, �1 ��3,�3 ��2, and m�1 � 65�.Find m�ABC. Justify each step.

1

�

2

�

3

�

�1 � �2 Trans. Prop. of � m�1 � m�2 Def. of � �

65� � m�2 Subst. Prop. of �

m�1 � m�2 � m�ABC � Add. Post. 65� � 65� � m�ABC Subst. Prop. of � 130� � m�ABC Simplify.


Name Date Class

LESSON Problem SolvingAlgebraic Proof

1. Because of a recent computer glitch, an airline mistakenly sold tickets for round-trip flights at a discounted price. The equation n (p � t ) � 3298.75 relates the number of discounted tickets sold n, the price of each ticket p, and the tax per ticket t.What was the discounted price of each ticket if 1015 tickets were sold and the tax per ticket was $1.39? Solve the equation for p. Justify each step.

n (p � t ) � 3298.75 Given equation 1015(p � 1.39) � 3298.75 Subst. Prop. of � p � 1.39 � 3.25 Div. Prop. of � p � $1.86 Subtr. Prop. of �

2. The equation C � 7.25s � 15.95a describes the total s � number of student tickets

a � number of adult tickets

C � total cost of admission

cost of admission C to the aquarium. How many student tickets were sold if the total cost for the entire class and 6 adults was $298.70? Solve the equation for s. Justify each step.

C � 7.25s � 15.95a Given equation 298.70 � 7.25s � 15.95(6) Subst. Prop. of � 298.70 � 7.25s � 95.7 Simplify. 203 � 7.25s Subtr. Prop. of � 28 � s Div. Prop. of � s � 28 students Sym. Prop. of �

Refer to the figure. Choose the best answer.

3. Which could be used to find the value of x?

��

��

�

�

�

�

A Segment Addition Postulate

B Angle Addition Postulate

C Transitive Property of Congruence

D Definition of supplementary angles

4. What is m�SQR?

F 28� H 61�

G 29� J 62�

2-5


Name Date Class

LESSON Reading StrategiesBuilding Vocabulary2-5

When you look into a mirror, you see your reflect ion. You see your exact, identical self. Similarly, when you use the Reflex ive Property of Equality,a quantity is equal to its exact, identical self.

5 � 52x � 3 � 2x � 3

m�C � m�C

The two halves of a football field are symmetrical to each other. When two football teams change sides, the second side of the field is the same as the first side. In the same way, the Symmetric Property of Equalityallows equal quantities to be written on either side of the equal sign.

If x � 2, then 2 � x.If m�A � 40°, then 40° � m�A.

Just as the cars on a train link the engine to the caboose, the Transitive Property of Equality links equal quantities to each other. If two expressions are equal to the same thing, they are equal to each other.

If AB � 8 and 8 � CD,

�� 8then AB � CD.

AB � CD

Sometimes, at school, a substitute teacher may replace your teacher for a day. Similarly, the Substitution Property of Equality allows equal expressions to be used interchangeably.

If b � 5 and a � b � 9, then a � 5 � 9.

Which properties do the following situations describe?

1. butterfly wings Symmetric Property of Equality

2. removing the quarterback of the football team and replacing him with another quarterback Substitution Property of Equality

3. the reflection of clouds on a still pond Reflexive Property of Equality

4. There are four quarters in one dollar and ten dimes in one dollar, so four quarters have the same value as ten dimes. Transitive Property of Equality

Documents

LESSON Reteach Algebraic Proof - CRHS Mathematics · PDF fileCopyright © by Holt, Rinehart and Winston. 68 Holt Geometry All rights reserved. Copyright © by Holt, Rinehart and Winston