Name Date Class Grade 9 2015 / 2016 - Funmath8's Blog · PDF fileName ___ Date _ Class ___ Grade 9 2015 / 2016 First Semester 2015- 2016

Name _______________________________________ Date __________________ Class __________________

Grade 9

2015 / 2016

First Semester 2015- 2016

Strand Theme / Area

Unit Chapter / Module lesson

Algebra 1

Volume 2

Exponential

Relationships

6 Chapter 14 : Rational Exponents and

Radicals

1 & 2

Polynomial operations 7 Chapter 17 : Adding and Subtracting

polynomials

1-2&3

Chapter 18 : Multiplying And Dividing

Polynomials

1-2&3

Geometry Volume 2

Trigonometry 5 Chapter 13 : Trigonometry with Right

Triangles

1-2&3

Geometry Volume 1

Quadrilaterals Coordinate Proof

3 Chapter 10 : Coordinate Proof Using Slope

and Distance

1&2

Chapter 9 : Properties of Quadrilaterals 1-2-3&4

Name _______________________________________ Date __________________ Class __________________

Understanding Rational Exponents and Radicals

Success for English Learners

Problem 1

1

nx means n x

1

3325 25

1. What is the index in the radical expression 3 25? __________________

2. What is the exponent in the radical expression 3 25? _________________

Example

1

12249 49 7

11

331000 1000 10 1

12216 16 4

2

2 2338 8 2 4 5

5 5224 4 2 32 3

3 3229 9 3 27

Simplify each expression. Show your steps.

1. 1

2100 2. 1

38 3. 1

29 4. 1

225

_______________ _______________ _______________ ________________

LESSON

14-1

n x

index

base

Name _______________________________________ Date __________________ Class __________________

Understanding Rational Exponents and Radicals

Practice and Problem Solving: A/B

Write the name of the property that is demonstrated by each equation.

1. 4 4(2 ) 16a a 2. 6 3 18(3 ) 3

_______________________________________ ________________________________________

Simplify each expression.

3. 2

38 4. 3

51 5. 1

29

_______________________ ________________________ ________________________

6. 3

225 7. 5

416 8. 1

327

_______________________ ________________________ ________________________

9. 1 1

4 281 4 10. 2 2

3 5343 • 32 11. 1

2100

_______________________ ________________________ ________________________

Find the value of the expression for the value indicated.

12. 2100m for 5m 13. 81a

a for 1

2a

_______________________________________ ________________________________________

Solve.

14. The equation1

20.25t d can be used to find the number of seconds,

t, that it takes an object to fall a distance of d feet. How long does it take an

object to fall 64 feet?

________________________________________________________________________________________

15. Show that

31

416

and 1

3 416 are equivalent.

________________________________________________________________________________________

16. The surface area, S, of a cube with volume V can be found using the

formula 2

36 .S V Find the surface area of a cube whose volume is

125 cubic inches.

LESSON

14-1

Name _______________________________________ Date __________________ Class __________________

Simplifying Expressions with Rational Exponents and Radicals Pra ctice and Problem Solving: Modified

Match each expression with a fractional exponent to an equivalent

radical expression. The first one is done for you.

1. 1

2x

A. 3( )x

2. 1

3x

____________

B. x

3. 2

3x

____________

C. 23( )x

4. 3

2x

____________

D. 3 x

Rewrite each expression using a fractional exponent. The first one is

done for you.

5. 5 x 6. 4 5x 7. 3 218 8. 2 610

_______________ _______________ _______________ ________________

Simplify each expression. The first one is done for you.

9. 1

249 10. 1

481 11. 1

31

_______________________ ________________________ ________________________

12. 1 1

3 28 100 13. 5

38 14. 16x

_______________________ ________________________ ________________________

Solve. The first one is started for you.

15. Given a square with area x, you can use the formula 1

21.4d x to

estimate the length of the diagonal of the square. Use the formula to

estimate the length of the diagonal of a square with area 100 cm2.

________________________________________________________________________________________

16. For a pendulum with a length of L meters, the time in seconds that it

takes the pendulum to swing back and forth is 1

22 .L How long does it

take a pendulum that is 9 meters long to swing back and forth?

7

1

5x

B

LESSON

14-2

LESSON

12-1

1

21.4(100 )d

Name _______________________________________ Date __________________ Class __________________

Simplifying Expressions with Rational Exponents and Radicals



1. 55 y 2. 4 12x y 3. 3 6 3a b

_______________________ ________________________ ________________________

4. 425y 5. 6 93 x y 6. 2 2(9 )y 2 2(9 )y

_______________________ ________________________ ________________________

7. 5 35 (32 )y 8. 1

3 2 23( )x y x y 9. 3 43 (27 )y 3 46 (27 )y

_______________________ ________________________ ________________________

10. 84 ( )xy 11. 1

4 62( )x x 12.

184

3 3

( )x

x

_______________________ ________________________ ________________________

Solve.

13. Given a cube with volume V, you can use the formula 1

34P V to find

the perimeter of one of the cube’s square faces. Find the perimeter of

a face of a cube that has volume 125 m3.

________________________________________________________________________________________

14. The Beaufort Scale measures the intensity of tornadoes. For a tornado

with Beaufort number B, the formula 3

21.9v B may be used to

estimate the tornado’s wind speed in miles per hour. Estimate the wind

speed of a tornado with Beaufort number 9.

________________________________________________________________________________________

15. At a factory that makes cylindrical cans, the formula

1

2

12

Vr

is used

to find the radius of a can with volume V. What is the radius of a can

with a volume of 192 cm3?

________________________________________________________________________________________

LESSON

14-2

Name _______________________________________ Date __________________ Class __________________

Rational Exponents and Radicals

Module Quiz: B

1. Describe the solution set of

_______________________________________

2. is equal to which of the following?

A 4 C 6

B 5 D 7

3. Does each expression equal 4 when

simplified?

A Yes No

B Yes No

C Yes No

D Yes No

4. What is simplified?

_______________________________________

5. is equal to which of the following?

A 2 C 96

B 16 D 512

6. Simplify .

_______________________________________

7. Which expression is equivalent to be?

A C

B D

8. What is simplified?

_____________________________________________

9. Does each of the following belong to the

set of integers?

A Yes No

B Yes No

C Yes No

D Yes No

10. Determine if belongs to each set. Choose True or False.

A Integers True False

B Rational numbers True False

C Irrational numbers True False

11. Does each set include only irrational

numbers?

A Yes No

B Yes No

C Yes No

12. Determine whether the set is closed under

subtraction.

A whole numbers Yes No

B rational numbers Yes No

C integers Yes No

13. Which statement shows that the product of

irrational numbers is not always an

irrational number?

A

B

C

D

2 5.

2 3

y x

x y

1

3343

1

216

1

381

1

4256

1

51,024

1 1

4 281 121

2

364

312216 25x

1

eb1

eb

1eb

eb

4 3

7

a a

a

1

381

1

364

1

316

1

811

23

7 7, 11, 4

7, 17,

5, , 144

9

2 2 4

3 3 3 3

5 2 10

1 77

7 7

MODULE

14

Name _______________________________________ Date __________________ Class __________________

Understanding Polynomial Expressions

Reteach

Polynomials have special names based on the number of terms.

POLYNOMIALS

No. of

Terms 1 2 3 4 or more

Name Monomial Binomial Trinomial Polynomial

The degree of a monomial is the sum of the exponents in the monomial. The degree of a

polynomial is the degree of the term with the greatest degree.

Examples

Find the degree of 8x2y3. Find the degree of 4ab 9a3.

8x2y3 The exponents are 2 and 3. 4

2

ab 39

3

a

The degree of the monomial

is 2 3 5.

Identify each polynomial. Write the degree of each expression.

1. 7m3n5 2. 4x2y3 y4 7 3. x5 x5y

_______________________ ________________________ ________________________

You can simplify polynomials by combining like terms.

The following are like terms:

The following are not like terms:

Examples

Add 3x2 4x 5x2 6x.

3x2 5x2 4x

6x Identify and rearrange like terms so they are together.

8x2 10x Combine like terms.


4. 2y2 3y 7y y2 5. 8m4 3m 4m4 6. 12x5 10x4 8x4

LESSON

17-1

The degree of

the binomial is 3.

same variable,

different exponent

one with variable,

one constant

same variable but

different power

3x2 and 3x 47 and 7y 8m and m5

4y and 7y 8x2 and 2x2 7m5 and m5

same variables raised to same power

Name _______________________________________ Date __________________ Class __________________

Understanding Polynomial Expressions

Practice and Problem Solving: Modified

Identify each expression as a monomial, a binomial, a trinomial, or

none of the above. Write the degree of each expression. The first one

is done for you.

1. 4w2 2. 9x3 2x

_______________________________________ ________________________________________

3. 35b6 4. 4p5 5p3 11

_______________________________________ ________________________________________

5. 12 3x4 x 6. 3m 1

_______________________________________ ________________________________________

Simplify each expression. The first one is done for you.

7. 6n2 3n n2 8. 5c3 2c 4c

_______________________________________ ________________________________________

9. 3b 1 2b 8 10. 7a4 9a3 3a4 4a

_______________________________________ ________________________________________

Find the value of each polynomial for the given value of x. Then

determine the polynomial that has the greater value. The first one is

started for you.

11. 4x2 5x 2 or 5x2 2x 4 for x 3 12. 6x3 4x2 7 or 7x3 6x2 4 for x 2

_______________________________________ ________________________________________


15. A firework is launched from the ground at a velocity of 180 feet per

second. Its height after t seconds is given by the polynomial

16t2 180t. What is the height of the firework after 2 seconds?

________________________________________________________________________________________

16. The volume of one box is 4x3 4x2 cubic units. The volume of the

second box is 6x3 18x2 cubic units. Write a polynomial for the total

volume of the two boxes.

___________________________________________________________

17. Antoine is making a banner in the shape of a triangle. He wants to

line the banner with a decorative border. How long will the border be?

___________________________________________________________

LESSON

17-1

monomial; degree 2

5n2 3n

19; 35; 5x2 2x 4

h 16 (2)2 180 (2)

Name _______________________________________ Date __________________ Class __________________

Adding Polynomial Expressions

Reteach

You can add polynomials by combining like terms.

These are examples of like terms: 4y and 7y 8x2 and 2x2 m5 and 7m5

These are not like terms: 3x2 and 3x 4y and 7 8m and 8n

Add (5y2 7y 2) (4y2 y 8).

(5y2 7y 2 ) (4y2 y 8 ) Identify like terms.

(5y2 4y2) (7y y ) ( 2 8 ) Rearrange terms so that like terms are together.

9y2 8y 10 Combine like terms.

Add (5y2 7y 2) (4y2 y 8).

(5y2 7y 2 ) (4y2 y 8 ) Identify like terms.

(5y2 4y2) (7y y ) ( 2 8 ) Rearrange terms so that like terms are together.

9y2 8y 10 Combine like terms.

Add.

1. (6x2 3x) (2x2 6x) ____________________________________________________

2. (m2 10m 5) (8m 2) ____________________________________________________

3. (6x3 5x) (4x3 x2 2x 9) ____________________________________________________

4. (2y5 6y3 1) (y5 8y4 2y3 1) ____________________________________________________

LESSON

17-2

These are like terms because they have

the same variables and same exponent.

one with a

variable, one

is a constant

same variable

but different

exponent

different

variables

Name _______________________________________ Date __________________ Class __________________

Adding Polynomial Expressions: Practice and Problem

Solving: Modified Add. The first one is done for you.

1. 2m 4 2. 3y2 y 3 3. 4z3 3z2 8

m 2 2y2 2y 9 2z3 z2 3

_______________________ ________________________ ________________________

4. 12k 3 5. 6s3 9s 10 6. 15a4 6a2a

4k 2 3s3 4s 10 6a4 2a2 a

_______________________ ________________________ ________________________

7.

2

2

( 13 4 )

(3 7 )

ab b a

ab a b

8.

2

2

( 8 )

( 12 2 8 )

r pr p

r pr p

_______________________________________ ________________________________________

Add the polynomial expressions using the horizontal format.

9. (3y2 y 3) (2y2 2y 9) 10. (4z3 3z2 8) (2z3 z2 3)

_______________________________________ _________________________________

11.(3x3 4) (x3 10) 12. (10g2 3g 10) (2g2 g 9)

_______________________________________ ________________________________________

13. (12p5 8) (8p5 6) 14. (11b2 3b 1) (2b2 2b 8)

_______________________________________ ________________________________________


15. Rebecca is building a pen for her rabbits against the side of her house.

The polynomial 4n 8 represents the length and the polynomial 2n 6

represents the width.

a. What polynomial represents the perimeter

of the entire pen?

________________________________________

________________________________________

b. What polynomial represents the perimeter

of the pen NOT including the side of the house.

________________________________________

13. A rectangular picture frame has the dimensions shown in

the figure. Write a polynomial that represents the perimeter

of the frame.

LESSON

17-2

3m 6

(4n 8) (4n 8 ) (2n 6) (2n 6)

Name _______________________________________ Date __________________ Class __________________

Subtracting Polynomial Expressions Reteach

To subtract polynomials, you must remember to add the opposites.

Find the opposite of (5m3 m 4).

(5m3 m 4)

(5m3 m 4) Write the opposite of the polynomial.

5m3 m 4 Write the opposite of each term in the polynomial.

Subtract (4x3 x2 7) (2x3).

(4x3 x2 7) (2x3) Rewrite subtraction as addition of the opposite.

(4x3 x2 7) (2x3) Identify like terms.

(4x3 2x3) x2 7 Rearrange terms so that like terms are together.

2x3 x2 7 Combine like terms.

Subtract (6y4 3y2 7) (2y4 y2 5).

(6y4 3y2 7) (2y4 y2 5) Rewrite subtraction as addition of the opposite.

(6y4 3y2 7 ) (2y4 y2 5) Identify like terms.

(6y4 2y4 ) (3y2 y2) (7 5) Rearrange terms so that like terms are together.

4y4 4y2 12 Combine like terms.

Subtract.

1. (9x 3 5x) (3x)

_______________________________________

2. (6t 4 3) (2t 4 2)

_______________________________________

3. (2x3 4x 2) (4x3 6)

_______________________________________

4. (t 3 2t) (t 2 2t 6)

_______________________________________

5. (4c5 8c2 2c 2) (c3 2c 5)

_______________________________________

LESSON

17-3

Name _______________________________________ Date __________________ Class __________________

Subtracting Polynomial Expressions


Subtract. The first one is done for you.

1. 8p 6 2. 9y2 6y 3 3. 5z3 8z2 5

p 2) (5y2 3y 2) (2z3 3z2 2)

_______________________ ________________________ ________________________

4. 20k 6 5. 7s3 4s 30 6. 25a4 9a2 6a

(10k 2) (5s3 2s 10) (10a4 2a2 a)

_______________________ ________________________ ________________________

7.

2

2

(10 5 2)

( 2 1 )

b b

b b

8.

3 2

3 2

( 7 5 2 )

( 3 2 2 )

c c c

c c c

_______________________________________ ________________________________________

9. (5x3 14) (2x3 1) 10. (15g2 6g 3) (10g2 2g 2)

_______________________________________ ________________________________________

11. (7p5 8) (3p5 6) 12. (4b2 8b 1) (2b2 3b 5)

_______________________________________ ________________________________________

Solve. The first problem is started for you.

11. The angle GEO is represented by 3w 7 and angle OEM is 2w 1.

Write a polynomial that represents the difference between angle

GEO and angle OEM.

_______________________________________________________

12. The polynomial 35p 300 represents the number of men enrolled in a

college and 25p 100 represents the number of women enrolled in the

same college. What polynomial shows the difference between the

number of men and women enrolled in the college?

________________________________________________________________________________________

LESSON

17-3

4p 4

(3w 7) (2w 1)

Name _______________________________________ Date __________________ Class __________________

________________________________________________________________________________________

Adding and Subtracting Polynomials

Module Quiz: B

1. Solve 3 316 2 .x x What is the value

of x?

_____________________________________________

2. What is 3(5x2 9x) evaluated for x 3?

_____________________________________________

3. For the expression 4xy

2z, determine if each statement is True or False.

A The expression is a monomial. True False

B The expression has a degree of 2. True False

C The expression has a coefficient of 4. True False

4. What is the degree of 5x3 2x2y3z?

_____________________________________________

5. Is each of the following a cubic binomial?

A 2x3 4x Yes No

B 3x2 x Yes No

C x3 2 Yes No

D x4 3x2 11 Yes No

6. Which of the following is the correct

classification of 3x3y2 9x 1?

A binomial with a degree of 3

B binomial with a degree of 2

C trinomial with a degree of 3

D trinomial with a degree of 5

7.Simplify 5mn2 8m mn2.

_____________________________________________

8. Find the sum of (4x3 2x) and

(8x3 5x 4).

_____________________________________________

9. What is (2x2 5x 7) (7x2 3)

simplified?

_____________________________________________

10. A rectangle has width w and its length is 2 units shorter than 3 times the width. Does each polynomial represent the perimeter of the rectangle?

A Yes No

B 3w2 2w 4 Yes No

C 8w 4 Yes No

D Yes No

11. The amount Tomas makes, in dollars, working h hours can be represented by the

expression 18h 8. Tomas hopes to get a raise that can be represented by the

expression 2h 32. Write an expression that represents how much Tomas will make working h hours if he gets the raise. How much will he make for working 8 hours?

_____________________________________________

w w w 2 3 2 3 2

w w w 2 3 3

MODULE

17

Name _______________________________________ Date __________________ Class __________________

Multiplying Polynomial Expressions by Monomials

Reteach

To multiply monomial expressions, multiply the constants, and then multiply

variables with the same base.

Example Multiply (3a2b) (4ab3).

(3a2b) (4ab3)

(3 × 4) (a2 × a) (b × b3) Rearrange so that the constants and the variables with the same

bases are together.

12a3b4 Multiply.

To multiply a polynomial expression by a monomial, distribute the monomial to each term

in the polynomial.

Example

Multiply 2x(x2 3x 7).

2x(x2 3x 7)

(2x)x2 (2x)3x (2x)7 Distribute.

2x3 6x2 14x Multiply.

Multiply.

1. (5x2y3) (2xy) 2. (2xyz) (4x2yz) 3. (3x) (x2y3)

_______________________ _______________________ ________________________

Fill in the blanks below. Then complete the multiplication.

4. 4(x 5) 5. 3x(x 8) 6. 2x(x2 6x 3)

_______________________ _______________________ ________________________

Multiply.

7. 5(x 9) 8. 4x(x2 8) 9. 3x2(2x2 5x 4)

_______________________ _______________________ ________________________

10. 3(5 x2 2) 11. (5a3b) (2ab) 12. 5y(y2 7y 2)

___________________ ___________________ ___________________

5x 8x 2 6 3x x

LESSON

18-1

Name _______________________________________ Date __________________ Class __________________

Multiplying Polynomial Expressions by Monomials

Practice and Problem Solving: Modified Find the product. The first one is done for you.

1. 7(3a2 2a 7) 2. 9(3x2 4x 3)

_______________________________________ ________________________________________

3. 6s3(2s2 4s 10) 4. 5a2(6a4 2a2 1)

_______________________________________ ________________________________________

Solve. The first one is done for you.

5. The length of a rectangle is 5 inches greater than the width.

a. Write a variable for the width of the rectangle. __________________________

b. Write an expression for the length of the rectangle. __________________________

c. Write a simplified expression for the area of the rectangle.

(area length width) __________________________

d. Find the area of the rectangle when the width is

3 inches. __________________________

6. The length of a rectangle is 3 inches greater than the width.

a. Write a polynomial expression that represents

the area of the rectangle. _____________________________________

b. Find the area of the rectangle when the

width is 4 inches. _____________________________________

7. The length of a rectangle is 8 centimeters less than 3 times the width.

a. Write a polynomial expression that represents

the area of the rectangle. _____________________________________

b. Find the area of the rectangle when the

width is 10 centimeters. _____________________________________

LESSON

18-1

21a2 14a 49

w

Name _______________________________________ Date __________________ Class __________________

Multiplying Polynomial Expressions

Reteach

Use the Distributive Property to multiply binomial and polynomial expressions.

Examples Multiply (x 3) (x 7).

(x 3) (x 7)

x(x 7) 3(x 7) Distribute.

(x)x (x)7 (3)x (3)7 Distribute again.

x2 7x 3x 21 Multiply.

x2 4x 21 Combine like terms.

Multiply (x 5) (x2 3x 4).

(x 5) (x2 3x 4)

x(x2 3x 4) 5 (x2 3x 4) Distribute.

(x)x2 (x)3x (x)4 (5)x2 (5)3x (5)4 Distribute again.

x3 3x2 4x 5x2 15x 20 Multiply.

x3 8x2 19x 20 Combine like

terms.

Multiplying Polynomial Expressions


Fill in the blanks by multiplying the First, Outer, Inner,

and Last terms. Then simplify. The first one is started for you.

1.(x 5) (x 2) 2. (x 4) (x 3)

_____ _____ _____ _____ _____ _____ _____ _____

F O I L F O I L

Simplify: ___________________________ Simplify: __________________________

3. (x 5)(x 6) 4. (a 7)(a 3) 5. (d 8)(d 4)

_______________________ ________________________ ________________________

LESSON

18-2

x2 2x 5x 10

first last

(x 3) (x 2)

inner outer

LESSON

18-2

Name _______________________________________ Date __________________ Class __________________

Fill in the blanks below. The first three are started for you.

6. (x 5)2 7. (x 10)2 8. (x 7) (x 7)

22 2 5 5 x x

22 2 1 0 1 0 x x 22

7 x

_______________________ ________________________ ________________________

9. (x 4)2 10. (b 2)2 11. (p 9)(p 9)

___________________________ ________________________ ________________________

Fill in the blanks below. Then simplify.

12. (x 3) (x2 4x 7) x (x2 4x 7) 3(x2 4x 7)

Distribute: _____ _____ _____ _____ _____ _____

Simplify: _____________________________________

13. (y 2)(y2 6y 5) 14. (p 4)(p2 3p 2) 15. (n 2)(n2 4n 1)

_______________________ ________________________ ________________________

Solve.

16. Zoe babysat for x 3 hours yesterday. She earned x 2 dollars per

hour. Write a polynomial expression that represents the amount

Zoe earned.

___________________________________________________________

Solve.

22. Write a polynomial expression that represents the volume of

the cube.

___________________________________________________

Name _______________________________________ Date __________________ Class __________________

Special Products of Binomials

Reteach

A perfect-square trinomial is a trinomial that is the result of squaring a binomial.

(a b)2 a2 2ab b2

(a b)2 a2 2ab b2

A difference of squares is a special product with no middle term.

(a b)(a b) a2 b2

State whether the products will form a difference of squares or a

perfect-square trinomial.

1. (x 10)(x 10) 2. (y 6)(y 6) 3. (z 3)(z 3)

_______________________ ________________________ ________________________

Multiply.

4. (x 8)2 5. (x 2)2 6. (7x 5)2

_______________________ ________________________ ________________________

7. (x 8)(x 8) 8. (10 x)(10 x) 9. (5x 2y)(5x 2y)

___________________ ___________________ ___________________

LESSON

18-3

Square a.

Add the product of 2, a, and b.

Square b.

Square a.

Subtract the product of 2, a, and b.

Square a. Square b.

Subtract.

Square b.

Name _______________________________________ Date __________________ Class __________________

Special Products of Binomials

Practice and Problem Solving: Modified Fill in the blanks. Then simplify. The first one is done for you.

1. (x + 5)2

2. (m+ 3)2 3. (2+a)2

x2 2(x)(5) 52 ____2 2(____)(____) ____2 ____2 2(____)(____) ____2

_______________________ ________________________ ________________________

4. (x + 4)2

5. (a+ 7)2

6. (8+b)2

_______________________ ________________________ ________________________

7. (y - 4)2

8. (y - 6)2

9. (9- x)2

y2 2(y)(4) 42 ____2 2(____)(____) ____2 ____2 2(____)(____) ____2

_______________________ ________________________ ________________________

Find the product.

10. (x -10)2

11. (b-11)2 12. (3- x)2

_______________________ ________________________ ________________________

Fill in the blanks. Then simplify. The first one is done for you.

13. (x +7)(x -7) 14. (4+ y)(4- y) 15. (x + 2)(x - 2)

x2 72 ____2 ____2 ____2 ____2

_______________________ ________________________ ________________________

Find the product.

16. ( 8)( 8)x x 17. (3 )(3 )y y 18. ( 1)( 1)x x

_______________________ ________________________ ________________________

Solve.

16. Write a simplified expression for each of the following.

a. area of the large rectangle

_____________________________________

b. area of the small rectangle

_____________________________________

c. area of the shaded area

_____________________________________

LESSON

18-3

x2 10x 25

y2 8y 16

x2 49

Name _______________________________________ Date __________________ Class __________________

Multiplying Polynomials

Module Quiz: B

1. Is each of the following expressions

equivalent to 25 2 6 12?x x

A Yes No

B Yes No

C Yes No

D Yes No

2. Find the product of 25x y and 52

5xy .

Determine if each statement about the

product is True or False.

A It is a monomial. True False

B It has a coefficient

of 2. True False

C Its degree is 7. True False

3. Find the product of 8xy and 3x 7y.

_______________________________________

4. Is each expression equivalent to

2x(3x 2) 4(3x 2)?

A (6x 4)(12x 8) Yes No

B (2x 4)(3x 2) Yes No

C Yes No

D 2(3x 2)(x 2) Yes No

5. What is the product of (3x 1)(3x 4)?

_______________________________________

6. Which product results in x2 100?

A (x 10)2

B (x 10)2

C x(x 100)

D (x 10)(x 10)

7. Is each of the following a perfect square

trinomial?

A x2 10x 25 Yes No

B x2 16x 16 Yes No

C x2 7x 49 Yes No

D x2 50x 625 Yes No

8. Multiply (x 1) (3x2 6x 12).

________________________________________

9. The area of a carpet is 36x2y 9xy

square inches. If the width is 3xy inches,

what is the length of the carpet?

________________________________________

10. Multiply (x 4)2.

________________________________________

x x 22 5

x x 25 2 18

x x 2 3 18

x x 25 2 0

x x 26 8 8

MODULE

18

Name _______________________________________ Date __________________ Class __________________

Tangent Ratio

Reteach

In a right triangle, the longest side, the side opposite

the right angle, is called the hypotenuse.

In the figure, the side opposite of X is .YZ

XY is the side adjacent to .X

The tangent (tan) ratio for X is opposite

.adjacent

If YZ 3 and XY 4, then 3

tan 0.75.4

X

Find the tangent of R and T.

1. tanR __________

2. tanT __________

If a tangent ratio is known, the inverse tangent 1tan

function on a calculator will calculate the angle

measurement.

In the figure,8

tan .16

X

So, 1 8m tan 26.6 .

16X

Find the measure of angle C. Round to the nearest tenth if necessary.

3. AB 3 and AC 4

4. AC 9 and AB 5

LESSON

13-1

Name _______________________________________ Date __________________ Class __________________

Tangent Ratio


For Problems 1–8, identify the features of the right triangle.

The first one is done for you.

1. the hypotenuse _________ 2. the legs __________

3. the side opposite A _________ 4. the side opposite B _________

5. the side adjacent to A _________ 6. the side adjacent to B _________

7. the tangent of A _________ 8. the tangent of B _________

Triangle RST is a right triangle with the right angle at R.

Answer the questions about the triangle. The first one is

done for you.

9. What is the relationship between the tangent of S and the tangent

of T? _____________________

10. If the tangent of T is x, what is tan1x (the inverse tangent of x)?

m _________

11. If RST is an isosceles triangle, what is the tangent of S? _________

What is the tangent of T? _________

12. What angle has a tangent of 1? tan _________ 1

Use a calculator to find each tangent. Round to the nearest

hundredth. The first one is done for you.

13. tan81 _____________ 14. tan38 _____________ 15. tan12 _____________

16. tan30 _____________ 17. tan72 _____________ 18. tan8 _____________

The inverse tangent of x is the angle whose tangent is x.

Use a calculator to find each inverse tangent. Round to the

nearest 0.1 degree. Check your work by finding the tangent

of each of your answers. The first one isw done for you.

19. tan10.65 _____________ 20. tan1

_____________ 21. tan10.4 _____________

tan _____________ 0.65 tan _____________

13

7 tan _____________ 0.4

22. tan1 _____________ 23. tan12 _____________ 24. tan110 _____________

tan _____________ 4

5 tan _____________ 2 tan _____________ 10

4

5

LESSON

13-1

reciprocals

6.31

33.0

33

AB

13

7

Name _______________________________________ Date __________________ Class __________________

Tangent Ratio


Identify the relationships in the figure to the right.

1. tanX WX

2. tanV

3. tan1 m_____ 4. tan1 m_____

5. tanX tanV _____ 6. tan1 tan1 _____

Use a calculator to find each tangent or inverse tangent. Round

tangents to the nearest 0.01 and angles to the nearest 0.1 degree.

Check the inverse tangents by finding the tangent of each angle.

7. tan23 _____________ 8. tan43 _____________ 9. tan47 _____________

10. tan10.14 _____________ 11. tan11 _____________ 12. tan16.1 _____________

tan _____________ 0.14 tan _____________ 1 tan _____________ 6.1

Solve Problems 13–16 using tangent ratios and a calculator. Refer to

the figure to the right of each problem.

13. To the nearest hundredth, what is tanM in ?LMN ________

14. Write a ratio that gives tanS. ________ Find the value of tanS to

the nearest hundredth. ________ Use the inverse tangent function

on your calculator to find the angle with that tangent. ________

15. Write and solve a tangent equation to find the distance from

C to E to the nearest 0.1 meter. ________ meters

16. The glide slope is the path a plane uses while it is landing

on a runway. The glide slope usually makes a 3angle with

the ground. A plane is on the glide slope and is 1 mile (5280 feet)

from touchdown. Find EF, the plane’s altitude, to the

nearest foot. Show your work.

________________________________________________________________________________________

________________________________________________________________________________________

LESSON

13-1

VW

WX

WX

VW

VW

WX

WX

VW

Name _______________________________________ Date __________________ Class __________________

Sine and Cosine Ratios

Reteach

In a right triangle, the sine of an angle is the ratio of the

length of the side opposite the hypotenuse.

The cosine of an angle is the ratio of the length of the side

adjacent to the hypotenuse.

In the figure:

sinYZ

XXZ

and cosXY

XXZ

sinXY

ZXZ

and cosYZ

ZXZ

Find the sine and cosine of angles A and B in the figure.

1. sin A _______ cos A _________

2. sin B _______ cos B _________

When the sine or cosine ratio of an angle is known, the

angle measure can be determined using inverse operations.

The sine of an angle is equal to the cosine of that angle’s

complement.

In the figure, the sin of 12

,13

A so

1 12m sin 67.4 .

13A

So, sin 67.4 is equal to cos 22.6 because 90 67.4 22.6.

Calculate the following values from triangle RST. Round to the

nearest tenth, if necessary.

3. cos R

4. m R

5. sin S

6. m S

LESSON

13-2

Name _______________________________________ Date __________________ Class __________________



For Problems 1–4, fill in the blanks to complete each definition. Then

use side lengths from the figure to complete the trigonometric ratios.


1. The sine (sin) of an angle is the ratio of the length of the leg _____________________

the angle to the length of the _____________________.

2. sinA c

sinB

3. The cosine (cos) of an angle is the ratio of the length of the leg _____________________

to the angle to the length of the _____________________.

4. cosA c

cosB

Use the figure to the right for Problems 5–12. Write the sines and

cosines as ratios and as decimals to the nearest hundredth. Then

find the measures of the angles to the nearest degree. The first one

is done for you.

5. sinX 14.4

16 ________ 6. sinY ________

7. cosX ________ 8. cosY ________

9. When you know the sine of an angle, you can find the measure of

the angle in degrees by using the inverse sine, sin1. Describe how

to find the inverse sine of the number n on your calculator.

_____________________________________________________

10. In Problem 5 you found the sine of X. Use your calculator to find the

inverse sine of X, which is the measure of X. __________________

11. Show how to use a different inverse to find mX. (Use your

answer from Problem 7.) __________________

12. If you calculated mX correctly, what is mY? ______________

Confirm your answer by using the inverse cosine. _____________

LESSON

13-2

opposite

hypotenuse

Name _______________________________________ Date __________________ Class __________________



After verifying that the triangle to the right is a right triangle,

use a calculator to find the given measures. Give ratios to the

nearest hundredth and angles to the nearest degree.

1. Use the Pythagorean Theorem to confirm that the triangle

is a right triangle. Show your work.

________________________________________________________________

2. sin1 _________________ 3. sin2 _______________________

4. cos1 _________________ 5. cos2 _______________________

6. Show how to find m1 using the inverse sine of 1.

________________________________________________________________________________________

7. Show how to find m2 using the inverse sine of 2.

________________________________________________________________________________________

Use a calculator and trigonometric ratios to find each length.

Round to the nearest hundredth.

8. 9. 10.

BD _________________ QP _________________ ST _________________

Use sine and cosine ratios to solve Problems 11–13.

11. Find the perimeter of the triangle. Round to the nearest

0.1 centimeter. _________________

12. To the nearest 0.1 inch, what is the length of the hypotenuse

of the springboard shown to the right? _________________

13. What is the height of the springboard (the dotted

line)? _________________

LESSON

13-2

Name _______________________________________ Date __________________ Class __________________

Special Right Triangles

Reteach

An isosceles-right triangle is a right triangle with two congruent legs.

The base angles of an isosceles-right triangle both measure 45, so

another name for this triangle is a 45-45-90 triangle. Both legs are

the same length. The hypotenuse length is the leg length times 2.

The sine, cosine, and tangent of 45 can be calculated from the

triangle, using the ratios.

1 2sin45 cos45

22 2

x

x tan45 1

x

x

Find the given side lengths and angle measurements for

triangle ABC.

1. BC

2. AC

3. m A

Another special right triangle is the 30-60-90 triangle like triangle

XYZ in the figure.

The length of the hypotenuse is double the length of the shorter

leg, and the other leg’s length is 3 times the length of the

shorter leg.

The sine, cosine, and tangent of 30 and 60 can be calculated

using these ratios.

Find the indicated values from the figure.

4. RT 5. RS _____________

6. sin 30 7. cos 30 _________

8. sin 60 9. cos 60 _________

10. tan 30 11. tan 60 __________

LESSON

13-3

Name _______________________________________ Date __________________ Class __________________



A Pythagorean triple is a set of three whole numbers that can be the

side lengths of a right triangle. Substitute the given numbers into the

Pythagorean theorem to see whether or not they make a Pythagorean

triple. ( yes; no) Show your work. The first one is done for you.

1. a 9; b 40; c 42 ___________________________________________________________________

2. a 7; b 24; c 25 _______________________________________________________

3. a 11; b 59; c 60 _______________________________________________________

4. Discover another Pythagorean triple by taking one of these sets that

works and multiplying each number by any positive integer. Show that

your new set of numbers works in the Pythagorean Theorem.

________________________________________________________________________________________

The table below shows trigonometric relationships in some

special right triangles (30-60-90 and 45-45-90). Use the table and

trigonometric ratios to find the missing measures. Show your work.


Angle Sine Cosine Tangent

30 1

2

3

2

3

3

45 2

2

2

2 1

60 3

2

1

2

3

5. RT _________ 6. ST _________

cos30 3

;6 2

RT RT

RS sin30 ;

2

ST ST

7. LK ______ 8. JK ______

tan45 1LK

JL ; cos45 ;

2

JL

JK

LESSON

13-3

sin opposite

hypotenuse

cos adjacent

hypotenuse

tan opposite

adjacent

3 3 ft

92 402 422; 81 1600 1681; 422 1764; 1681 1764 ?

Name _______________________________________ Date __________________ Class __________________



Use the figure to the right for Problems 14. Write each

trigonometric ratio as a simplified fraction and as a decimal

rounded to the nearest hundredth.

1. sinL 2. cosL

_________________ _________________

3. tanM 4. sinM

_________________ _________________

Write each trigonometric ratio as a simplified fraction.

5. sin 30 _______________ 6. cos 30 _______________ 7. tan 45 _______________

8. tan 30 _________________ 9. cos 45 _______________ 10. tan 60 _______________

11. Fill in the side lengths for these special right triangles with a

hypotenuse of 1. Use decimals to the nearest 0.01, and be sure that

your answers make sense, for example that the hypotenuse is longer

than the legs.

Use special right triangle relationships to solve Problems 12–14.

12. If cos A 0.28, which angle in the triangles to the

right is A? _______________

If sin B 0.22, which angle is B? _______________

13. What is EF, the measure of the longest side of the sail

on the model? Round to the nearest inch. _________________ in.

What is the measure of the shortest side? _________________ in.

14. If the small sail is similar to the larger one and is 11

inches high, about how wide is it? _________________ in.

LESSON

13-3

Name _______________________________________ Date __________________ Class _________________

Trigonometry with Right Triangles

Module Quiz: B

PQR is shown.

13. What are the missing side lengths in ?PQR Explain.

_______________________________________

_______________________________________

_______________________________________

TSU is shown.

14. What are the missing side lengths in ?TSU Explain. Keep your answer in simplified

radical form.

_______________________________________

_______________________________________

15. What is a Pythagorean Triple?

_______________________________________

_______________________________________

16. State whether the following are Pythagorean Triples or multiples of Pythagorean Triples.

A 6, 8, 10 Yes No

B 14, 48, 50 Yes No

C 25, 25, 100 Yes No

MODULE

13

Name _______________________________________ Date __________________ Class _________________

Slope and Parallel Lines

Reteach

Parallel lines have the same slope.

In the figure, the slope of lines a and b is 2

.3

Slope can be used to classify quadrilaterals.

If only one set of opposite sides have the same

slope, the quadrilateral is a trapezoid.

If both pairs of opposite sides have the same slope,

the quadrilateral is a parallelogram.

Remember, the slope (m) of a line that passes through the points

1 1,x y and 2 2,x y is computed using the formula 2 1

2 1

.y y

mx x

Prove that ABCD is a parallelogram.

1. ABCD is a parallelogram if ____________ ____________ and

____________ ____________.

2. Names the coordinates of A, B, C, and D.

___________________________________________________

3. Find the slope of .AB ___________________________

4. Find the slope of .BC ___________________________

5. Find the slope of .CD ___________________________

6. Find the slope of .DA ______________________

7. Do you have enough information to prove that ABCD is a parallelogram?

Why or why not?

________________________________________________________________________________________

LESSON

10-1

Name _______________________________________ Date __________________ Class __________________



Find the slope. The first one is done for you.

1. Line AB

_______________________________________

2. Line CD

___________________________________________

3. Are AB and CD parallel? Explain your reasoning.

________________________________________________________________________________________

________________________________________________________________________________________

Line A contains the points (1, 2) and (2, 5). Line B contains the points

(7, 1) and (9, 5). Find the slope. The first one is done for you.

4. Line A 5. Line B

_______________________________________ ________________________________________

6. Are Line A and Line B parallel? Explain your reasoning.

________________________________________________________________________________________

________________________________________________________________________________________

For Problems 7–10, use the graph. The first one is done for you.

7. Describe a method you can use to show that figure

MNPQ is a trapezoid.

________________________________________________________

_________________________________________________________

8. Which two sides should you choose to see if they are parallel?

Explain why you chose those sides.

________________________________________________________________________________________

________________________________________________________________________________________

9. What is the slope of the first side you chose? 10. What is the slope of the second side?

_______________________________________ ____________________________________

3

A trapezoid has one pair of opposite sides that are

parallel. So, show that two sides of MNPQ are parallel.

slope 2

3

LESSON

10-1

Name _______________________________________ Date __________________ Class __________________



Line A contains the points (2, 6) and (4, 10). Line B contains the

points (2, 3) and (3, 13).

1. Are the lines parallel? Explain your reasoning.

________________________________________________________________________________________

________________________________________________________________________________________

Figure JKLM has as its vertices the points J(4, 4), K(2, 1), L(3, 2),

and M(1, 5).

Find each slope.

2. JK 3. KL 4. LM 5. MJ

_______________ _______________ _______________ ________________

6. Is JKLM a parallelogram? Explain your reasoning.

________________________________________________________________________________________

________________________________________________________________________________________

For Problems 7–10, use the graph at the right.

7. Find the slope of line .

_____________________________________________

8. Explain how you found the slope.

______________________________________________

______________________________________________

______________________________________________

9. Line m is parallel to line and passes through point M.

Find the slope of line m.

______________________________________________

10. Find the equation of line m. Explain how you found the equation.

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

LESSON

10-1

Name _______________________________________ Date __________________ Class _________________

Slope and Perpendicular Lines

Reteach

Two non-vertical lines are perpendicular to each other if the

product of their slopes is 1.

The slope of line a is 2. The slope of line b is 1

.2

Since 1

2 1,2

the lines are perpendicular.

If a parallelogram on the coordinate plane has four right

angles, that parallelogram is a rectangle.

To prove this, show the consecutive sides of the parallelogram are

perpendicular. So, show the product of the slopes is 1.

Prove that WXYZ is a rectangle.

1. Name the coordinates of W, X, Y, and Z.

W ___________ X ___________ Y ___________ Z ___________

2. Calculate the slopes of each side of the parallelogram.

WX ___________ XY ___________

YZ ___________ ZW ___________

3. Find the products of the slopes of these segments:

WX and XY _________ XY and YZ _________

YZ and ZW _________ ZW and YZ _________

4. Is WXYZ a rectangle? Why or why not?

________________________________________________________________________________________

LESSON

10-2

Name _______________________________________ Date __________________ Class __________________



Find the slope. The first one is done for you.

1. Line segment AB

_______________________________________

2. Line segment CD

_______________________________________

3. Are AB and CD perpendicular? Explain your reasoning.

________________________________________________________________________________________


points (7, 7) and (3, 1). Find the slope. The first one is done for you.

4. Line A 5. Line B

_______________________________________ ________________________________________

6. Are Line A and Line B perpendicular? Explain your reasoning.

________________________________________________________________________________________

________________________________________________________________________________________

For Problems 7–13, use the graph. The first one is done for you.

7. Describe a method you can use to show that

Figure GHJK is a rectangle.

__________________________________________________

_________________________________________________

Find each slope.

8. GH 9. HJ 10. JK 11. KG

_______________ _______________ _______________ ________________

12. Is Figure GHJK a rectangle? Explain your reasoning.

________________________________________________________________________________________

________________________________________________________________________________________

A rectangle has four right angles. So, show that

slope 1

2

adjacent line segments are perpendicular.

2

LESSON

10-2

Name _______________________________________ Date __________________ Class __________________




points (2, 3) and (2, 2).

1. Are the lines perpendicular? Explain your reasoning.

________________________________________________________________________________________

________________________________________________________________________________________

Figure WXYZ has as its vertices the points W(2, 7), X(5, 6), Y(5, 4),

and Z(1, 2).

Find each slope.

2. WX 3. XY 4. YZ 5. ZW

_______________ _______________ _______________ ________________

6. Is Figure WXYZ a rectangle? Explain your reasoning.

________________________________________________________________________________________

________________________________________________________________________________________

For Problems 7–10, use the graph at the right.

7. Find the slope of line .

_________________________________________________

8. Explain how you found the slope.

_________________________________________________

_________________________________________________

_________________________________________________

9. Line t is perpendicular to line and passes through point K.

Find the slope of line t.

_________________________________________________

10. Find the equation of line t. Explain how you found the equation.

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

LESSON

10-2

Name _______________________________________ Date __________________ Class __________________

Properties of Parallelograms


Fill in the blanks with words from the Word Bank to complete each

definition or theorem. The first one is done for you.

1. If a quadrilateral is a parallelogram, then its consecutive angles are

____________________.

2. If a quadrilateral is a parallelogram, then its opposite sides are

____________________.

3. If a quadrilateral is a parallelogram, then its diagonals

____________________ each other.

4. If a quadrilateral is a parallelogram, then its opposite angles are

____________________.

Find each measure. The first one is done for you.

5. RS 6. mK

_______________________________________ ________________________________________

7. Angle Y of parallelogram WXYZ 8. Side GD of parallelogram DEFG

_______________________________________ _________________

9. QP _________________

10. NQ _________________

11. OR _________________

12. QO _________________

LESSON

9-1

Word Bank

bisect

congruent

parallel

supplementary

2.5 cm

Name _______________________________________ Date __________________ Class __________________

Properties of Parallelograms


PQRS is a parallelogram. Find each measure.

1. RS __________

2. mS __________

3. mR __________

The figure shows a swing blown to one side by a

breeze. As long as the seat of the swing is parallel to

the top bar, the swing makes a parallelogram. In

ABCD, DC 2 ft, BE 41

2ft, and mBAD 75.

Find each measure.

4. AB ___________ 5. ED ___________ 6. BD ___________

7. mABC ___________ 8. mBCD ___________ 9. mADC ___________

Three vertices of GHIJ are G(0, 0), H(2, 3), and J(6, 1).

Use the grid to the right to complete Problems 10–16.

10. Plot vertices G, H, and J on the coordinate plane.

11. Find the rise (difference in the y-coordinates) from

G to H. _________________

12. Find the run (difference in the x-coordinates) from

G to H. _________________

13. Using your answers from Problems 11 and 12, add the rise to the

y-coordinate of vertex J and add the run to the x-coordinate of vertex J.

The coordinates of vertex I are (_________________, _________________).

14. Plot vertex I. Connect the points to draw GHIJ.

15. Check your answer by finding the slopes of IH and .JG

slope of IH _________________ slope of JG _________________

16. What do the slopes tell you about IH and ?JG __________________________________

LESSON

9-1

Name _______________________________________ Date __________________ Class __________________

Conditions for Parallelograms


For each definition or theorem, tell what information you would

need about figure WXYZ to conclude that it is a parallelogram.

For some problems, there is more than one correct answer, but give

only one example per problem. The first one is done for you.

1. If both pairs of opposite angles of a quadrilateral are

congruent, then the quadrilateral is a parallelogram. __________________________________

2. If both pairs of opposite sides of a quadrilateral are

parallel, then the quadrilateral is a parallelogram. ___________________________________

3. If an angle of a quadrilateral is supplementary to both of its

consecutive angles, then the quadrilateral is a parallelogram.

________________________________________________________________________________________

4. If one pair of opposite sides of a quadrilateral are parallel and

congruent, then the quadrilateral is a parallelogram.

________________________________________________________________________________________

5. If the diagonals of a quadrilateral bisect each other, then the

quadrilateral is a parallelogram. (Hint: The diagonals of the

figure are WY and .XZ )

________________________________________________________________________________________

6. If both pairs of opposite sides of a quadrilateral are

congruent, then the quadrilateral is a parallelogram.

_______________________________________

This desk lamp has a circular base and a movable arm in the shape of

a parallelogram. Use the figure to solve Problems 7–10. The first one

is done for you.

7. AD is vertical. Name another side of parallelogram ABCD that is

also vertical. _________________

8. Because AD is attached to the base, AD stays vertical as the arm is

moved. Tell what happens to BC as the arm is moved up or down.

________________________________________________________________________________________

9. What happens to the size of A as the lamp is raised?

LESSON

9-2

W Y and X Z

BC

Name _______________________________________ Date __________________ Class __________________

Conditions for Parallelograms


Determine whether each figure is a parallelogram for the given values

of the variables. Explain your answers.

1. x 9 and y 11 2. a 4.3 and b 13

_______________________________________ ________________________________________

_______________________________________ ________________________________________

_______________________________________ ________________________________________

A quadrilateral has vertices E(1, 1), F(4, 5), G(6, 6), and H(3, 2).

Complete Problems 3–6 to determine whether EFGH is a

parallelogram.

3. Plot the vertices and draw EFGH.

4. Use the Pythagorean Theorem to find the lengths of

sides EF and .HG EF ______________ HG ______________

5. Use the Slope Formula to find the slopes of sides EF and

.GH slope of EF ______________ slope of HG ______________

6. The answers to Problems 4 and 5 reveal important information about

figure EFGH. State the theorem that uses the information found to

prove that EFGH is a parallelogram.

________________________________________________________________________________________

________________________________________________________________________________________

Use the given method to determine whether the quadrilateral with the

given vertices is a parallelogram.

7. Find the slopes of all four sides: J(4, 1), K(7, 4), L(2, 10), M(5, 7).

________________________________________________________________________________________

________________________________________________________________________________________

8. Find the lengths of all four sides: P(2, 2), Q(1, 3), R(4, 2), S(3, 7).

________________________________________________________________________________________

________________________________________________________________________________________

LESSON

9-2

Name _______________________________________ Date __________________ Class _________________

166

Properties of Rectangles, Rhombuses, and Squares


Match each figure with the letter of one of the vocabulary terms.

Use each term once. The first one is done for you.

1. 2. 3.

_______________________ _____________________ _______________________________

Fill in the blanks to complete each theorem. The first one is done for you.

4. If a parallelogram is a rhombus, then its diagonals are ___________________________.

5. If a parallelogram is a rectangle, then its diagonals are ___________________________.

6. If a quadrilateral is a rectangle, then it is a ___________________________.

7. If a parallelogram is a rhombus, then each diagonal ___________________________ a pair of opposite angles.

The part of a ruler shown is a rectangle with AB 3 inches and

BD 31

4 inches. Find each length. The first one is done for you.

8. DC _____________________________________

9. AC _____________________________________

10. DE _____________________________________

Use the phrases and theorems from the Word Bank to

complete this two-column proof. The first step is done for you.

11. Given: GHIJ is a rhombus.

Prove: 1 3

LESSON

9-3

Alternate Interior Thm.

Trans. Prop. of

2 3

Given

parallel

3 in.

perpendicular

B

A. rectangle B. rhombus C. square

Name _______________________________________ Date __________________ Class _________________

167

Conditions for Rectangles, Rhombuses, and Squares


Use the figure for Exercises 1–4. Determine whether each conclusion

is valid. Then fill in the blanks to make a true statement. The first one

has been done for you.

1. Given: , AC BD AB BC

Conclusion: ABCD is a rhombus.

The conclusion is valid. When diagonals of a parallelogram are perpendicular,

the parallelogram must be a rhombus.

2. Given: || , , AB DC AD BC AC BD

Conclusion: ABCD is a parallelogram.

The conclusion is _________________ . When opposite sides of a quadrilateral are

_________________ and _________________ , the quadrilateral is a _________________ .

3. Given: || , , AB DC AD BC AC BD

Conclusion: ABCD is a rectangle.

The conclusion is _____. You need to know that _____ and AD are _________________ .

4. Given: , , m m 45AB DC AD BC ADB ABD

Conclusion: ABCD is a square.

The conclusion is _____. You need to know that AC and _____ are _________________ .

Complete Exercises 5–8 to show that the conclusion is valid. The first

one has been done for you.

Given: , , and .JK ML JM KL JK KL

Conclusion: JKLM is a square.

5. Because and ,JK ML JM KL JKLM is a parallelogram.

6. Because JKLM is a parallelogram and M is a right angle, JKLM is a

___________________________.

7. Because JKLM is a parallelogram and ,JK KL JKLM is a

___________________________.

8. Because JKLM is a _________________ and a _________________,

JKLM is a square.

LESSON

9-4

Name _______________________________________ Date __________________ Class _________________

168

Conditions for Rectangles, Rhombuses, and Squares


Fill in the blanks to complete each theorem.

1. If one pair of consecutive sides of a parallelogram are congruent, then

the parallelogram is a _____________________.

2. If the diagonals of a parallelogram are _____________________, then

the parallelogram is a rhombus.

3. If the _____________________ of a parallelogram are congruent, then

the parallelogram is a rectangle.

4. If one diagonal of a parallelogram bisects a pair of opposite angles,

then the parallelogram is a _____________________.

5. If one angle of a parallelogram is a right angle, then the parallelogram

is a _____________________.

Use the figure for Problems 6–7. Determine whether each conclusion is

valid. If not, tell what additional information is needed to make it valid.

6. Given: AC and BD bisect each other. AC BD

Conclusion: ABCD is a square.

________________________________________________________________________________________

7. Given: ,AC BD AB BC

Conclusion: ABCD is a rhombus.

________________________________________________________________________________________

Complete Problems 8–11 to show that the conclusion is valid.

Given: , ,JK ML JM KL and .JK KL M is a right angle.

Conclusion: JKLM is a square.

8. Because JK ML and ,JM KL JKLM is a _____________________.

9. Because JKLM is a parallelogram and M is a right angle, JKLM is a

_____________________.

10. Because JKLM is a parallelogram and ,JK KL JKLM is a _____________________.

11. Because JKLM is a _____________________ and a _____________________,

JKLM is a square.

LESSON

9-4

Name _______________________________________ Date __________________ Class _________________

Properties of Quadrilaterals

Module Quiz: B

ABCD is a parallelogram.

1. Determine whether each statement is true

or false.

A AE BE True False

B AB CD True False

C AD BC True False

D AE EC True False

2. Compare the diagonals of a rhombus and

rectangle, where neither is a square. How

are they the same? How are they different?

_______________________________________

_______________________________________

Use the figure for 3–4.

3. If the side lengths of quadrilateral KLMN

are known, what theorem can you use to

prove the figure is a parallelogram?

_______________________________________

4. For what value of x and y is KLMN a

parallelogram?

_______________________________________

_______________________________________

5. JKLM is a parallelogram. State whether

each additional single condition will make

JKLM a rhombus. Explain your reasoning.

A Opposite sides are congruent.

Yes No

___________________________________

B Diagonals bisect.

Yes No

___________________________________

C Diagonals are perpendicular.

Yes No

___________________________________

D Opposite angles are congruent.

Yes No

___________________________________

6. Determine the word belonging in each

blank. Explain your reasoning.

A A square is _____?_____ a rectangle.

always sometimes never

___________________________________

B A parallelogram is _____?_____ a

rectangle.


___________________________________

C A kite is _____?_____ a parallelogram.


___________________________________

7. State whether the diagonals of each figure

bisect each other.

A Parallelogram Yes No

B Square Yes No

C Kite Yes No

D Isosceles Trapezoid Yes no

MODULE

9

Name _______________________________________ Date __________________ Class ______________

Documents

Name Date Class Grade 9 2015 / 2016 - Funmath8's Blog · PDF fileName _____ Date _____ Class _____ Grade 9 2015 / 2016 First Semester 2015- 2016

Name Date Class Grade 9 2015 / 2016 - Funmath8's Blog · PDF fileName ___ Date _ Class ___ Grade 9 2015 / 2016 First Semester 2015- 2016