chapter 5€¦ · · 2015-01-07each figure in exs. 11 and 12 is a parallelogram with its...

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GEOMETRY

CHAPTER 5: QUADRILATERALS

NAME

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Section 5-1: Properties of Parallelograms

Definition of a ____________________ ( ) – a quadrilateral with both pairs of opposite sides parallel

I. Record the length of each side (in cm) and the measure of each angle of this parallelogram. Use the

measurements to make three conclusions.

AB = ________ BC = ________

DC = ________ AD = ________

m A = _______ m B = _______

m C = _______ m D = _______

1. ____________________________________________________________________

2. ____________________________________________________________________

3. ____________________________________________________________________

II. Using a ruler, draw in ̅̅ ̅̅ and ̅̅ ̅̅ . Label the intersection of these segments X. Find the following

measures. Make a fourth conclusion.

AC = ________ DB = ________

AX = ________ DX = ________

CX = ________ BX = ________

4. ____________________________________________________________________

III. Given: AB = 8, BC = 6, AC = 6.1, BD = 11.8, m A = 130°

Mark the given information in the parallelogram below. Draw in the diagonals and label their intersection “X”.

Find the following without using a ruler. (Base your answers on the 4 conclusions you made above).

DC = _______ AD = ________

m B = ______ m C = _______

m D = ______ AX = ________

DX = ________ BX = ________

CX = ________

A B

D C

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THEOREM

Opposite sides of a parallelogram are __________________.

THEOREM

Opposite __________ of a parallelogram are congruent.

THEOREM

Diagonals of a parallelogram ___________ each other.

Examples:

Classify each statement as true or false.

1. Every parallelogram is a quadrilateral. 1. ______________________

2. Every quadrilateral is a parallelogram. 2. ______________________

3. All angles of a parallelogram are congruent. 3. ______________________

4. All sides of a parallelogram are congruent. 4. ______________________

5. In RSTU, ̅̅̅̅ ̅̅ ̅̅ . 5. ______________________

6. In ABCD, if m A = 50°, then m C = 130°. 6. ______________________

7. In XWYZ, ̅̅ ̅̅ ̅̅ ̅̅ ̅. 7. ______________________

8. In ABCD, ̅̅ ̅̅ and ̅̅ ̅̅ bisect each other. 8. ______________________

In Exercises 9 and 10, quad RSTU is a parallelogram. Find the values of x, y, a, and b.

9. 10.

Each figure in Exs. 11 and 12 is a parallelogram with its diagonals drawn. Find the values of x and y.

11. 12.

x°

45° 35°

y°

12

R S

T U

9

b

3y + 4

8 4x

13

18 22

4y – 2 2x + 8

y°

R

9

U

80°

x°

6

a T

b

S

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Section 5-2: Ways to Prove that Quadrilaterals Are Parallelograms

1. Show that both pairs of opposite sides are ______________.

(definition)

2. Show that both pairs of opposite sides are _______________.

(theorem)

3. Show that one pair of opposite sides are ____________

and _______________. (theorem)

4. Show that both pairs of opposite angles are ________________.

(theorem)

5. Show that the diagonals __________ each other.

(theorem)

Examples:

Complete with always, sometimes, or never.

1. The diagonals of a quadrilateral ___________________ bisect each other.

2. If the measures of two angles of a quadrilateral are equal, then the quadrilateral is ____________________ a

parallelogram.

3. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is

_________________ a parallelogram.

4. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is _________________

a parallelogram.

5. To prove a quadrilateral is a parallelogram, it is ____________ enough to show that one pair of opposite

sides is parallel.

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Decide if each quadrilateral must be a parallelogram. If the answer is yes, state the definition or theorem that

applies.

6. 7.

__________________________________ __________________________________________

__________________________________ __________________________________________

8. 9.

__________________________________ __________________________________________

__________________________________ __________________________________________

10.

_______________________________

_______________________________

State the principal definition or theorem that enables you to deduce, from the information given, that quad.

ABCD is a parallelogram.

11. BE = ED; CE = EA

12. BAD DCB; ADC CBA

13. _____

BC || _____

AD ; _____

AB || _____

DC

14. _____

BC _____

AD , _____

AB _____

DC

15. _____

BC ||_____

AD ; _____

BC _____

AD

4 4

17

17

5 5

100°

100°

80°

80°

6 6

5

5 8

8

A

B C

D

E

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A D Opposite sides are parallel.

Opposite sides are congruent.

Opposite angles are congruent.

B C Consecutive angles are

supplementary.

A diagonal of a divides it into 2 triangles.

The diagonals of a bisect each other.

(Note: Bisect means to cut in half.)

PAUL

L U

P A

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Section 5-3: Theorems Involving Parallel Lines

Theorem 1: If 2 lines are parallel, then all points on one line are _________________ from

the other line.

1. What do you think equidistant means?

_________________________________________________

2. According to the theorem above, x = ____ and y = _____.

3. Why do you think the 3 lines were drawn perpendicularly to the 2 parallel lines?

________________________________________________

Theorem 2: If 3 parallel lines cut off congruent segments on one transversal, then they cut

off _________________ segments on every transversal.

4. According to Theorem 5-9 on the previous page, x = _____, y = _____,

and z = _____.

4 x y

7

x

y

6

8

z

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5. Solve for a.

Theorem 3: A line that contains the midpoint of one side of a triangle and is parallel to another

side passes through the ___________________ of the third side.

6. According to Theorem 3, if AB = BC and ̅̅ ̅̅ ̅̅ ̅̅ , then _____________.

7. If AC = 14, then AB = _____ and BC = _____.

8. If AD = 14, then DE = _____ and AE = _____.

Theorem 4: The segment that joins the midpoints of 2 sides of a triangle:

1._________________________________________

2._________________________________________

9. Using the diagram above, if B is the midpoint of ̅̅ ̅̅ and D is the midpoint

of ̅̅ ̅̅ , then _____ =

(_____) and ̅̅ ̅̅ || _____.

10. If AB = BC = 6 and AD = DE = 8 and BD = 12, then CE = _____.

10

10

5a

15

A

B

C

D

E

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B

AC

R S

T

B

C

T

S

R

A

B

CA

X

Z

11. Given: R, S, and T are midpoints of the sides of ABC.

Complete the table.

12. Given: ⃡ || ⃡ || ⃡ ; ̅̅̅̅ ̅̅̅̅

Complete.

a. If RS = 12, then ST = ______.

b. If AB = 8, then BC = ______.

c. If AC = 10x, then BC = ______.

13. Given: Points X, Y, and Z are midpoints of ̅̅ ̅̅ , ̅̅ ̅̅ , and ̅̅ ̅̅ .

Complete.

a. If AC = 24, then XY = ______.

b. If AB = k, then YZ = ______.

Y c. If XZ = 2k + 3, then BC = _______. d. If AB = 9, BC = 8, AC = 6, then the perimeter

of XYZ = ______.

e. If the perimeter of XYZ = 24, then the

perimeter of ABC = ______.

f. Name all congruent triangles.

AB BC AC ST RT RS

a. 12 14 18

b. 15 22 10

c. 10 9 7.8

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R

11

11

C

B A

L M

N

• •

•

•

•

•

•

• U

T

S

R

14. Name all points that must be midpoints of the sides of ABC.

15. Given: AB = BC = CD

Complete.

a. If RS = 6, then SU = ______.

b. If RT = 6x + 2 and TU = 10, then x = ____.

D C

B A

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Review for quiz on 5-1 to 5-3

I. Quadrilateral KLMN is a parallelogram. Complete each statement.

1. ̅̅̅̅̅ _____

2. NML _____

3. ̅̅̅̅̅ _____

4. 1 _____

5. KNM is supplementary to _____ or _____.

_________________________________________________

II. Decide if it is possible to prove that the quadrilateral below is a

parallelogram. Answer yes if it is possible. Answer no if it is not possible.

6. ̅̅ ̅̅ ̅̅ ̅̅ ; ̅̅ ̅̅ ̅̅ ̅̅ _____

7. ̅̅ ̅̅ ̅̅ ̅̅ ; ̅̅ ̅̅ ̅̅ ̅̅ _____

8. ̅̅ ̅̅ ̅̅ ̅̅ ; ̅̅ ̅̅ ̅̅ ̅̅ _____

9. 1 3; 2 4 _____

10. ̅̅ ̅̅ ̅̅ ̅̅ ; ̅̅ ̅̅ ̅̅ ̅̅ _____

_________________________________________________

K

5 6

4 3

2 1 8

7

X

N M

L

F

2 1

3 4

G

A B

E

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III. Complete each statement with the number that makes the statement true.

11. DE = _____

12. m ADE = _____

13. m ABC = __

14. m C = ____

15. EC = _____

16. AC = _____

17. DF = _____

18. m DFC = _____

19. If PQ = 5, then PS = _____.

20. If YZ = 7, then XY = _____.

21. If QX = 12, then PW = _____.

22. If WZ = 24, then WX = _____.

12

54°

A

E D

B F C

18

41°

10

T

X

P

R

S

Y

Z

Q

W

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IV. Solve for x and y in each parallelogram.

23. x = _____ y = _____ 24. x = _____ y = _____

(6x + 2) (2y)

26 20

(4y)°

(3x + 4)° (5x + 16)°

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Section 5-4: Special Parallelograms

I. Write the briefest definition that you can for each of the following

figures. Imagine that someone is using your definition to draw the

figure.

Parallelogram - __________________________________________

______________________________________________________

Rectangle - _____________________________________________

______________________________________________________

Rhombus - ______________________________________________

______________________________________________________

Square - _______________________________________________

______________________________________________________

PARALLELOGRAM RECTANGLE

RHOMBUS SQUARE

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II. THEOREMS

THEOREM 5-12

The ______________ of a rectangle are congruent.

THEOREM 5-13

The diagonals of a rhombus are ___________________.

THEOREM 5-14

Each diagonal of a rhombus ___________ two angles of the rhombus.

THEOREM 5-15

The midpoint of the hypotenuse of a right triangle is __________________ from the

three vertices.

THEOREM 5-16

If an angle of a parallelogram is a right angle, then the parallelogram is a

________________________.

THEOREM 5-17

If two consecutive sides of a parallelogram are congruent, then the parallelogram is a

______________________.

III. Answer the following questions about the 4 figures in section I.

1. Which ones are parallelograms? __________________________

_________________________________________________

2. Is a rectangle a square? If not, why? _____________________

_________________________________________________

3. Is a square a rectangle? If not, why? _____________________

_________________________________________________

4. Is a rhombus a square? If not, why? ______________________

_________________________________________________

5. Is a square a rhombus? If not, why? ______________________

_________________________________________________

6. Is a rhombus a rectangle? If not, why? ____________________

_________________________________________________

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Examples:

I. Quadrilateral ABDC is a rectangle.

AD = 12

BC = _____

BE = _____

EC = _____

mEAC = 60

mBDA = _____

mBAE = _____

mACE = _____

II. Quadrilateral ABCD is a rhombus.

mBAE = 50; AD = 4

mAEB = _____

mDAE = _____

mACD = _____

mABE = _____

DC = _____

A B

C D

E

A B

C D

E

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A

C B

M

III. Quadrilateral ABCD is a square.

AC = 12

EB = _____

ED = _____

mBEC = _____

mEBC = _____

IV. ABC is a right ; M is the midpoint of ̅̅ ̅̅ .

If AM = 7, then MB = _____,

AB = _____, and CM = _______.

If AB = x, then AM = _____,

MB = ______, and MC = ______.

A B

C D

E

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Rectangles, Rhombuses and Squares

Rectangle Rhombus Square

with 4 right s with 4 sides with 4 right s &

4 sides

Diagonals are Diagonals are Diagonals are &

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Section 5-5: Trapezoids

1. Trapezoid - ___________________________________

____________________________________________

2. The parallel sides of a trapezoid are the ______________ .

3. The other sides are the _____________ .

4. A trapezoid with congruent legs is called an ___________

________________.

Theorem : _________________________________________

________________________________________________ .

Diagonals of an isosceles trapezoid are ______________.

Example: Given trapezoid ABCD is isosceles, find mA, mB, and mC.

A B

C 110

D

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The median of a trapezoid is the segment that ______________

________________________________________________ .

Theorem: The median of a trapezoid

1) __________________________________

2) _________________________________

__________________________________

Given: EF is a median of trapezoid ABCD.

a. If AB = 5 and CD = 15, find EF.

b. If AB = x – 3, CD = 2x - 4, and EF = 10, find x.

A B

E F

C D

A

A B

E F

C D

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Trapezoids

Exactly one pair of parallel sides

Exactly two pairs of consecutive angles are

supplementary.

1 & 2,

3 & 4 are supplementary.

Parallel sides are bases; non-parallel sides are legs.

Isosceles trapezoids have congruent legs.

4

3

2

1

base 2

legleg

base 1

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Review for Chapter 5 Test

I. Write the letter of every special quadrilateral that has the given property.

A. Parallelogram B. Rectangle C. Rhombus D. Square E. Trapezoid

1. All angles are right angles. __________________

2. All sides are congruent. __________________

3. Diagonals are congruent. __________________

4. Diagonals bisect each other. __________________

5. Diagonals are perpendicular. __________________

6. The quadrilateral is equiangular & equilateral. __________________

7. Both pairs of opposite sides are parallel. __________________

8. Exactly one pair of opposite sides is parallel. __________________

9. Both pairs of opposite sides are congruent. __________________

10. Each diagonal bisects two angles. __________________

II. Quadrilateral DKRT is a parallelogram.

11. If DK = 12 and KR = 8, then TR = _____.

12. If DR = 28 and KT = 18, then HR = _____.

13. If ̅̅ ̅̅ ̅̅ ̅̅ , then the parallelogram must be a

______________. (rectangle, rhombus, square)

14. If m 1 = 30 and m 8 = 40, then m RTD = _____.

15. If m 2 = 45 and m 3 = 55, then m 6 = _____.

16. If TH = 2x + 1 and KH = 4x, then x = _____.

D

K

T

R

H

1

8

6

7

3

2

4

5

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III. State whether the given information is sufficient to prove that quad. MNOP is a

parallelogram.(yes or no)

17. ̅̅ ̅̅ ̅̅̅̅ ; ̅̅ ̅̅ ̅̅̅̅ ____________

18. 1 5; 4 8 ____________

19. ̅̅ ̅̅ ̅̅ ̅̅ ̅; ̅̅ ̅̅ ̅̅ ̅̅ ̅ ____________

20. ____________

21. 7 3; 4 8 ____________

22. 1 2; 3 4 ____________

IV. Given: ̅̅ ̅̅ ̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ; ̅̅ ̅̅ ̅ ̅̅ ̅̅ ̅̅ ̅̅

23. If RS = 8, then TV = _____.

24. If RV = 21, then RT = _____.

25. If RT = 3x and SV = x + 8, then x = _____.

V. Given: ̅̅̅̅̅ is the median of trapezoid HIJK.

26. If KJ = 7 and HI = 15, then LM = _____.

27. If HI = 22 and LM = 17, then KJ = _____.

28. If trapezoid HIJK is isosceles

and m I = 85, then m K = _____.

29. If HI = 4x, LM = 2x + 3,

and KJ = x – 2, then x = _____.

M N

R W

S X

T Y

K J

L M

H I

S

1 4

2 3

7 6

8 5

O P

V Z

chapter 5€¦ · · 2015-01-07each figure in exs. 11 and 12 is a parallelogram with its...

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