about parallelograms. . . the diagonals of proving that a … · 2019. 1. 10. · a parallelogram...

63 Proving That a QuadrilateralIs a Parallelogram

% Mathematics Florida StandardsMAFS.912.G-C0.3.11 Prove theorems

about parallelograms... the diagonals ofa parallelogram bisect each other and itsconverse... Also MAFS.912.G-SRT.2.5

MP 1.1viP3

Objective To determine whether a quadrilateral is a parallelogram

^ ^ ^

V/ 'It Getting Ready! ► X C 4

Can you visualizeparallelogramscomposed oftriangles in thepattern?

coch section of glass in the exteriorof a building in Macau, China, formsan equilateral triangle. Do you thinkthe window washer's feet stay parallelto the ground as he lands at each levelof windows? Explain. (Assume that thebases of the lowest triangles oreparallel to the ground.)

▼

MATHEMATICALPRACTICES

In the Solve It, you used angle properties to show that lines are parallel. In this lesson,you will apply the same properdes to show that a quadrilateral is a parallelogram.

Ess6ntial UndGrstanding You can decide whether a quadrilateral is aparallelogram if its sides, angles, and diagonals have certain properties.

In Lesson 6-2, you learned theorems about the properties of parallelograms. In thislesson, you will learn the converses of those theorems. That is, if a quadrilateral hascertain properties, then it must be a parallelogram. Theorem 6-8 is the converse ofTheorem 6-3.

Theorem 6-8

TheoremIf both pairs of opposite sides of aquadrilateral are congruent, thenthe quadrilateral is a parallelogram.

If. . . Then . ..C AB^'^ ABCD is a O

\ DBC^DA to;

—'D

You will prove Theorem 6-8 In Exercise 20.

Theorems 6-9 and 6-10 are the converses of Theorems 6-4 and 6-5, respectively. Theyuse angle relationships to conclude that a quadrilateral is a parallelogram.

C PowerGeometry.com I Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram 367

Which theorem

should you use?The diagram gives youinformation about sides.

Use Theorem 6-8 because

it uses sides to conclude

that a quadrilateral is:aparallelogram. i

Theorem 6-9

Theorem

If an angle of a quadrilateral

is supplementary to both

of its consecutive angles,

then the quadrilateral is a

parallelogram.

If...

•r Im/LA -h mZ-B = 180

m^A + m/.D = 180

Then ...

ABCD is a O

—'d

You will prove Theorem 6-9 in Exercise 21.

Theorem 6-10

Theorem

If both pairs of opposite angles of a

quadrilateral are congruent, then

the quadrilateral is a parallelogram.

If...

B C

Then ...

ABCD is a O

fl - ^ r

'D

You will prove Theorem 6-W in Exercise 18.

You can use algebra together with Theorems 6-8,6-9, and 6-10 to find segment lengthsand angle measures that assume that a quadrilateral is a parallelogram.

Problem 1 Finding Values for Paroiielogranis

For what value of y must PQRS be a parallelogram?

Step 1 Find x.

3a: — 5 = 2a: 4-1If opp. sides are =, then the quad.

IgiRilDDEO RESPONSE

3x - 5

X + 2'

IS a o.

Subtract 2x from each side.

Add 5 to each side.

a:-5 = 1

a: = 6

Step 2 Findy.

y = a: -h 2 If opp. sides are s, then the quad, is a o.

= 6 + 2 Substitute 6 for x.

= 8 Simplify.

For PQRS to be a parallelogram, the value of y must be 8.

^ Got It? 1 . Use the diagram at the right. For what values of x and ymust EFGH be a parallelogram?

2x + 1

(4x + 13)=

'(y + 10)°^^ (12x + 7)° ̂

368 Chapter 6 Polygons and Quadrilaterals

You know that the converses of Theorems 6-3,6-4, and 6-5 are true. Using what you

have learned, you can show that the converse oflheorem 6-6 is also true.

Theorem 6-11

Theorem

If the diagonals of a

quadrilateral bisect each

other, then the quadrilateral

is a parallelogram.

mZmA fJ D

Then...

ABCD is a C7

fi ^ ^ r

fi

Proof Proof of Theorem 6-11

Given: i4C and BD bisect each other at E.

Prove: ABCD is a parallelogram.D

LAEB s lCED

Vertical A are s.

AC and BD bisect each other at E.

Given

61 = OKBE = DE

Def. of segment bisector

ABEC s z,DEA

Vertical A are s.

LAEB = ACED

SAS

ABAE = ADCE

Corresp. parts of = A are s

If alternate Interior A s,then lines are ||.

ABEC s ADEA

SAS

AECB = AEAD

Corresp. parts of s A are

^11

If alternate interior A s

then lines are ||.

ABCD is a parallelogram.

Def. of parallelogram


^hinkHow do you decideIf you have enoughInformation?

If you can satisfy everycondition of a theorem

about parallelograms,then you have enoughinformation.

Theorem 6-12 suggests that if you keep two objects of the same length parallel, suchas cross-country skis, then the quadrilateral formed by connecting their endpoints isalways a parallelogram.

Theorem 6-12

Theorem

If one pair of opposite sides

of a quadrilateral is both

congruent and parallel,

then the quadrilateral is a

parallelogram.

If...

g I, c W = m

/.' / ^1154

Then ...

ABCD is a O

'cn'a'-^—'dYou will prove Theorem 6-12 in Exercise 19.

J\

Problem 2 Deciding Whether a Quadrilateral Is a Parallelogram

Can you prove that the quadrilateral is a parallelogram based on the given

information? Explain.

O Given: AB = 5, CD = 5,m/.A = 50, mLD = 130

Prove: ABCD is a parallelogram.

0 Given: HI = HK, JI = JK

Prove: HIJK is a parallelogram.

H

Yes. Same-side interior angles yl and

D are supplementary, so AB || CD.Since AB = CD, ABCD is a

No. By Theorem 6-8, you need to show that

both pairs of opposite sides are congruent,

not consecutive sides.

parallelogram by Theorem 6-12.

Got It? 2. Can you prove that the quadrilateral is a parallelogram based on the given~' information? Explain.

a. Given: EF = '^,W\\FG b. Given; ̂ ALN = ADNL, ̂ANL = ̂DLNProve: DEFG is a parallelogram. Prove: LAND is a parallelogram.

N


As the arms of the

lift move, what

changes and whatstays the same?The angles the arms formwith the ground and theplatform change, but thelengths of the arms andthe platform stay thesame.

Identifying Parallelograms

Vehicle Lifts A truck sits on the platform of a vehicle lift. Two moving arms raise the

platform until a mechanic can fit underneath. Why will the truck always remain! parallel to the ground as it is lifted? Explain.

The angles of PQRS change as platform QR rises, but its side lengths remain the same.Both pairs of opposite sides are congruent, so PQRS is a parallelogram by Theorem 6-8.

By the definition of a parallelogram, PS || QR. Since the base of the lift PS lies along theground, platform QR, and therefore the truck, will always be parallel to the ground.

© Got It? 3. Reasoning What is the maximum height that the vehicle lift can elevate thetruck? Explain.

Concept Summary Proving That a Quadrilateral Isa Parallelogram

SourceMethod

Prove that both pairs of opposite sides are parallel. Definition ofparallelogram

Diagram

oProve that both pairs of opposite sides are

congruent.

Theorem 6-8

Prove that an angle is supplementary to both of its Theorem 6-9consecutive angles.

75

75° 105°

Prove that both pairs of opposite angles arecongruent.

Theorem 6-10

Prove that the diagonals bisect each other. Theorem 6-11

Prove that one pair of opposite sides is congruent Theorem 6-12and parallel.


&Lesson CheckDo you know HOW?

1. For what value of y must

LMNP be a parallelogram?

For Exercises 2 and 3, is the given information enough

to prove that ABCD is a parallelogram? Explain.

y 68°

2. B

'2y'

irD

MATHEMATICAL

Do you UNDERSTAND? USf PRACTICES

4. Vocabulary Explain whyyou can now tvrite abiconditional statement regarding opposite sides of a

parallelogram.

5. Compare and Contrast HowisTheorem 6-11 in this

lesson different from Theorem 6-6 in the previous

lesson? In what situations should you use each

theorem? Explain.

6. Error Analysis Your friend says, "If a quadrilateral

has a pair of opposite sides that are congruent and

a pair of opposite sides that are parallel, then it is a

parallelogram." What is your friend's error? Explain.

MATHEMATICAL

PRACTICESPractice and Problem-Solving Exercises

Practice Algebra For what values of x and y must ABCD be a parallelogram? ^ See Problems 1 and 2.

10.

3x+ 1

6x-4

B

n 5x - 8 C

3x° (y + 78)

3y° (4x - 21)

12. B

A 2x + 7 ®

X + 38

4x - 1 °

Can you prove that the quadrilateral is a parallelogram based on the giveninformation? Explain.

13.

16. Fishing Quadrilaterals are formed on

the side of this fishing tackle box bythe adjustable shelves and connecting

pieces. Explain why the shelves are

always parallel to each other no matter

what their position is.

^ See Problem 3.


Apply ^^17. Writing Combine each of Theorems 6-3, 6-4,6-5, and 6-6 with its converse fromthis lesson into biconditional statements.

18. Developing Proof Complete this two-column proof of Theorem 6-10.

Given: LA = LC, LB = LD


Statements Reasons

B

D

1) x + y + :c-l-y= 360 1) The sum of the measures of theangles of a quadrilateral is 360.

2) 2(A: + y]=360 2) a. ?

3) x + y=180 3) b. ?

4) LA and LB are supplementary. 4) Definition of supplementary

LA and LD are supplementary.

5) c. ? II ? , ? II ? 5) d. ?

6) /IBCD is a parallelogram. 6) e. ?

19. Think About a Plan Prove Theorem 6-12.

Given: \\DA,^ = m


• How can drawing diagonals help you?

• How can you use triangles in this proof?

20. Prove Theorem 6-8.

Proo^Given: AB = CD, EC = DA


D

2 A Prove Theorem 6-9.Proof

Given: Z./1 is supplementary to Z.5

LA is supplementary to LD.


B C

A

Algebra For what values of the variables must ABCD be a parallelogram?

22. 4 B 24.

Proof

B

(3x + ior /(2x+ 158x + sr 3x + 6

(4x - 33)°

25. Given: ATRS = ARTW

Prove: ilSTlVisaparallelogram.

26. Open-Ended Sketch two noncongruent parallelogramsABCD and EFGH such that AC = EG and BD = FH.

Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram 373

Challenge! Construction In the figure at the right, point D is constructed byProof drawing two arcs. One has center C and radius AB. The other has center

B and radius AC. Prove that AM is a median of AABC.

28. Probability If two opposite angles of a quadrilateral measure 120and the measures of the other angles are multiples of 10, what is theprobability that the quadrilateral is a parallelogram?

M

X?

SAT/Aa

Standardized Test Prep

29. From which set of information can you conclude that RSTW is a parallelogram?

RS WT, RS = ST

Wr,'^ = mv

Short

^e^ponse

Extended

^Response

30. Write a proof using the diagram.

Given: ANRf = ACPT, Jn || CT

Prove: JNTC is a parallelogram.

31. Use the figure at the right.a. Write an equation and solve for x

b. Is AF II DEI Explain.c. Is BDfF a parallelogram? Explain.

CC:^ RS = ST,RW= WT

^=TZ,^=WZ

N

7X-11)

Mixed Review

Algebra Find the value of each variable in each parallelogram. ^ See Lesson 6-2.

32. 33. 3m-12 34.

+ 15 (8x+ 15)/-3'

1102'm + 7

D35. Explain how you can use overlapping

congruent triangles to prove AC = BD.

Get Ready! To prepare for Lesson 6-4, ^do Exercises 36-44.

PACE is a parallelogram and mLPAC = 124. Complete the following.

36. AC=m 37. CE=m 38. PA = m

39. RE =m 40. CP =■ 41. m^CEP=«

42. m/LEPA = ■ 43. mAECA = ■ 44. mAACR = ■

^ See Lessons 4-4 and 4-7.

^ See Lessons 5-2 and 6-2.

^ c


about parallelograms. . . the diagonals of proving that a … · 2019. 1. 10. · a parallelogram...

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