Geometry

28. Think About a Plan  What are the values of x and y in the parallelogram? yЊ 3xЊ
• How are the angles related? 3yЊ
• Which variable should you solve for first?

Algebra  Find the value of a. Then find each side length or angle measure.

29. B a Ϫ 3.5 C 30. G hsm11gJ mse_0602_t06078.ai
H
2a Ϫ 20.4 a ϩ 1.6 (20a ϩ 30)Њ

A 18.5 D 5aЊ
(17a ϩ 48)Њ K

31. Studio Lighting  A pantograph is an expandable device shown at the

right. Pantographs are used in the television industry iEhn,spFmo, as1int1idognGminasrgee_th0e602_t06080.ai

lhigshmti1n1g ganmdsoet_h0er6e0q2u_ipt0m6e0n7t.9I.naithe photo, points D,
vertices of a parallelogram. ▱DEFG is one of many parallelograms that

change shape as the pantograph extends and retracts. E
a. If DE = 2.5 ft, what is FG? b. If m∠E = 129, what is m∠G? DF
c. What happens to m∠D as m∠E increases or decreases? Explain.
G
32. Prove Theorem 6-4. BC
Proo f Given:  ▱ABCD

Prove:  ∠A is supplementary to ∠B. A D
∠A is supplementary to ∠D.

Use the diagram at the right for each proof. hs 3m4.1 G1igvemns: e▱_0L6E0N2S_atn0d6▱08N4G.aTiH G T
Pr3o3o.f Given:  ▱LENS and ▱NGTH N H
Prove:  ∠L ≅ ∠T Prove:  LS } GT
E
35. Given:  ▱LENS and ▱NGTH
Prove:  ∠E is supplementary to ∠T . LS

Use the diagram at the right for each proof. SY T
Pr3o6o.f Given:  ▱RSTW and ▱XYTZ
Prove:  ∠R ≅ ∠X X Z hsm11gmse_0602_t06086.ai
R
37. Given:  ▱RSTW and ▱XYTZ W
Prove:  XY } RS

Find the measures of the numbered angles for each parallelogram.

38. 3 39. 28Њ hsm311gmse_06024_0t.0 6087385.aЊ i1
1 2 38Њ 110Њ 81Њ 2
1

48Њ 2

41. Algebra  The perimeter of ▱ABCD is 92 cm. AD is 7 cm more

thhsamn 1tw1igcemAsBe._F0in6d0t2h_etl0e6n1gt3h1s.oafiallhfosmur1si1dgems osfe▱_0A6B0C2D_.t06132.ai hsm11gmse_0602_t06134.ai

Lesson 6-2  Properties of Parallelograms 365

C Challenge 42. Writing  Is there an SSSS congruence theorem for parallelograms? Explain.

43. Prove Theorem 6-7. Use the diagram at the right. A B
Proo f Given:  <AB> } <CD> } <EF>, AC ≅ CE C1
E4 3D
Prove:  BD ≅ DF to <AE> and G 2 6F
i(nHtienrts:eDctrianwg l<CinDe>satthGroaungdh<EBF>aantdHD.)parallel
H5

44. Measurement  Explain how to separate a blank card
into three strips that are the same height by using lined
paper, a straightedge, and Theorem 6-7.

Standardized Test Prep hsm11gmse_0602_t06136.ai

SAT/ACT 45. PQRS is a parallelogram with m∠Q = 4x and m∠R = x + 10. Which statement P Q
explains why you can use the equation 4x + (x + 10) = 180 to solve for x? S R
The measures of the interior angles of a quadrilateral have a sum of 360.
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Consecutive angles of a parallelogram are supplementary.

46. In the figure of DEFG at the right, DE } GF . Which statement must be true? DE

m∠D + m∠E = 180 DE ≅ GF hsm11gmse_0602_t12794

m∠D + m∠G = 180 DG ≅ EF GF

Short 47. An obtuse triangle has side lengths of 5 cm, 9 cm, and 12 cm. What is the length of
the side opposite the obtuse angle?
Response
5 cm 9 cm 12 cm not enough information

48. Find the measure of one exterior angle of a regular hexagon. Explain your method. hsm11gmse_0602_t06139.ai

Mixed Review

Find the sum of the measures of the interior angles of each polygon. See Lesson 6-1.
52. 40-gon
49. decagon 50. 16-gon 51. 25-gon

53. What additional information do you A See Lesson 4-6.
See Lesson 6-2.
need to prove △ADC ≅ △ABC by the

HL Theorem? DCB

G et Ready!  To prepare for Lesson 6-3, do Exercise 54.

54. Two consecutive angles in a parallelogram have measures x + 5 and
4x - 10. Find the measure of the smaller angle.
hsm11gmse_0602_t06140.ai

366 Chapter 6  Polygons and Quadrilaterals

6-3 Proving That a Quadrilateral CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Is a Parallelogram
GM-ACFOS.C.9.121. GP-rCovOe.3th.1e1o rePmrosvaebtohuetorems
paabroaullteploagrarallmelsog. r.a.mthse.d. i.atghoendailasgoofnaals of
paaprarllaelloeglorgarmambisbeiscetcetaecahcohtohtehrearnadnidtsits
converse . . . Also GM-ASFRST.9B1.52.G-SRT.2.5

MP 1, MP 3

Objective To determine whether a quadrilateral is a parallelogram

Can you visualize Each section of glass in the exterior
parallelograms of a building in Macau, China, forms
composed of an equilateral triangle. Do you think
triangles in the the window washer’s feet stay parallel
pattern? to the ground as he lands at each level
of windows? Explain. (Assume that the
bases of the lowest triangles are
parallel to the ground.)

MATHEMATICAL

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

Essential Understanding  You can decide whether a quadrilateral is a
parallelogram if its sides, angles, and diagonals have certain properties.

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

Theorem 6-8 If . . . C AB ≅ CD Then . . .
B BC ≅ DA ABCD is a ▱
Theorem
If both pairs of opposite sides of a A D BC
quadrilateral are congruent, then
the quadrilateral is a parallelogram. AD

You will prove Theorem 6-8 in Exercise 20.

Theorems 6-9 and 6-10 are the converseshosfmTh1e1ogremmsse6_-04 6an0d2_6t-50,6r4es7p3e.catiivheslmy.1T1hgemy se_0603_t06433.a

use angle relationships to conclude that a quadrilateral is a parallelogram.

Lesson 6-3  Proving That a Quadrilateral Is a Parallelogram 367

Theorem 6-9

Theorem If . . . C  m∠A + m∠B = 180 Then . . .
If an angle of a quadrilateral B m∠A + m∠D = 180 ABCD is a ▱
is supplementary to both
of its consecutive angles, A D BC
then the quadrilateral is a
parallelogram. AD

You will prove Theorem 6-9 in Exercise 21.

hsm11gmse_0603_t06432.ai

Theorem 6-10 hsm11gmse_0603_t06433.ai

Theorem If . . . Then . . .
If both pairs of opposite angles of a B
quadrilateral are congruent, then C ∠A ≅ ∠C ABCD is a ▱
the quadrilateral is a parallelogram. A  ∠B ≅ ∠D BC

D AD

You will prove Theorem 6-10 in Exercise 18.

hsm11gmse_0603_t06434.ai

hsm11gmse_0603_t06433.ai
You can use algebra together with Theorems 6-8, 6-9, and 6-10 to find segment lengths
and angle measures that assume that a quadrilateral is a parallelogram.

Problem 1 Finding Values for Parallelograms

Which theorem For what value of y must PQRS be a parallelogram? P 3x Ϫ 5 Q
should you use?
The diagram gives you Step 1 Find x. xϩ2 y
information about sides. 3x - 5 = 2x + 1 Iisf oap ▱p. s. ides are ≅, then the quad. S
Use Theorem 6-8 because x - 5 = 1 Subtract 2x from each side. 2x ϩ 1 R
it uses sides to conclude
that a quadrilateral is a x = 6 Add 5 to each side. 8
parallelogram.

Step 2 Find y.

y = x + 2 If opp. sides are ≅, then the quad. is a ▱. hsm11g01m10se10_010 6010310_t06439.ai

22222 2

= 6 + 2 Substitute 6 for x. 33333 3
44444 4

= 8 Simplify. 55555 5
66666 6

77777 7

For PQRS to be a parallelogram, the value of y must be 8. 88888 8
99999 9

Got It? 1. Use the diagram at the right. For what values of x and y E (3y Ϫ 2)Њ F
must EFGH be a parallelogram? hsm11gmse_(04x60ϩ3_13t0)Њ6440.ai

(y ϩ 10)Њ
H (12x ϩ 7)Њ G

368 Chapter 6  Polygons and Quadrilaterals hsm11gmse_0603_t06442.ai

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 of Theorem 6‑6 is also true.

Theorem 6-11

Theorem If . . . C AE ≅ CE Then . . . C
If the diagonals of a B BE ≅ DE ABCD is a ▱
quadrilateral bisect each
other, then the quadrilateral AE D B
is a parallelogram.
AD

Proof Proof of Theorem 6-11 hsm11gmse_0603_t06443.ai B hsm11gCmse_0603_t06433

Given: AC and BD bisect each other at E. E
D
Prove: ABCD is a parallelogram. A

AC and BD bisect each other at E.
Given

hsm11gmse_0603_t06445.ai

∠AEB ≅ ∠CED AE ≅ CE ∠BEC ≅ ∠DEA
Vertical ⦞ are ≅. BE ≅ DE Vertical ⦞ are ≅.
Def. of segment bisector

△AEB ≅ △CED △BEC ≅ △DEA
SAS SAS

∠BAE ≅ ∠DCE ∠ECB ≅ ∠EAD
Corresp. parts of ≅ are ≅. Corresp. parts of ≅ are ≅.

AB ʈ CD BC ʈ AD
If alternate interior ⦞ ≅, If alternate interior ⦞ ≅,

then lines are ʈ. then lines are ʈ.

ABCD is a parallelogram.
Def. of parallelogram

hsm11gmse_0603_t06446.ai

Lesson 6-3  Proving That a Quadrilateral Is a Parallelogram 369

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

Theorem 6-12

Theorem If . . . C BC ≅ DA Then . . . C
If one pair of opposite sides B BC } DA ABCD is a ▱
of a quadrilateral is both
congruent and parallel, A D B
then the quadrilateral is a
parallelogram. AD

You will prove Theorem 6-12 in Exercise 19.

hsm11gmse_0603_t06448.ai hsm11gmse_0603_t06433.ai

Problem 2 Deciding Whether a Quadrilateral Is a Parallelogram

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

if you have enough information? Explain.
information?
I f you can satisfy every A Given:  AB = 5, CD = 5, B Given:  HI ≅ HK , JI ≅ JK

condition of a theorem     m∠A = 50, m∠D = 130 Prove:  HIJK is a parallelogram.
about parallelograms,
th en you have enough Prove:  ABCD is a parallelogram.
HI
information. A 5B

50Њ

130Њ C KJ
D5

Yes. Same-side interior angles A and No. By Theorem 6-8, you need to show that

D are supplementary, so AB } CD. both pairs of opposite sides are congruent,

SincehAsmB 1≅1CgDm,sAeB_C0D60is3a_ t06451.a i not chosnmse1cu1tgivme ssied_e0s.603_t06453.ai

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 ≅ GD, DE } FG b. Given:  ∠ALN ≅ ∠DNL, ∠ANL ≅ ∠DLN
Prove:  DEFG is a parallelogram. Prove:  LAND is a parallelogram.

DE AN

GF LD

hsm11gmse_0603_t06455.ai hsm11gmse_0603_t06456.ai

370 Chapter 6  Polygons and Quadrilaterals

As the arms of the Problem 3 Identifying Parallelograms
lift move, what
changes and what Vehicle Lifts  A truck sits on the platform of a vehicle lift. Two moving arms raise the
stays the same? platform until a mechanic can fit underneath. Why will the truck always remain
The angles the arms form parallel to the ground as it is lifted? Explain.
with the ground and the
platform change, but the Q Q 26 ft R
lengths of the arms and R 26 ft 6 ft
the platform stay the 6 ft
same. P 6 ft S
6 ft
26 ft
26 ft SP

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 the
ground, 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 the
truck? Explain.

Concept Summary  P roving That a Quadrilateral Is
a Parallelogram

Method Source Diagram
Prove that both pairs of opposite sides are parallel. Definition of
parallelogram

Prove that both pairs of opposite sides are Theorem 6-8
congruent.

Prove that an angle is supplementary to both of its Theorem 6-9 hsm117g5mЊ se_0603_t064
consecutive angles. Theorem 6-10
75Њ 105Њ
Prove that both pairs of opposite angles are
congruent. hsm11gmse_0603_t064

Prove that the diagonals bisect each other. Theorem 6-11 hsm11gmse_0603_t120

Prove that one pair of opposite sides is congruent Theorem 6-12 hsm11gmse_0603_t064
and parallel. hsm11gmse_0603_t064

Lesson 6-3  Proving That a Quadrilateral Is a Phasrmall1el1oggrmamse _0360731_t064

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

1. For what value of y must L M 4. Vocabulary  Explain why you can now write a
LMNP be a parallelogram? 68Њ biconditional statement regarding opposite sides of a

yЊ 68Њ parallelogram.
PN 5. Compare and Contrast  How is Theorem 6-11 in this

For Exercises 2 and 3, is the given information enough lesson different from Theorem 6-6 in the previous
to prove that ABCD is a parallelogram? Explain. lesson? In what situations should you use each
theorem? Explain.

2. B C 3. hsm11gmse_06C03_t06142 .a6i. Error Analysis  Your friend says, “If a quadrilateral
2y Њ xЊ B has a pair of opposite sides that are congruent and

xЊ 2y Њ H D a pair of opposite sides that are parallel, then it is a
A D A parallelogram.” What is your friend’s error? Explain.

hsPmr1a1cgtmicsee_a06n0d3_Pt0r6o1b4h3les.ammi 1-1Sgomlsvei_n0g603E_xt0e6r1c4i5s.eais MATHEMATICAL

PRACTICES

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

7. B 4x Ϫ 5 C 8. D 2y Ϫ 7 C 9. B C
4 3xЊ (y ϩ 78)Њ
2x

3x yϪ1 3yЊ (4x Ϫ 21)Њ
A AD
D AB

10. D C 11. D 5x Ϫ 8 C 12. B C

6x Ϫ 4 hsm11gmse_0603_t06149.ai (x ϩ 38)Њ

hsm3x1ϩ11gmse_0603_t06146.ai hsm11gmse_0603_t06150.ai

AB A 2x ϩ 7 B (4x Ϫ 1)Њ D
A

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

i1n3fo. r mhsamtio1n1?gEmxpslea_in0.603_t061511.a4i. hsm11gmse_0603_t061521.a5i. hsm11gmse_0603_t06153.ai

16. Fishing  Quadrilaterals are formed on A B See Problem 3.
the side of this fishing tackle box by hsm11gmse_0603_t06157.ai

thhsema1d1jgumstsaeb_0le60s3h_etl0v6e1s5a4n.adi connectinhgsm11gmse_0603_t06155.ai
pieces. Explain why the shelves are
DC
always parallel to each other no matter

what their position is.

372 Chapter 6  Polygons and Quadrilaterals

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

18. Developing Proof  Complete this two-column proof of Theorem 6-10. B C
xЊ yЊ
Given:  ∠A ≅ ∠C, ∠B ≅ ∠D
Prove:  ABCD is a parallelogram. A D

Statements Reasons

1) x + y + x + y = 360 1) The sum of the measures of the

2) 2(x + y) = 360 angles of a quadrilateral is 360.
3) x + y = 180
4) ∠A and ∠B are supplementary. 2) a. ? hsm11gmse_0603_t06158.ai
∠A and ∠D are supplementary.
5) c. ? } ? , ? } ? 3) b. ?
6) ABCD is a parallelogram.
4) Definition of supplementary

5) d. ?
6) e. ?

19. Think About a Plan  Prove Theorem 6-12. B C
Proof Given:  BC } DA, BC ≅ DA A D
Prove:  ABCD is a parallelogram.

• How can drawing diagonals help you?
• How can you use triangles in this proof?

20. Prove Theorem 6-8. 21. Prove Theorem 6-9.
Proof
Proof
Given:  AB ≅ CD, BC ≅ DA Given:  ∠A is supplementary to ∠B
Prove:  ABCD is a parallelogram.   hs m11gm ∠seA_0i6s0s3u_pt0p6l1e5m9e.ani tary to ∠D.
B C Prove:  ABCD is a parallelogram.
BC

AD

AD

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

22.h sAm11gmse_0603_t0B6430.ai 23. A 2y ϩ 2 B 24. B C
3x ϩ 6 hsm11gmse_0603_t06581.ai
(3x ϩ 10)Њ D (2x ϩ 15)Њ
(8x ϩ 5)Њ yϩ4 (4x Ϫ 33)Њ

5yЊ 3y Ϫ 9 C A D
DC

25. Given:  △TRS ≅ △RTW S T
Proof Prove:  RSTW is a parallelogram.
hsm11gmse_0603_t06162.ai
hsm11gmse_0603_t06160.ai hsm11gmse_0603_t06161.ai W
26. Open-Ended  Sketch two noncongruent parallelograms
R
ABCD and EFGH such that AC ≅ EG and BD ≅ FH.

Lesson 6-3  Proving That a Quadrilateral Is a Parallelogram 373
hsm11gmse_0603_t06164.ai

C Challenge 27. Construction  In the figure at the right, point D is constructed by A
Proof drawing two arcs. One has center C and radius AB. The other has center
B MC
B and radius AC. Prove that AM is a median of △ABC. D

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

Standardized Test Prep

SAT/ACT 29. From which set of information can you conclude that RSTW is a parallelogrhasmm?11gmse_0603_t06165.ai

RS } WT, RS ≅ ST RS ≅ ST, RW ≅ WT
RS } WT, ST ≅ RW RZ ≅ TZ, SZ ≅ WZ

Short 30. Write a proof using the diagram. N PT
Response Given: △NRJ ≅ △CPT, JN } CT
Prove: JNTC is a parallelogram. JR C

Extended 31. U se the figure at the right. A
Response a. Write an equation and solve for x.
b. Is AF } DE? Explain. (7x Ϫ 11)Њ CD
c. Is BDEF a parallelogram? Explain. B
hsm11gmse_0603_t0616(57x.aϪi 7)Њ
6x Њ
FE

Mixed Review

Algebra Find the value of each variable in eachhpsamr1a1llgemloseg_r0a6m03. _t06168.ai See Lesson 6-2.

32. 23 4hЊ a ϩ 15 33. 3m Ϫ 12 34. 6e Q2cR 8
2hЊ kЊ ƒ
(8x ϩ 15)Њ 3
102
3x Њ
mϩ7

35. Explain how you can use overlapping DC See Lessons 4-4 and 4-7.

hcsomn1g1rgumenset_t0ri6a0n3g_lte0s61to69p.aroi ve AC ≅ BD. E hsm11gmse_0603_t06171.ai
B
G et Ready!  hsm11gmse_A0603_t06170.ai
To prepare for Lesson 6-4,

do Exercises 36–44.

PACE is a parallelogram and mjPAC = 124. Complete the following. See Lessons 5-2 and 6-2.

36. AC = ■ 37. CE = ■ 38. PA = ■ A C
39. RE = ■ 40. CP = ■ 41. m∠CEP = ■ 3.5
42. m∠EPA = ■ 43. m∠ECA = ■ 44. m∠ACR = ■
R 6.6

P 7.47 E

374 Chapter 6  Polygons and Quadrilaterals

6-4 Properties of Rhombuses, MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Rectangles, and Squares
MG-ACFOS.C.9.121. GP-rCovOe.3th.1e1o rePmrosvaebtohuetorems about
parallelograms . . . rectangles are parallelograms
wpaitrhalcleolnogruamenst wdiathgocnoanlgsr.uAelnstodGia-gSoRnTa.lBs..5Also
MMPAF1S,.M91P2.3G-SRT.2.5
MP 1, MP 3

Objectives To define and classify special types of parallelograms
To use properties of diagonals of rhombuses and rectangles

Can you make a Fold a piece of notebook paper in half. Fold it in half
good argument again in the other direction. Draw a diagonal line from
to justify your one vertex to the other. Cut through the folded paper
observations? along that line. Unfold the paper. What do you notice
about the sides and about the diagonals of the figure
you formed?

MATHEMATICAL In the Solve It, you formed a special type of parallelogram with characteristics that you

PRACTICES

will study in this lesson.

Lesson Essential Understanding  The parallelograms in the Take Note box below have
basic properties about their sides and angles that help identify them. The diagonals of
Vocabulary these parallelograms also have certain properties.
• rhombus
• rectangle
• square

Key Concept  Special Parallelograms Diagram

Definition
A rhombus is a parallelogram with four congruent sides.

A rectangle is a parallelogram with four right angles.

A square is a parallelogram with four congruent sides and hsm11gmse_0604_t06018
four right angles. hsm11gmse_0604_t06019

hsm11gmse_0604_t06020
Lesson 6-4  Properties of Rhombuses, Rectangles, and Squares 375

The Venn diagram at the right Special Parallelograms
shows the relationships among Rhombuses Squares Rectangles
special parallelograms.

How do you decide Problem 1 Classifying Special Parallelograms A B
whether ABCD is a E
rhombus, rectangle, Is ▱ABCD a rhombus, a rectangle, ohrsamsq1u1agrem?sEex_p0la6i0n4. _t06021
or square? F
▱ABCD is a rectangle. Opposite angles of a parallelogram H
Use the definitions of are congruent so m∠D is 90. By the Same-Side Interior
Angles Theorem, m∠A = 90 and m∠C = 90. Since G
rhombus, rectangle, and ▱ABCD has four right angles, it is a rectangle. You cannot DC
conclude that ABCD is a square because you do not know
square along with the its side lengths.

markings on the figure.

Got It? 1. Is ▱EFGH a rhombus, a rectangle, or a
square? Explain.

Theorem 6-13 If . . . Then . . . D
ABCD is a rhombus AC # BD
Theorem
If a parallelogram is a AD A
rhombus, then its diagonals are
perpendicular.

BC BC

Theorem 6-14

Theorem If . . . Then . . .
If a parallelogram is a ABCD is a rhhosmmb1u1sgmse_06 04_At 060 22 hsm1D1gm∠s1e≅_0∠6024_t06023
rhombus, then each 1 2 3 4 ∠3 ≅ ∠4
diagonal bisects a pair A D ∠5 ≅ ∠6
of opposite angles.
7 8 6 5 ∠7 ≅ ∠8

BC

BC

You will prove Theorem 6-14 in Exercise 45.

hsm11gmse_0604_t06024
hsm11gmse_0604_t06022

376 Chapter 6  Polygons and Quadrilaterals

Proof Proof of Theorem 6-13 A D
Given: ABCD is a rhombus.
Prove: The diagonals of ABCD are perpendicular.

Statements Reasons B C

1) A and C are equidistant 1) All sides of a rhombus
from B and D; B and D are ≅.
are equidistant from A
and C. hsm11gmse_0604_t06025

2) A and C are on the 2) Converse of the
perpendicular bisector Perpendicular Bisector
of BD; B and D are Theorem
on the perpendicular
bisector of AC. 3) Through two points,
there is one unique
3) AC # BD line perpendicular to a
given line.

You can use Theorems 6-13 and 6-14 to find angle measures in a rhombus.

Problem 2 Finding Angle Measures

How are the What are the measures of the numbered angles in rhombus ABCD? BC
numbered angles
formed? m∠1 = 90 The diagonals of a rhombus are #. 58Њ
The angles are formed
by diagonals. Use what m∠2 = 58 Alternate Interior Angles Theorem 1 2
you know about the 3
diagonals of a rhombus m∠3 = 58 Each diagonal of a rhombus bisects a 4
to find the angle pair of opposite angles. AD
measures.
m∠1 + m∠3 + m∠4 = 180 Triangle Angle-Sum Theorem
90 + 58 + m∠4 = 180 Substitute.

148 + m∠4 = 180 Simplify. hsm11gmse_0604_t06026

m∠4 = 32 Subtract 148 from each side.

Got It? 2. What are the measures of the numbered angles in QR
rhombus PQRS? 104Њ 3 4

1 S
2

P

hsm11gmse_0604_t06027
Lesson 6-4  Properties of Rhombuses, Rectangles, and Squares 377

The diagonals of a rectangle also have a special property.

Theorem 6-15

Theorem If . . . D Then . . . D
If a parallelogram is ABCD is a rectangle AC ≅ BD
a rectangle, then its
diagonals are congruent. A A

B CB C

You will prove Theorem 6-15 in Exercise 41.

hsm11gmse_0604_t06028 hsm11gmse_0604_t06031

Problem 3 Finding Diagonal Length

How can you find the Multiple Choice  In rectangle RSBF, SF = 2x + 15 and RB = 5x − 12. S B
length of a diagonal? What is the length of a diagonal?
Since RSBF is a rectangle
a nd its diagonals are 1 9 18 33
congruent, use the
expressions to write an You know that the diagonals of RF
equation. a rectangle are congruent, so hsm11gmse_0604_t06032
their lengths are equal.
SF = RB
Set the algebraic expressions for
SF and RB equal to each other 2x + 15 = 5x − 12
and find the value of x. 15 = 3x − 12
27 = 3x
Substitute 9 for x in the 9=x
expression for RB. RB = 5x − 12

= 5(9) − 12
= 33
The correct answer is D.

Got It? 3. a. If LN = 4x - 17 and MO = 2x + 13, what are M N

the lengths of the diagonals of rectangle LMNO?

b. Reasoning  What type of triangle is △PMN ? Explain. P

LO

hsm11gmse_0604_t06033

378 Chapter 6  Polygons and Quadrilaterals

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

Is each parallelogram a rhombus, rectangle, or PRACTICES
square? Explain.
5. V ocabulary  Which special parallelograms are

equiangular? Which special parallelograms are

1. 2. equilateral?

130Њ 6. Error Analysis  Your class D (9x Ϫ 6)Њ G
F
needs to find the value of

x for which ▱DEFG is a

2 50Њ rectangle. A classmate’s

3. What are the measures of the work is shown below. E

numbered angles in the rhombhussm? 11gmse1_03604_t05911 What is the error? Explain. (2x ϩ 8)Њ

4. Ahlsgmeb1r1ag  mJKsLeM_0i6s0a4r_etc0ta5n9g1l0e. If JL = 4x - 12 and 2x + 8 = 9x - 6
MK = x, what is the value of x? What is the length of 14 = 7hxsm11gmse_0604_t05913
each diagonal? 2=x

hsm11gmse_0604_t05912

Practice and Problem-Solving Exercises PMRATAHhCEMsTmAITCI1CEA1SL gmse_0604_t06659

A Practice Decide whether the parallelogram is a rhombus, a rectangle, or a See Problem 1.
square. Explain.

7. 8.

Find the measures of the numbered angles in each rhombus. See Problem 2.
11. 1
9. 3 4 10. 26Њ 2 1
2
3 3

1 118Њ
2 106Њ
14. 3 4
12. 3 2 13. hsm 11gm se_0604_t05916.ai
4 hsm111gms2e_30604_t05915 1
4 58Њ 30Њ 2
hsm1 11gmse_0604_t05914
113Њ

Lesson 6-4  Properties of Rhombuses, Rectangles, and Squares 379
hsm 11gm se_0604_t05918.ai hsm 11gm se_0604_t05919.ai

Find the measures of the numbered angles in each rhombus.

15. 3 16. 3 17. 12
42 2 3
35Њ

35Њ 1 1 60Њ

Algebra  LMNP is a rectangle. Find the value of x and the length of each See Problem 3.
diagonal.

18. LhNsm=1x1gamndseM_P06=024x_t-0549 20.ai hsm 11gm se_1096.0 L4N_t0=559x2-1.a8iand MhPs=m21x1g+m1se_0604_t05935.ai

20. LN = 3x + 1 and MP = 8x - 4 21. LN = 9x - 14 and MP = 7x + 4

22. LN = 7x - 2 and MP = 4x + 3 23. LN = 3x + 5 and MP = 9x - 10

B Apply Determine the most precise name for each quadrilateral.

24. 25. 5 26. 7 27.

4 4 7 7
5 7

List the quadrilaterals that have the given property. Choose among hsm 11gm se_0604_t06189.ai

parallelogram, rhombus, rectangle, and square.

28. Ahlsl msid1e1sgamrese≅_.0 60 4 _ t05 9 3 7h.sami 1 1 g m s e _ 0 6 0 4 _ t05939.ai 4≅_.t0
29. Ohps pmo1s1itgemsisdee_s0a6re0
5 9 4 1 .a i

30. Opposite sides are }. 31. Opposite ⦞ are ≅.
32. All ⦞ are right ⦞. 33. Consecutive ⦞ are supplementary.

34. Diagonals bisect each other. 35. Diagonals are ≅.

36. Diagonals are #. 37. Each diagonal bisects opposite ⦞.

Algebra  Find the values of the variables. Then find the side lengths.

38. rhombus  15 39. square  2x Ϫ 7

3y 5x yϪ1 2y Ϫ 5

4x ϩ 3 3y Ϫ 9

40. Think About a Plan  Write a proof. PL
Proo f Given:  Rectangle PLAN

Prove:  △LhTsPm≅11△gmNTsAe_0604_t05942.ai hsm 11gm se_0604T_t05943.ai

• What do you know about the diagonals of rectangles?
• Which triangle congruence postulate or theorem can you use? N A


380 Chapter 6  Polygons and Quadrilaterals hsm11gmse_0604_t06262.ai

41. Developing Proof  Complete the flow proof of Theorem 6-15. A D

Given:  ABCD is a rectangle.

Prove:  AC ≅ BD BC

ABCD is a ▱. e.

b. Opposite sides of
a ▱ are ≅.

hsm 11gm se_0604_t06239.ai

ABCD is BC ≅ BC f. AC ≅ BD
a rectangle. c. SAS h.

a.

∠ABC and ∠DCB ∠ABC ≅ ∠DCB
are right ⦞. g.

d.

Algebra  Find the value(s) of the variable(s) for each parallelogram.

42. RZ = 2x + 5, 43. m∠1 = 3y - 6 44. BD = 4x - y + 1
B
SW = 5x - 20
R hsm11gmse_06W04_t06243.ai

Z 9xЊ 1 6zЊ A 2x Ϫ 1 3y ϩ 5 C
ST

D

45. hPsrmov1e1Tghmesoere_m0660-41_4t.06245.ai hsm 11gm se_0604_t06246.ai AD
Proo f Given:  ABCD is a rhombus. 34
hsm 11g2m1se_0604_t06248.ai
Prove:  AC bisects ∠BAD and ∠BCD. BC

46. Writing  Summarize the properties of squares that follow from a
square being (a) a parallelogram, (b) a rhombus, and (c) a rectangle.

47. Algebra  Find the angle measures and the side lengths of the K 4b Ϫ 6r J
rhombus at the right. r ϩhsm1 1x1Њ gmse_060(b24Ϫx_tϩ03662)5Њ 0.ai
H 2r Ϫ 4 G
48. Open-Ended  On graph paper, draw a parallelogram that is neither
a rectangle nor a rhombus.

Algebra  ABCD is a rectangle. Find the length of each diagonal.

49. AC = 2(x - 3) and BD = x + 5 50. AC = 2(5a + 1) and BD = 2(a + 1)
51. AC = 35y and BD = 3y - 4 hsm11gmse_0604_t06254.ai
3c
52. AC = 9 and BD = 4 - c

Lesson 6-4  Properties of Rhombuses, Rectangles, and Squares 381

C Challenge Algebra  Find the value of x in the rhombus. 54.

53. (6x2 Ϫ 3x)Њ
(7x2 Ϫ 10)Њ

(2x2 Ϫ 25x)Њ

(3x2 ϩ 60)Њ

Standardized Test Prep hsm 11gm se_0604_t05945.ai

hsm 11gm se_0604_t05944.ai

SAT/ACT 55. Which statement is true for some, but not all, rectangles?

Opposite sides are parallel. Adjacent sides are perpendicular.

It is a parallelogram. All sides are congruent.

56. A part of a design for a quilting pattern consists of a regular pentagon and five 1

isosceles triangles, as shown. What is m∠1?

18 72

36 108

57. Which term best describes AD in △ABC? A

altitude median

Short angle bisector an indirect proof that △PQR is perpendicular bisectorhsmB11gDmse_0604_tC12846
Response 58. Write the first step of
not a right triangle.

hsm 11gm se_0604_t05947.ai

Mixed Review

Can you conclude that the quadrilateral is a parallelogram? Explain. See Lesson 6-3.
C
59. 5 60. 6 61. B
D
4 4 25Њ
5 25Њ
A
6

In △PQR, points S, T, and U are midpoints. Complete each statement. R See Lesson 5-1.

62. ThQsm=11g?m s e_0604_t06539.4 8P.Qai = ? 64. TU = ? hs m 1 1 g8m s6e _ 06 0 4 _ t0 5 9 5 0 .a i
65. SU } ? 66. TU } hsm 11gm se_0604_t05949.ai S T
? 67. PQ } ? 5

P UQ
Get Ready!  To prepare for Lesson 6-5, do Exercises 68 and 69.

68. Draw a rhombus that is not a square. See Lesson 6-4.

69. Draw a rectangle that is not a square.

hsm 11gm se_0604_t05951.ai

382 Chapter 6  Polygons and Quadrilaterals

6-5 Conditions for Rhombuses, CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Rectangles, and Squares
MG-ACFOS.C.9.121. GP-rCovOe.3th.1e1o rePmrosvaebtohuetopreamrasllaebloogurtams
.p.a.rarellcetlaonggralemssa.re. .praercatlalenlgolgersaamres pwaitrhallceolnogruamenst
dwiiatghocnoanlgs.ruAelnsot dGia-gSoRnTa.lBs.5Also
MMPAF1S,.M91P2.3G, -MSRPT4.2.5
MP 1, MP 3, MP 4

Objective To determine whether a parallelogram is a rhombus or rectangle

Which vertices form a square? A rhombus? yB C
A rectangle? Justify your answers.
6A J F D
It’s OK to look 4G L
up the definitions x
for these 2 H
quadrilaterals
if you need to. O 2 4E6

MATHEMATICAL

PRACTICES

Essential Understanding  You can determine whether a parallelogram is a
rhombus or a rectangle based on the properties of its diagonals. hsm11gmse_0605_t14262

Theorem 6-16

Theorem If . . . Then . . .
If the diagonals of ABCD is a ▱ and AC # BD ABCD is a rhombus
a parallelogram are
perpendicular, then AD AD
the parallelogram is a
rhombus. BC BC

Proof Proof of Theorem 6-16 AD

Given: ABCD is a parallelogram, AhCsm#11gBmDse_0605_t06034.ai hsm11gmse_0605_t06274.ai

Prove: ABCD is a rhombus. B E
C
Since ABCD is a parallelogram, AC and BD bisect each other, so
BE ≅ DE. Since AC # BD, ∠AED and ∠AEB are congruent right

angles. By the Reflexive Property of Congruence, AE ≅ AE. So △AEB ≅ △AED by SAS.

Corresponding parts of congruent triangles are congruent, so AB ≅ AD. Since opposite

sides of a parallelogram are congruent, AB ≅ DC ≅ BC ≅ AD. By definition, ABCD is a

rhombus. hsm11gmse_0605_t06035.ai

Lesson 6-5  Conditions for Rhombuses, Rectangles, and Squares 383

Theorem 6-17

Theorem If . . . Then . . .
If one diagonal ABCD is a ▱, ∠1 ≅ ∠2, and ∠3 ≅ ∠4 ABCD is a rhombus
of a parallelogram
bisects a pair AD AD
of opposite 3
angles, then the 4 BC
parallelogram is a
rhombus. 12 C
B

You will prove Theorem 6-17 in Exercise 23.

Theorem 6-18

Theorem Ihfsm. .11. gmse_0605_t06036.ai Thshme1n1g.m. s.e_0605_t06274.ai
If the diagonals of ABCD is a ▱, and AC ≅ BD ABCD is a rectangle
a parallelogram are AD
congruent, then AD
the parallelogram
is a rectangle.

BC BC

You will prove Theorem 6-18 in Exercise 24.

You can use Theorems 6-1h6s,m61-17g,masne_d066-0158_tt0o6c0l3a7s.saiify parallelograms. Nohtsicme1t1hgamt ifsae_0604_t06028
parallelogram is both a rectangle and a rhombus, then it is a square.

Problem 1 Identifying Special Parallelograms

How can you Can you conclude that the parallelogram is a rhombus, a rectangle, or a

determine whether square? Explain. B
a figure is a special A
p arallelogram?

See if you can satisfy

every condition of a

definition or theorem

about rhombuses,

r ectangles, or squares. Yes. A diagonal bisects two angles. By Yes. The diagonals are congruent, so by

Theorem 6-17, this parallelogram is a Theorem 6-18, this parallelogram is a

h rhsmom11bgumss.e _0605_t06038. ai rectangle. The diagonals are perpendicular,
shosmb1y1Tghmesoe_re0m6056_-t1066,0i3t 9is.aai rhombus.

Therefore, this parallelogram is a square.

Got It? 1. a. A parallelogram has angle measures of 20, 160, 20, and 160. Can you
conclude that it is a rhombus, a rectangle, or a square? Explain.

b. Reasoning  Suppose the diagonals of a quadrilateral bisect each other.

Can you conclude that it is a rhombus, a rectangle, or a square? Explain.

384 Chapter 6  Polygons and Quadrilaterals

Problem 2 Using Properties of Special Parallelograms A D
Algebra  For what value of x is ▱ABCD a rhombus?

For ▱ABCD to be a m∠ABD = m∠CBD (6x Ϫ 2)Њ
rhombus, its diagonals B (4x ϩ 8)Њ C

must bisect a pair of

opposite angles.

Set the expressions for 6x - 2 = 4x + 8 hsm11gmse_0605_t06040.ai
m∠ABD and m∠CBD
equal to each other. 2x - 2 = 8
   2x = 10
Solve for x.      x = 5

DG

Got It? 2. For what value of y is ▱DEFG a rectangle? 5y ϩ 3 7y Ϫ 5
4
4
E
F

Problem 3 Using Properties of Parallelograms hsm11gmse_0605_t06041.ai

Community Service  Builders use properties of diagonals
to “square up” rectangular shapes like building frames
and playing-field boundaries. Suppose you are on the
volunteer building team at the right. You are helping to lay
out a rectangular patio for a youth center. How can you
use properties of diagonals to locate the four corners?

You can use two theorems.

• Theorem 6-11: If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.
Is there only one
r ectangle that can • Theorem 6-18: If the diagonals of a parallelogram are congruent, then the
be formed by pulling parallelogram is a rectangle.
the ropes taut?
No, you can change the Step 1 Cut two pieces of rope that will be the diagonals of the foundation rectangle.
shape of the rectangle. Cut them the same length because of Theorem 6-18.
Have two of the people
move closer together. Step 2 Join the two pieces of rope at their midpoints because of Theorem 6-11.
Then the other two
people move until the Step 3 Pull the ropes straight and taut. The ends of the ropes will be the corners of a
ropes are taut again. rectangle.

Got It? 3. Can you adapt this method slightly to stake off a square play area? Explain.

Lesson 6-5  Conditions for Rhombuses, Rectangles, and Squares 385

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

Can you conclude that the parallelogram is a rhombus, PRACTICES
a rectangle, or a square? Explain.
5. Name all of the special parallelograms that have

1. T O 2. each property.
a. Diagonals are perpendicular.
S SO Х TP P b. Diagonals are congruent.
c. Diagonals are angle bisectors.
d. Diagonals bisect each other.
e. Diagonals are perpendicular bisectors of

each other.

For what value of x is the figure the given special 6. E rror Analysis  Your friend says, “A parallelogram
with perpendicular diagonals is a rectangle.” What
parallelogram? hsm11gmse_0605_t05958.ai is your friend’s error? Explain.

3. rhhsmom11bgumss e_0605_t05957.ai 4. rectangle

3x ϩ 9 7. R easoning  When you draw a circle and two of
8x Ϫ 1 3x Ϫ 5 its diameters and connect the endpoints of the
xϩ1 diameters, what quadrilateral do you get? Explain.

hsPm1r1agmcstei_c0e605a_tn05d959P.ari obhlsemm11g-mSsoe_l0v60i5n_tg059E6x0.aei rcises MATHEMATICAL

PRACTICES

A Practice Can you conclude that the parallelogram is a rhombus, a rectangle, or a See Problem 1.
square? Explain.

8. 9. 10.

For what value of x is the figure the given special parallelogram? See Problem 2.
13. rectangle
11. rhhsmom11bgumss e_0605_t05961.ai 12. rectangle
hsm11gmse_0605_t05962.ai hsm11gmse_0605_t05963.ai
L O
(6x Ϫ 9)Њ 8x ϩ 3 4x ϩ 7
4 4 LN ϭ 4x Ϫ 7
(2x ϩ 39)Њ MO ϭ 2x ϩ 13
MN

14. Carpentry  A carpenter is building a bookcase. How can the carpenter use a See Problem 3.

tape measure to check that the bookshelf is rectangular? Justify your answer
hsm11gmse_0605_t05965.ai
hasnmd1n1agmmsee_a0n6y0t5h_et0o5r9e6m4s.aui sed. hsm11gmse_0605_t05966.ai

386 Chapter 6  Polygons and Quadrilaterals

B Apply STEM 15. Hardware  You can use a simple device called a turnbuckle
to “square up” structures that are parallelograms. For the
gate pictured at the right, you tighten or loosen the
turnbuckle on the diagonal cable so that the rectangular
frame will keep the shape of a parallelogram when it sags.
What are two ways you can make sure that the turnbuckle
works? Explain.

16. Reasoning  Suppose the diagonals of a parallelogram are
both perpendicular and congruent. What type of special
quadrilateral is it? Explain your reasoning.

Algebra  For what value of x is the figure the given special parallelogram?

17. rectangle 18. rhombus 19. rectangle

(5x ϩ 2)Њ (4x Ϫ 12)Њ
(3x ϩ 4)Њ
(3x ϩ 6)Њ
3xЊ (8x ϩ 7)Њ

Open-Ended  Given two segments with lengths a and b (a ≠ b), what special

parallelograms meet the given conditions? Show each sketch.

20. Bhsomth11dgimagsoe_n0a6l0s5h_atv0e59l6e7n.gatih a. hsm11gmse_060251_.t 0T5h9e6t8w.aoi diagonalshhsamve1l1enggmthssea_a0n6d0b5._t05969

22. One diagonal has length a, and one side of the quadrilateral has length b.

23. Prove Theorem 6-17. A D
Proo f Given:  ABCD is a parallelogram. 34

AC bisects ∠BAD and ∠BCD. 21
BC
Prove:  ABCD is a rhombus.

24. Prove Theorem 6-18. A B
Proo f Given:  ▱ABCD, AC ≅ BD D

Prove:  ABCD is a rectangle.

hsm11gCmse_0605_t05970

Think About a Plan  Explain how to construct each figure given its diagonals.

• What do you know about the diagonals of each figure?

• How can you apply constructions to what you know about the diagonals?
26. rehcstamng1l1e gmse_0605_t0597127. rhombus
25. parallelogram

C Challenge Determine whether the quadrilateral can be a parallelogram. Explain.

28. The diagonals are congruent, but the quadrilateral has no right angles.
29. Each diagonal is 3 cm long and two opposite sides are 2 cm long.
30. Two opposite angles are right angles, but the quadrilateral is not a rectangle.

Lesson 6-5  Conditions for Rhombuses, Rectangles, and Squares 387

31. In Theorem 6-17, replace “a pair of opposite angles” with “one angle.” Write a
Proof paragraph that proves this new statement to be true, or give a counterexample to

prove it to be false.

Standardized Test Prep

SAT/ACT 32. Each diagonal of a quadrilateral bisects a pair of opposite angles of the
quadrilateral. What is the most precise name for the quadrilateral?

parallelogram rhombus rectangle not enough information

33. Given a triangle with side lengths 7 and 11, which value could NOT be the length of
the third side of the triangle?

Short 13 7 5 2

Response 34. What is the sum of the measures of the exterior angles in a pentagon?

180 360 540 108

35. The midpoint of PQ is ( -1, 4). One endpoint is P( -7, 10). What are the coordinates
of endpoint Q? Explain your work.

Mixed Review

Find the measures of the numbered angles in each rhombus. See Lesson 6-4.

36. 128Њ 37. 1 3 38.
3 57Њ 32Њ
2
21 31
2

Write the two conditionals as a biconditional. See Lesson 2-3.

39. If a parallelogram is a rhombus, then its diagonals are perpendicular.

Ihf sthme1d1iaggmonsaels_o0f6a0p5a_rat0ll5el9o7gr2am arhespmer1p1egnmdicsuel_a0r,6th0e5n_tth0e5p9a7ra3lleloghrasmm1is1agmse_0605_t05974

rhombus.

40. If a parallelogram is a rectangle, then its diagonals are congruent.

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

G et Ready!  To prepare for Lesson 6-6, do Exercises 41–43.

Algebra  Find the values of the variables. Then find the lengths of the sides. See Lesson 1-3.

41. 4.5 a Ϫ 1.4 42. 4.8 43. 3m 7m Ϫ 14

b Ϫ 2.3 2a Ϫ 7 4y ϩ 6 7y Ϫ 3

5y ϩ 1.4 n nϩ6

hsm11gmse_0605_t05975 hsm11gmse_0605_t05978

388 Chapter 6  Polygons and Quadrilaterals

hsm11gmse_0605_t05976

6-6 Trapezoids and Kites CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

GM-ASFRST.9B1.52 .GUs-SeRcoT.n2g.r5u eUnsce c.o. n. gcriuternicaeto. .s.olve
pcriotebrlieamtsoasnodlveprporvoebrlemlatsioansdhpiprsovine greeloamtioentrsihcips
finigguereosm. etric figures.

MP 1, MP 3, MP 4, MP 6

Objective To verify and use properties of trapezoids and kites

Make a sketch and Two isosceles triangles form the figure at
number the angles the right. Each white segment is a
to help make sense midsegment of a triangle. What can you
of the problem. determine about the angles in the orange
region? In the green region? Explain.

MATHEMATICAL In the Solve It, the orange and green regions are trapezoids. The entire figure is a kite. In
PRACTICES this lesson, you will learn about these special quadrilaterals that are not parallelograms.

Essential Understanding  The angles, sides, and diagonals of a trapezoid have
certain properties.

A trapezoid is a quadrilateral with exactly one pair of parallel base leg
sides. The parallel sides of a trapezoid are called bases. The
Lesson nonparallel sides are called legs. The two angles that share a leg base angles
base of a trapezoid are called base angles. A trapezoid has base angles
Vocabulary two pairs of base angles. base
• trapezoid
• base An isosceles trapezoid is a trapezoid with legs that are BC
• leg congruent. ABCD at the right is an isosceles trapezoid. The
• base angle angles of an isosceles trapezoid have some unique properties. hAsm11gmse_0606_Dt06314
• isosceles
Theorem 6-19
trapezoid
• midsegment of a

trapezoid
• kite

Theorem If . . . Then . . .
If a quadrilateral is an TRAP is an isosceles trapezoid hsm1∠1Tgm≅s∠eP_,0∠60R6≅_t∠06A315
isosceles trapezoid, then
each pair of base angles with bases RA and TP RA
is congruent.
RA

TP
TP

You will prove Theorem 6-19 in Exercise 45.

hsm11gmse_0606_t06316 hsm11gmse_0606_t06317

Lesson 6-6  Trapezoids and Kites 389

Problem 1 Finding Angle Measures in Trapezoids

What do you know CDEF is an isosceles trapezoid and mjC = 65. What are mjD, DE

about the angles mjE, and mjF ?

of an isosceles m∠C + m∠D = 180 Two angles that form same-side interior angles 65Њ
trapezoid? along one leg are supplementary. C

You know that each pair

of base angles is 65 + m∠D = 180 Substitute. F
congruent. Because the m∠D = 115 Subtract 65 from each side.
bases of a trapezoid are Q

parallel, you also know Since each pair of base angles of an isosceles trapezoid is congruent, R
that two angles that m∠C = m∠F = 65 and m∠D = m∠E = 115.
share a leg are hsm11gmse_0610066Њ_t06318

supplementary.

Got It? 1. a. In the diagram, PQRS is an isosceles trapezoid and

m∠R = 106. What are m∠P, m∠Q, and m∠S?
b. Reasoning  In Problem 1, if CDEF were not an isosceles trapezoid,
S
would ∠C and ∠D still be supplementary? Explain.
P

Problem 2 Finding Angle Measures in Isosceles Trapezoids

Paper Fans  The second ring of the paper fan shown hsm11gmse_0606_t06320
at the right consists of 20 congruent isosceles trapezoids
What do you notice that appear to form circles. What are the measures of
about the diagram? the base angles of these trapezoids?
Each trapezoid is part of
an isosceles triangle with Step 1 Find the measure of each angle at the center
base angles that are the
acute base angles of the of the fan. This is the measure of the vertex angle
trapezoid.
of an isosceles triangle.

  m∠1 = 360 = 18
20

Step 2 Find the measure of each acute base angle of an

isosceles triangle.

18 + x + x = 180 Triangle Angle-Sum Theorem

18 + 2x = 180 Combine like terms.

2x = 162 Subtract 18 from each side.

x = 81 Divide each side by 2.

Step 3 Find the measure of each obtuse base angle of
the isosceles trapezoid.

81 + y = 180 Two angles that form same-side interior
angles along one leg are supplementary.

y = 99 Subtract 81 from each side.

Each acute base angle measures 81. Each obtuse base angle measures 99.

Got It? 2. A fan like the one in Problem 2 has 15 angles meeting at the center. What are
the measures of the base angles of the trapezoids in its second ring?

390 Chapter 6  Polygons and Quadrilaterals

Theorem 6-20 If . . . Then . . . C
ABCD is an isosceles AC ≅ BD
Theorem trapezoid
If a quadrilateral is an B
isosceles trapezoid, then its BC
diagonals are congruent.

AD
AD

You will prove Theorem 6-20 in Exercise 54.

hsm11gmse_0606_t06324
hsm11gmse_0606_t06321

In Lesson 5-1, you learned about midsegments of triangles. Trapezoids also have
midsegments. The midsegment of a trapezoid is the segment that joins the midpoints

of its legs. The midsegment has two unique properties.

Theorem 6-21  Trapezoid Midsegment Theorem

Theorem If . . . Then . . .
If a quadrilateral is a trapezoid, TRAP is a trapezoid with
then midsegment MN (1) MN } TP, MN } RA, and
(1) the midsegment is parallel ( )(2)
to the bases, and RA MN = 1 TP + RA
(2) the length of the 2
midsegment is half the sum of MN
the lengths of the bases.
TP

You will prove Theorem 6-21 in Lesson 6-9.

hsm11gmse_0606_t06325

Problem 3 Using the Midsegment of a Trapezoid

Algebra  QR is the midsegment of trapezoid LMNP. 4x Ϫ 10
What is x?
M
QR = 1 (LM + PN) Trapezoid Midsegment Theorem L
2 Q
P
How can you check x + 2 = 1 [(4x - 10) + 8] Substitute. xϩ2
your answer? 2
Find LM and QR. Then R
see if QR equals half x + 2 = 1 (4x - 2) Simplify.
of the sum of the base 2
lengths.
x + 2 = 2x - 1 Distributive Property

3=x Subtract x and 8 N
add 1 to each side.

hsm11gmse_0606_t06328

Lesson 6-6  Trapezoids and Kites 391

Got It? 3. a. Algebra  MN is the midsegment of trapezoid PQRS. Q 10 R
What is x? What is MN? M 2x ϩ 11 N
P 8x Ϫ 12 S
b. Reasoning  How many midsegments can a triangle

have? How many midsegments can a trapezoid

have? Explain.

A kite is a quadrilateral with two pairs of consecutive sides congruent

and no opposite sides congruent. hsm11gmse_0606_t06329

Essential Understanding  The angles, sides, and diagonals of

a kite have certain properties.

Theorem 6-22 hsm11gmse_0606_t06333

Theorem If . . . C Then . . .
If a quadrilateral is a kite, ABCD is a kite AC # BD
then its diagonals are
perpendicular. B B

A AC

D D

Proof Proof of Theorem 6-22 B

Given: Kite ABCD with AB ≅ ADhsamnd11CgBm≅sCeD_0606_t0633A5 hsm11gmse_0606_t06336

Prove: AC # BD C

Statements Reasons D
1) Given
1) Kite ABCD with AB ≅ AD
and CB ≅ CD 2) Converse of Perpendicuhlsamr 11gmse_0606_t06338.ai

2) A and C lie on the Bisector Theorem
perpendicular bisector
of BD. 3) Two points determine a
line.
3) AC is the perpendicular
bisector of BD. 4) Definition of perpendicular
bisector
4) AC # BD

392 Chapter 6  Polygons and Quadrilaterals

Problem 4 Finding Angle Measures in Kites

Quadrilateral DEFG is a kite. What are mj1, mj2, and mj3? D
3 52Њ
m∠1 = 90 Diagonals of a kite are #. E 12 G

90 + m∠2 + 52 = 180 Triangle Angle-Sum Theorem F
2L
142 + m∠2 = 180 Simplify.
hKsm113g1mse_0606_t06M340.ai
How are the triangles m∠2 = 38 Subtract 142 from each side.
36Њ N
congruent by SSS? △DEF ≅ △DGF by SSS. Since corresponding parts of

DE ≅ DG and FE ≅ FG congruent triangles are congruent, m∠3 = m∠GDF = 52.

because a kite has

scR ioednfelgesr.xuiDveFent≅PcrooDpnFeserbtcyyuottihfvee Got It? 4. Qmu∠a2d,railnatdermal∠K3L?MN is a kite. What are m∠1,

Congruence.

Concept Summary  Relationships Among Quadrilaterals

Npoarpaalliersl of Quadrilateral hsm11gmse_0606_t06341.ai
sides
Only 1 pair of par2alplealirssidoefs
parallel sides Parallelogram

Kite Trapezoid Rhombus

Rectangle

Isosceles Square
trapezoid

hsm11gmse_0606_t06342.ai Do you UNDERSTAND? MATHEMATICAL
Lesson Check
PRACTICES
Do you know HOW?
What are the measures of the numbered angles? 4. V ocabulary  Is a kite a parallelogram? Explain.

1. 1 3 2. 1 2 5. C ompare and Contrast  How is a kite similar to a
2 86Њ 48Њ rhombus? How is it different? Explain.

78Њ 6. E rror Analysis  Since a parallelogram has two

3. What is the length of the midsegment of a trapezoid pairs of parallel sides, it certainly has one pair of
parallel sides. Therefore, a parallelogram must

with bases of length 14 and 26? also be a trapezoid. What is the error in this

hsm11gmse_0606_t0597h9s.mai 11gmse_0606_t05980.ai reasoning? Explain.

Lesson 6-6  Trapezoids and Kites 393

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find the measures of the numbered angles in each isosceles trapezoid. See Problems 1 and 2.
23
7. 3 2 8. 1 9.
77Њ 1 111Њ 2 49Њ
3 1

10. Z 11. Q R 12. B C
hYs1m11gm2 se_0606_t05981.ai
12 12
X 105Њ 3
Phsm65Њ11gmse3_06S 06_t05982.ai hsm601Њ 1gmse_03606_t05983.ai

AD

W

Find EF in each trapezoid. See Problem 3.
13. A x ϩ 3 D
14. h smA 141gDmse_0606_t05985.a15i . hsm1A1gx Ϫm1seD_0606_t05986.ai

hsmE 11gm3xse_06F06_t05984.ai E 3x ϩ 5 F E 2x ϩ 1 F
7x ϩ 4 B 15 C
B 12 C B C

Find the measures of the numbered angles in each kite. See Problem 4.

16. hsm1112gmse2_20Њ 606_t06516.1a7i . h sm113 g1m42s5eЊ_0606_t06518.a1i8. hsm541Њ 1gm1s9e0_Њ0606_t06519.ai

2

19. 2 20. 21. 2

h64sЊm113gm1 se_0606_t05990.ai hsm401Њ 11g3mse_0606_t05991.ai hsm315Њ1gm1se_0606_t05992.ai
2 53

4

22. 23. 9 10 24. 2
24
hsm381Њ1gm1 s5e3_Њ 0606_t05993.ai
34Њ 1 2 46Њ h1sm11gmse_0606_t05995.ai
35
hsm611gm3se4_06506_t05994.ai 46Њ

78

B Apply 25. Open-Ended  Sketch two noncongruent kites such that the diagonals of one are
congruent to the diagonals of the other.
hsm11gmse_0606_t05996.ai hsm11gmse_0606_t05997 hsm11gmse_0606_t05998

394 Chapter 6  Polygons and Quadrilaterals

26. Think About a Plan  The perimeter of a kite is 66 cm. The length of one of its sides is
3 cm less than twice the length of another. Find the length of each side of the kite.

• Can you draw a diagram?
• How can you write algebraic expressions for the lengths of the sides?

27. Reasoning  If KLMN is an isosceles trapezoid, is it possible for KM to bisect ∠LMN
and ∠LKN ? Explain.

Algebra  Find the value of the variable in each isosceles trapezoid.

28. T W 29. B C 30. Q R
60Њ
(3x ϩ 15)Њ

60Њ 5xЊ AD PS
S R QS ϭ x ϩ 5
RP ϭ 3x ϩ 3

Algebra  Find the lengths of the segments with variable expressions.

31. hsmA1x1Ϫgm5 Dse_0606_t05999 32. hsmE x11F gmse_0606_t0600033. H x Ϫ 3G

Ex F C 2x ϩ 4 D hsmC 11gmxse_0606D_t06001
H 4x ϩ 7 G
2x Ϫ 4 2x Ϫ 2
BC EF

Algebra  Find the value(s) of the variable(s) in each kite.

34. 35. (3x ϩ 5)Њ 36.
hsm11gmse_0606_t06521.ai hsm11gmse_0606_t06522.ai hsymЊ 116xgЊmse_0606_t06524.ai

yЊ

(x ϩ 6)Њ (2x Ϫ 4)Њ (2y Ϫ 20)Њ 3xЊ
2xЊ (4x Ϫ 30)Њ 2

STEM Bridge Design  The beams of the bridge at the right form

quadrilateral ABCD. △AED @ △CDE @ △BEC and hsm11gmse_0606_t06007

mjDhCsBm=1112g0m. se_0606_t06005 hsm11gmse_0606_t06006 EB

37. Classify the quadrilateral. Explain your reasoning. A DC

38. Find the measures of the other interior angles of
the quadrilateral.

Reasoning  Can two angles of a kite be as follows? Explain.

39. opposite and acute 40. consecutive and obtuse

41. opposite and supplementary 42. consecutive and supplementary

43. opposite and complementary 44. consecutive and complementary

Lesson 6-6  Trapezoids and Kites 395

45. Developing Proof  The plan suggests a proof of Theorem 6-19. Write a proof that
follows the plan.

Given:  Isosceles trapezoid ABCD with AB ≅ DC A D

Prove:  ∠B ≅ ∠C and ∠BAD ≅ ∠D

Plan:  Begin by drawing AE } DC to form parallelogram AECD so that B 1 C
AE ≅ DC ≅ AB. ∠B ≅ ∠C because ∠B ≅ ∠1 and ∠1 ≅ ∠C. E

Also, ∠BAD ≅ ∠D because they are supplements of the congruent

angles, ∠B and ∠C.

46. Prove the converse of Theorem 6-19: If a trapezoid has a pair of congruent base
hsm11gmse_0606_t06008
Proof angles, then the trapezoid is isosceles.

Name each type of special quadrilateral that can meet the given condition.
Make sketches to support your answers.

47. exactly one pair of congruent sides 48. two pairs of parallel sides

49. four right angles 50. adjacent sides that are congruent

51. perpendicular diagonals 52. congruent diagonals

53. Prove Theorem 6-20. BC
Proo f Given:  Isosceles trapezoid ABCD with AB ≅ DC
Prove:  AC ≅ DB AD
TP
54. Prove the converse of Theorem 6-20: If the diagonals of a
Proof trapezoid are congruent, then the trapezoid is isosceles. hsm11gmse_0606_t06010

55. Given:  Isosceles trapezoid TRAP with TR ≅ PA RA
Proo f Prove:  ∠RTA ≅ ∠APR

56. Prove that the angles formed by the noncongruent sides of a
Proof kite are congruent. (Hint: Draw a diagonal of the kite.)

Determine whether each statement is true or false. Justify your response.

57. All squares are rectangles. 58. A trapezoid is a parallelogram.

59. A rhombus can be a kite. 60. Some parallehlosgmra1m1sgamresseq_u0ar6e0s.6_t06009

61. Every quadrilateral is a parallelogram. 62. All rhombuses are squares.

C Challeng e 63. Given:  Isosceles trapezoid TRAP with TR ≅ PA; T P
Proof BI is the perpendicular bisector of RA, R A
intersecting RA at B and TP at I.

Prove:  BI is the perpendicular bisector of TP.

hsm11gmse_0606_t15810

396 Chapter 6  Polygons and Quadrilaterals

For a trapezoid, consider the segment joining the midpoints of the two given
segments. How are its length and the lengths of the two parallel sides of the
trapezoid related? Justify your answer.

64. the two nonparallel sides 65. the diagonals

66. <BN> is the perpendicular bisector of AC at N. Describe the set of points, B
D, for which ABCD is a kite. N

A C

PERFORMANCE TASK hsPMmRATA1H1CEMgTAImTCICEsASeL _0606_t06011

Apply What You’ve Learned MP 4
B
Look back at the information about Alejandro’s kite on page
17 in.
351. His sketch is shown again at the right, with the missing 17 in.
vertical support drawn in. C

Choose from the following words to complete the sentences A AC ϭ 30 in.
below. 25 in.

diagonals sides opposite 25 in.

consecutive congruent perpendicular

D

parallel bisects right

Alejandro's kite fits the geometric definition of a kite because it has two pairs of
congruent a. ? sides and no pairs of congruent b. ? sides.

The vertical and horizontal supports of the kite are its c. ? . Vertices B and D
are each equidistant from vertices A and C, so the vertical support d. ? the
horizontal support.

Because the diagonals of a kite are e. ? to each other, they divide the kite
into four f. ? triangles. The kite's vertical support divides it into two g. ?
triangles.

Lesson 6-6  Trapezoids and Kites 397

6 Mid-Chapter Quiz MathX

OLMathXL® for School
R SCHO Go to PowerGeometry.com
L®

FO

Do you know HOW? Classify the quadrilateral as precisely as possible.
Find the value of each variable. Explain your reasoning.

1. 58Њ xЊ wЊ 2. 10. 11.
50Њ

(2w Ϫ 25)Њ 89Њ

(x ϩ 6)Њ aЊ mЊ 64Њ 12. 13.
125Њ 100Њ

Algebra  Find the values of the variables for which hsm11gmse_06mq_t05891.ai hsm11gmse_06mq_t05892.ai

ABCD is a parallelogram. hsm11gmse_06mq_t05882 14. I<AnBt>h}e<CfiDg>u}re<EaF>t.the right,
Fhisnmd1A1Egm. se_06mq_t05893.ai
3. hsAm11gmse_06mq_Dt05881 A 10.3 C E
(2x ϩ 10)Њ
hBsm11g1m0 seD_06mq10_t0589F4.ai
(y ϩ 20)Њ

(2x Ϫ 10)Њ

BC hsmA11gmse_210B6mDq43_t05895.Cai

4. A D 15. Given:  ▱ABCD,
AC bisects ∠DAB
2y
2x Prove:  AC bisects ∠DCB.

hsm115xgϪm1 se_06m4qx _Ϫt02 5883

BC Do you UNDERSTAND?

Algebra  Classify the quadrilateral. Then find the Decide whether the statement is true or false. If true,
value(s) of the variable(s). explain why. If false, show a countehrsemx1a1mgmplsee._06mq_t05896.ai

5. 3 xhϩsm5 11gmse_065mx qϪ_1t065. 85884Њ xЊ 16. A quadrilateral with congruent diagonals is either an
yЊ isosceles trapezoid or a rectangle.

17. A quadrilateral with congruent and perpendicular
diagonals must be a kite.

7. 6x ϩ 1 2y ϩ 6 8. 6 2y 2 18. Each diagonal of a kite bisects two angles of the kite.

2x 19. Reasoning  Can you fit all of the interior angles of a
y quadrilateral around a point without overlap? What
hsm11gmse_06mq_t05886.ai about the interior angles of a pentagon? Explain.
hsm11gmse_06mq_t05885.ai
3x ϩ 6 4y Ϫ 3 20. Writing  Explain two ways to show that a
parallelogram is a rhombus.
9. Find AB, CD, and EF. A 5x Ϫ 4 B
21. Draw a diagram showing the relationships among
Ehsm11gm4sxe_06mq_t0F5889.ai the special quadrilaterals that you have learned.
hsm11gmse_06mq_t05887.ai

C 6x Ϫ 2 D

398 Chapter 6  Mid-Chapter Quiz
hsm11gmse_06mq_t05890.ai

Algebra Simplifying Radicals
Review

Use With Lesson 6-7

A radical expression is in simplest form when all of the following are true.

• The radicand has no perfect square factors other than 1. S radical symbol
• The radicand does not contain a fraction.
• A denominator does not contain a radical expression. ͙a d radicand

#Example 1

Simplify the expressions 12 18 and 1294 ÷ 13.
# #12 18 = 12 8
Write both numbers under one radical. 1294 , 13 = 294

Simplify the expression under the 53
radical.
  = 116     = 198

  = 4 Factor out perfect squares and simplify. #    = 149 2

  = 712

Example 2

Write 4 in simplest form.

53

4 = 14 Rewrite the single radical as a quotient.
53 13

= 2 Simplify the numerator.
13

#= 2 13 Multiply by 13 (a form of 1) to remove the radical from the denominator.
13 13 13

This is called rationalizing the denominator.

= 2 13
3

Exercises

#Simplify each expression. 3. 1128 , 12 4. 51245 #5. 16 18
1. 15 110 2. 1243
#8. 13 112
6. 11336 7. 111244 #13. 112 12 9. 172 , 12 10. 1169
11. 28 , 18 12. 1300 , 15 #14. #15.
16 13 13 115
19 12

Algebra Review  Simplifying Radicals 399

6-7 Polygons in the CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Coordinate Plane
MG-AGFPSE..9B1.27. GU-sGePcEo.o2r.d7in  aUtsees ctoocrodminpautetes to
pcoemrimpueterpseorfimpeotlyegrsoonfs p. o. l.ygons . . .

MP 1, MP 3, MP 4, MP 8

Objective To classify polygons in the coordinate plane

Apply what you You and a friend are playing a board game.
learned “B-4” Players place rubber bands on their own
about classifying square grid to form different shapes. The
polygons. object of the game is to guess the vertices
of your opponent’s shape. How would you
MATHEMATICAL place pieces on the grid shown to complete
a right isosceles triangle? Sketch the
PRACTICES triangle and justify the placement of
each piece.

In the Solve It, you formed a polygon on a grid. In this lesson, you will classify polygons
in the coordinate plane.

Essential Understanding  You can classify figures in the coordinate plane using
the formulas for slope, distance, and midpoint.

The chart below reviews these formulas and tells when to use them.

Formula Key Concept  Formulas and the Coordinate Plane
When to Use It

Distance Formula To determine whether
d = 2(x2 - x1)2 + (y2 - y1)2 • sides are congruent
• diagonals are congruent

Midpoint Formula To determine
• the coordinates of the midpoint of a side
( )M = x1 + x2, y1 + y2 • whether diagonals bisect each other
2 2

Slope Formula To determine whether
y2 - y1 • opposite sides are parallel
m = x2 - x1 • diagonals are perpendicular
• sides are perpendicular

400 Chapter 6  Polygons and Quadrilaterals

What formulas do Problem 1 Classifying a Triangle y
you use? Is △ABC scalene, isosceles, or equilateral? 4B
The terms scalene, 2
isosceles, and equilateral The vertices of the triangle are A(0, 1), B(4, 4), and C(7, 0). Ax
have to do with the O 2 4 6C
side lengths of a triangle. Find the lengths of the sides using the Distance Formula.
Use the Distance hsm11gmse_0607_t06565.ai
Formula. AB = 2(4 - 0)2 + (4 - 1)2

= 116 + 9 Simplify within parentheses.
Then simplify the powers.

= 125 Simplify the radicand.

= 5 Simplify.

BC = 2(7 - 4)2 + (0 - 4)2 CA = 2(0 - 7)2 + (1 - 0)2

= 19 + 16 = 149 + 1

= 125 = 150

= 5 = 512

Since AB = BC = 5, △ABC is isosceles.

Got It? 1. △DEF has vertices D(0, 0), E(1, 4), and F(5, 2). Is △DEF scalene, isosceles,
or equilateral?

Problem 2 Classifying a Parallelogram

Is ▱ABCD a rhombus? Explain. 6y C(4, 5)
B(0, 4)
How can you Step 1 Use the Slope Formula to find the slopes of the
determine whether diagonals.
ABCD is a rhombus?
You can find the length slope of AC = 4 5-0 = 5
of each side using - ( - 2) 6
the Distance Formula, D(2, 1) x
or you can determine slope of BD = 1 - 4 = - 3 A(؊2, 0) O 24
whether the diagonals 2 - 0 2
are perpendicular.
The second way # ( )Step 2 Find the product of the slopes.5-3=-15
requires fewer steps. 6 2 12

Since the product of the slopes is not -1, the diagonals are not perpendicular.

So ABCD is not a rhombus. hsm11gmse_0607_t06566.ai

Got It? 2. ▱MNPQ has vertices M(0, 1), N(-1, 4), P(2, 5), and Q(3, 2).
a. Is ▱MNPQ a rectangle? Explain.
b. Is ▱MNPQ a square? Explain.

c. Reasoning  Is the triangle in Problem 1 a right triangle? Explain.

Lesson 6-7  Polygons in the Coordinate Plane 401

Problem 3 Classifying a Quadrilateral Iy
A kite is shown at the right. What is the most precise 2
classification of the quadrilateral formed by
connecting the midpoints of the sides of the kite? K Ϫ2 O 24 x
Ϫ2 T
K(- 4, 0), I(0, 4), T(8, 0), and E(0, - 4)

E

The midpoints of the Use the Midpoint Formula to find the coordinates
sides of the kite of the vertices of the inner quadrilateral. Draw a
diagram to see it. Then classify the figure.

hsm11gmse_0607_t06567.ai

Step 1 Find the midpoint of each side of the kite.

( ) -4 + 0, 0 + 4
A = midpoint of KI = 2 2 = ( -2, 2)

( ) 0 + 8, 4 + 0 = (4, 2)
B = midpoint of IT = 2 2

( ) 8 + 0, 0 + ( - 4)
C = midpoint of TE = 2 2 = (4, -2)

( ) 0 + ( - 4), -4 + 0 = ( -2, -2)
D = midpoint of EK = 2 2

Step 2 Draw a diagram of ABCD. Iy
A
B

K O2 x
DC T

Step 3 Classify ABCD. E BC = 0 -2 - 2 0 = 4
DA = 0 2 - (-2) 0 = 4
AB = 0 4 - ( -2) 0 = 6 Use the definition of distance
CD = 0 -2 - 4 0 = 6 on a number line.

Since opposite sides are congruhesnmt1, 1AgBmCsDe_i0s6a07p_atr0a6ll5e6l8o.gariam.

Since AB and CD are both horizontal, and BC and DA are both vertical, the
segments form right angles. So, ABCD is a rectangle.

Got It? 3. An isosceles trapezoid has vertices A(0, 0), B(2, 4), C(6, 4), and
D(8, 0). What special quadrilateral is formed by connecting the
midpoints of the sides of ABCD?

402 Chapter 6  Polygons and Quadrilaterals

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. △TRI has vertices T( -3, 4), R(3, 4), and I(0, 0). Is
3. Writing  Describe how you would y
△TRI scalene, isosceles, or equilateral? determine whether the lengths of 4E
2. Is QRST below a rectangle? Explain. the medians from base angles D
and F are congruent.

2yR S 4. Error Analysis  A student says that D Fx
O2 x the quadrilateral with vertices Ϫ2 O 2
4
Q D(1, 2), E(0, 7), F(5, 6), and G(7, 0)
T
is a rhombus because its diagonals

are perpendicular. What is the student’s error?

hsm11gmse_0607_t06587.ai

Practicehsamn11dgmPser_o06b07l_et0m658-6S.aoi lving Exercises MATHEMATICAL

PRACTICES

A Practice Determine whether △ABC is scalene, isosceles, or equilateral. Explain. See Problem 1.

5. A y 6. y A 7. y A

O B CO x 2B
C Ϫ2 Ϫ2 B Ox
x
2 Ϫ2 Ϫ2 2

C Ϫ2

Determine whether the parallelogram is a rhombus, rectangle, square, or See Problem 2.

none. Explain.

8. hPs(m-111,g2m),seO_(006,007)_, St0(645, 808),.aTi(3, 2) hsm11gmse_06097_. tL0(615,829).,aMi (3, 3), N(h5s,m21),1Pgm(3s,e1_)0607_t06590.ai

10. R( -2, -3), S(4, 0), T(3, 2), V( -3, -1) 11. G(0, 0), H(6, 0), I(9, 1), J(3, 1)

12. W( -3, 0), I(0, 3), N(3, 0), D(0, -3) 13. S(1, 3), P(4, 4), A(3, 1), T(0, 0)

What is the most precise classification of the quadrilateral formed by See Problem 3.
connecting in order the midpoints of each figure below?

14. parallelogram PART 15. rectangle EFGH 16. isosceles trapezoid JKLM

2 y A y E y
P x JK
F x
Ϫ4 Ϫ2 O Ϫ2 O 2 2
M Ϫ2 O
R H x
T Ϫ4 2L
Ϫ4 G

hsm11gmse_0607_t06582 hsm11gmse_0607_t06583 hsm11gmse_0607_t06584
Lesson 6-7  Polygons in the Coordinate Plane 403

B Apply Graph and label each triangle with the given vertices. Determine whether each
triangle is scalene, isosceles, or equilateral. Then tell whether each triangle is a
right triangle.

17. T(1, 1), R(3, 8), I(6, 4) 18. J( -5, 0), K(5, 8), L(4, -1)

19. A(3, 2), B( -10, 4), C( -5, -8) 20. H(1, -2), B( -1, 4), F(5, 6)

Graph and label each quadrilateral with the given vertices. Then determine the
most precise name for each quadrilateral.

21. P( -5, 0), Q( -3, 2), R(3, 2), S(5, 0) 22. S(0, 0), T(4, 0), U(3, 2), V( -1, 2)

23. F(0, 0), G(5, 5), H(8, 4), I(7, 1) 24. M( -14, 4), N(1, 6), P(3, -9), Q( -12, -11)

25. A(3, 5), B(7, 6), C(6, 2), D(2, 1) 26. N( -6, 4), P( -3, 1), Q(0, 2), R( -3, 5)

27. J(2, 1), K(5, 4), L(8, 1), M(2, -3) 28. H( -2, -3), I(4, 0), J(3, 2), K( -3, -1)

29. W( -1, 1), X(0, 2), Y(1, 1), Z(0, -2) 30. D( -3, 1), E( -7, -3), F(6, -3), G(2, 1)

31. Think About a Plan  Are the triangles at the right y T W
congruent? How do you know? Q R6
2 x
• Which triangle congruence theorem can you use? S
• Which formula should you use? OP 2

32. Reasoning  A quadrilateral has opposite sides with equal
slopes and consecutive sides with slopes that are negative
reciprocals. What is the most precise classification of the
quadrilateral? Explain.

Determine the most precise name for each quadrilateral. Then fihndsmits1a1rgema. se_0607_t06585

33. A(0, 2), B(4, 2), C( -3, -4), D( -7, -4) 34. J(1, -3), K(3, 1), L(7, -1), M(5, -5)

35. DE is a midsegment of △ABC at the right. Show that the Triangle yB D
Midsegment Theorem holds true for △ABC.
2C
36. a. Writing  Describe two ways you can show whether a
quadrilateral in the coordinate plane is a square. AE x

b. Reasoning  Which method is more efficient? Explain. O 2 46

37. Interior Design  Interior designers often use grids hsm11gmse_0607_t06591
to plan the placement of furniture in a room. The
design at the right shows four chairs around a coffee
table. The designer plans for cutouts of chairs on
lattice points. She wants the chairs oriented at the
vertices of a parallelogram. Does she need to fix her
plan? If so, describe the change(s) she should make.

404 Chapter 6  Polygons and Quadrilaterals

38. Use the diagram at the right. A 4y F
a. What is the most precise classification of ABCD? D 2 G
b. What is the most precise classification of EFGH? x
c. Are ABCD and EFGH congruent? Explain.
46
C Challenge 39. Coordinate Geometry  The diagonals of quadrilateral Ϫ6 O
H
EFGH intersect at D( -1, 4). EFGH has vertices at E(2, 7) and B
F( -3, 5). What must be the coordinates of G and H to ensure
that EFGH is a parallelogram? C E
Ϫ4

The endpoints of AB are A(−3, 5) and B(9, 15). Find the coordinates of
the points that divide AB into the given number of congruent segments.

40. 4 41. 6 42. 10 43. 50h sm11gmse_4046. 0n7_t06592

Standardized Test Prep

SAT/ACT 45. K( -3, 0), I(0, 2), and T(3, 0) are three vertices of a kite. Which point could be the
fourth vertex?

E(0, 5) E(0, 0) E(0, -2) E(0, -10)

46. In the diagram, lines / and m are parallel. What is the value of x? 155Њ ᐉ
5 13 (x2 ϩ 11)Њ m
12 25

Short 47. In the diagram, which segment is shortest? P Q
57Њ
Response PS PQ
61Њ
PR QR hsm11gmse_0607_t06593
60Њ
62Њ
48. A( -3, 1), B( -1, -2), and C(2, 1) are three vertices of quadrilateral ABCD. S R
Could ABCD be a rectangle? Explain.

Mixed Review hsm11gmse_0607_t12944

49. Algebra  Find the measure of each angle and xϩ2 2 3 5x Ϫ 8 See Lesson 6-6.
the value of x in the isosceles trapezoid. 62Њ
1
Find the circumcenter of △ABC.
50. A(1, 1), B(5, 3), C(5, 1) See Lesson 5-3.

51. A( -5, 0), B( -1, -8), C( -1, 0)

Get Ready!  To prepare for Lesson 6-8, do Exerhcsismes115g2m–5s4e._0607_t06594

Find the slope of XY . See Lesson 3-7.

52. X(0, a), Y( -a, 2a) 53. X( -a, b), Y(a, b) 54. X(a, 0), Y(c + d, b)

Lesson 6-7  Polygons in the Coordinate Plane 405

6-8 Applying Coordinate CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Geometry
Prepares for GM-AGFPSE..9B1.24. GU-sGePcEo.o2r.d4in  aUtsees to
cporovrdeinsiamtepsletogeporomveetrsiicmtphleeogreemoms aetlgriecbtrhaeicoarellmy. s
aMlgPeb1r,aMicaPll3y., MP 4
MP 1, MP 3, MP 4

Objective To name coordinates of special figures by using their properties

The points shown are three y
vertices of a parallelogram. What
are all the possible coordinates of 6B
D, the fourth vertex? How do
you know? 4 C
x
Knowing previously 2A
established
properties of Ϫ4 Ϫ2 O 2468
parallelograms will
help with this one.

MATHEMATICAL

PRACTICES In the Solve It, you found coordinates of a point and named it using numbers for the

x- and y-coordinates. In this lesson, you will learhnstmo 1us1egvmarsiaeb_l0es6f0o8r_tht1e 3co0o6r8dinates.

Lesson Essential Understanding  You can use variables to name the coordinates of a
figure. This allows you to show that relationships are true for a general case.
Vocabulary
• coordinate proof In Chapter 5, you learned about the segment joining the midpoints of two sides of a
triangle. Here are three possible ways to place a triangle and its midsegment.

y Q(c, d) y y
Q(a, b) Q(0, 2b)
MN
M
P(a, b) R(e, b) x N M N
O P(0, 0) x P(؊2a, 0) O
x
R(c, 0) R(2c, 0)

Figure 1 Figure 2 Figure 3

Figure 1 does not use the axes, so it requires more variables. Figures 2 and 3 have

( ) ( )gIhnosoFmdigpu1lr1aegc3em,mthseeen_ct.0oIo6nr0Fd8iign_uatrt0ee6s25a, 3rthe9eMm(-idhaps,obmi)n1at 1ncodgomNrd(scien, b_a)t0e. 6Yso0aur8e_cMat0n6as2h5e, eb2s4mt0ha1ant1dFgNigmuasree2+_3c0,isb26t0h.8e_t06541

easiest to work with.

406 Chapter 6  Polygons and Quadrilaterals

To summarize, to place a figure in the coordinate plane, it is usually helpful to place at
least one side on an axis or to center the figure at the origin. For the coordinates, try to
anticipate what you will need to do in the problem. Then multiply the coordinates by
the appropriate number to make your work easier.

Problem 1 Naming Coordinates

How do you start the What are the coordinates of the vertices of each figure?

Lp orookbaletmth?e position of A SQRE is a square where SQ = 2a. B TRI is an isosceles triangle where TI = 2a.

the figure. Use the given The axes bisect each side. The y-axis is a median.

fih nroofwomrmftahareteioxa-ncahtnovdedryet-teaexxrmeissi.ne S y Q Ry

O x TO x
E R I

Since SQRE is a square centered at The y-axis is a median, so it bisects TI .

the origin and SQ = 2a, S and Q are TI = 2a, so T and I are both a units from the

eshaamscmhea1isu1tngruimtes sfforeor_mt0h6eea0oc8thh_eatrx0vi6se.5rTt4ihc2ee s. yh-asxmis1. T1hgemhesieg_h0t o6f0T8R_Itd0o6e5s4n3ot depend

on a, so use a different variable for R.

S(؊a, a) y Q(a, a) R(0, b) y

x T(؊a, 0) O x
O I(a, 0)

E(؊a, ؊a) R(a, ؊a)

Got It? 1. What are the coordinates of the vertices of each figure?

a. RECT is a rectangle with height a b. KITE is a kite where IE = 2a, KO = b,

hsm11gmasned_l0e6ng0t8h_2tb0.6T5h4e4y-axis bisects hsma1n1dgOmTs=e_c.0T6h0e8x_-atx0i6s5b4is5ects IE.

EC and RT .

y y
E C I

x

x KO T

RO T E

hsm11gmse_0608_t06546 hsm11gmse_0608_t06547

Lesson 6-8  Applying Coordinate Geometry 407

Problem 2 Using Variable Coordinates y

The diagram shows a general parallelogram with a vertex C(2b, 2c) B(2a ؉ 2b, 2c)
at the origin and one side along the x-axis. What are D
the coordinates of D, the point of intersection of the x
diagonals of ▱ABCO? How do you know?
O A(2a, 0)
• The coordinates of the vertices of ▱ABCO
• OB bisects AC and AC bisects OB

The coordinates of D Since the diagonals of a parallelogram bisect each

ointtheerrs,etchteiomn.idUpsoeinthteofMeiadcphoisnetgFmoermntuilsatthoehfirisnpmdoti1nhte1ogf mse_0608_t06548

midpoint of one diagonal.

Use the Midpoint Formula to find the midpoint of AC.

( )D = midpoint of AC = 2a + 2b, 0 + 2c = (a + b, c)
2 2

The coordinates of the point of intersection of the diagonals of ▱ABCO
are (a + b, c).

Got It? 2. a. Reasoning  In Problem 2, explain why the x‑coordinate of B is the sum of
2a and 2b.

b. The diagram below shows a trapezoid with the base centered at the origin.

Is the trapezoid isosceles? Explain.
y

R(؊b, c) A(b, c)

T(؊a, 0) O x
P(a, 0)

You can use coordinate geohmsemtr1y1agndmaslgee_b0r6a0to8p_rto0v6e5t4h9eorems in geometry. This kind

of proof is called a coordinate proof. Sometimes it is easier to show that a theorem is
true by using a coordinate proof rather than a standard deductive proof. It is useful to
write a plan for a coordinate proof. Problem 3 shows you how.

408 Chapter 6  Polygons and Quadrilaterals

Problem 3 Planning a Coordinate Proof

How do you start? Plan a coordinate proof of the Trapezoid Midsegment Theorem (Theorem 6-21).
Start by drawing a (1)  The midsegment of a trapezoid is parallel to the bases.
diagram. Think about (2)  The length of the midsegment of a trapezoid is half the sum of the lengths of
how you want to
place the figure in the the bases.
coordinate plane.
Step 1 Draw and label a figure. Step 2 Write the Given and Prove statements.

Midpoints will be involved, so use Use the information on the

multiples of 2 to name coordinates. diagram to write the statements.

y Given:  MN is the midsegment
R(2b, 2c) A(2d, 2c)
of trapezoid ORAP.

M N Prove:  MN } OP, MN } RA,
= 1 +
x MN 2 (OP RA)

O P(2a, 0)

Step 3 Determine the formulas you will need. Then write the plan.

•  First, use the Midpoint Formula to find the coordinates of M and N.

•  h Thsemn1, u1sgemthseeS_lo0p6e0F9o_rtm0u6l3a3to0 determine whether the slopes of MN, OP,

and RA are equal. If they are, MN, OP, and RA are parallel.

•  F inally, use the Distance Formula to find and compare the lengths of
MN, OP, and RA.

Got It? 3. Plan a coordinate proof of the Triangle Midsegment Theorem
(Theorem 5-1).

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Use the diagram at the right. 4. Reasoning  How do variable coordinates generalize

1. In ▱KLMO, OM = 2a. y figures in the coordinate plane?
What are the coordinates K L(2a ؉ 2b, c) 5. Reasoning  A vertex of a quadrilateral has

of K and M? coordinates (a, b). The x-coordinates of the other

2. What are the slopes of the O x three vertices are a or -a, and the y-coordinates are b
diagonals of KLMN? M or -b. What kind of quadrilateral is the figure?

3. What are the coordinates 6. Error Analysis  A classmate says the endpoints of
the midsegment of the trapezoid in Problem 3 are
of the point of intersection of KM and OL? ( ) ( )b2, +
c and d 2 a, c . What is your classmate’s
2 2
hsm11gmse_0608_t06550.ai error? Explain.

Lesson 6-8  Applying Coordinate Geometry 409

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Algebra  What are the coordinates of the vertices of each figure? See Problem 1.

7. rectangle with base b 8. square with sides of 9. square centered at the origin,
and height h
y length a with side length b

ST S y T T y W

x x O x
OW OW S Z

10. parallelogram where S is 11. rhombus centered at the 12. isosceles trapezoid with base

a units from the origin and origin, with SW = 2r and centered at the origin, with
Zhissmby1u1ngitsmfrsoem_0th6e0o8r_igti0n6 551.ai TZh=sm2t 1y1gmse_06 08_t06552. abihassem21a1agnmdy OseR_=06c 08_t06553.ai
T T RW
ST SW

OZ x Ox SO x
W Z Z

13. The diagram below sthheowpas raapllealroagllrealmogrisamahr.shWmoim1th1boguusmt. uHsseoinw_g0dt6ho0ey8oD_uitsk0tan6no5cw5e?5F.oarimula, See Problem 2.
determine whether
hsm11gmse_0608_t06554.ai
A(؊a, a) y hsm11gmse_0608_t06556.ai

B(b, b)

x

O

D(؊b, ؊b) C(a, ؊a)



14. Plan a coordinate proof to show that the midpoints of the sides of an isosceles See Problem 3.

trapezoid form a rhombus. y
EA
a. Nrigahmt,ewthitehcboootrtdominabtaessehoslfeminsog1st1hceg4lmaes, sttoerap_p0bea6zso0ei8dle_TntR0gAt6hP545abt7,t.ahaneid R
F
EG = 2c. The y-axis bisects the bases. D
b. Write the Given and Prove statements. x
GP
c. How will you find the coordinates of the midpoints of

each side? T

d. How will you determine whether DEFG is a rhombus?

B Apply 15. Open-Ended  Place a general quadrilateral in the coordinate plane.

16. Reasoning  A rectangle LMNP is centered at the origin with M(r, -s). What are the
hsm11gmse_0608_t06558.ai
coordinates of P?

410 Chapter 6  Polygons and Quadrilaterals

Give the coordinates for point P without using any new variables.

17. isosceles trapezoid 18. trapezoid with a right ∠ 19. kite
y (a, b)
y P y
(a, b) P (0, a)

(0, c) x
O (b, 0)
x
O (c, 0) x (0, ؊c)
OP

20. a. Draw a square whose diagonals of length 2b lie on the x- and y-axes.

dbch... sFGCmioinvm1de1ptthhgueetmecslotsoheoper_edl0seinn6oag0fttt8ehws_oootf0afadt6hsj5aied5cvee9eno.rattfiistchiedesehssosqfomutfhat1ehre1es.gqsuqmaursaeer.e_.0608_t06560.ai hsm11gmse_0608_t06561.ai

e. Writing  Do the slopes show that the sides are perpendicular? Explain.

21. Make two drawings of an isosceles triangle with base length 2b and height 2c.
a. In one drawing, place the base on the x-axis with a vertex at the origin.
b. In the second, place the base on the x-axis with its midpoint at the origin.
c. Find the lengths of the legs of the triangle as placed in part (a).
d. Find the lengths of the legs of the triangle as placed in part (b).
e. How do the results of parts (c) and (d) compare?

22. W and Z are the midpoints of OR and ST , respectively. In parts (a)–(c), find the
coordinates of W and Z.

a. y R(a, b) S(c, d) b. y R(2a, 2b) S(2c, 2d) c. y R(4a, 4b) S(4c, 4d)

W (?, ?) Z W (?, ?) Z W (?, ?) Z
O x O x O x

T(e, 0) T(2e, 0) T(4e, 0)

d. You are to plan a coordinate proof involving the midpoint of WZ. Which of the
figures (a)–(c) would you prefer to use? Explain.

hsm11gmse_0608_t06562.ahi sm11gmse_0608_t06563.ai hsm11gmse_0608_t06564.ai

Plan the coordinate proof of each statement.

23. Think About a Plan  The opposite sides of a parallelogram are congruent
(Theorem 6-3).

• How will you place the parallelogram in a coordinate plane?
• What formulas will you need to use?

24. The diagonals of a rectangle bisect each other.

25. The consecutive sides of a square are perpendicular.

Classify each quadrilateral as precisely as possible.

26. A(b, 2c), B(4b, 3c), C(5b, c), D(2b, 0) 27. O(0, 0), P(t, 2s), Q(3t, 2s), R(4t, 0)

28. E(a, b), F(2a, 2b), G(3a, b), H(2a, -b) 29. O(0, 0), L( -e, f), M(f - e, f + e), N(f, e)

Lesson 6-8  Applying Coordinate Geometry 411

30. What property of a rhombus makes it convenient to place its diagonals on the
x‑ and y‑axes?

STEM 31. Marine Archaeology  Marine archaeologists sometimes use a coordinate system
on the ocean floor. They record the coordinates of points where artifacts are found.
Assume that each diver searches a square area and can go no farther than b units
from the starting point. Draw a model for the region one diver can search. Assign
coordinates to the vertices without using any new variables.

C Challenge Here are coordinates for eight points in the coordinate plane (q + p + 0).
A(0, 0), B(p, 0), C(q, 0), D(p + q, 0), E(0, q), F(p, q), G(q, q), H(p + q, q). Which
four points, if any, are the vertices for each type of figure?

32. parallelogram 33. rhombus 34. rectangle

35. square 36. trapezoid 37. isosceles trapezoid

Standardized Test Prep

SAT/ACT 38. Which number of right angles is NOT possible for a quadrilateral to have?

exactly one exactly two exactly three exactly four

39. The vertices of a rhombus are located at (a, 0), (0, b), ( -a, 0), and (0, -b), where
a 7 0 and b 7 0. What is the midpoint of the side that is in Quadrant II?

Short a( ) ,b ( ) -a, b ( ) -a, - b a( ) , - b
2 2 2 2 2 2 2 2
Response
40. In ▱PQRS, PQ = 35 cm and QR = 12 cm. What is the perimeter of ▱PQRS?

23 cm 47 cm 94 cm 420 cm

41. In △PQR, PQ 7 PR 7 QR. One angle measures 170. List all possible whole
number values for m∠P.

Mixed Review

42. Let X( -2, 3), Y(5, 5), and Z(4, 10). Is △XYZ a right triangle? Explain. See Lesson 6-7.

Write (a) the inverse and (b) the contrapositive of each statement. See Lesson 2-2.

43. If x = 51, then 2x = 102. 44. If a = 5, then a2 = 25.

45. If b 6 -4, then b is negative. 46. If c 7 0, then c is positive.

47. If the sum of the interior angle measures of a polygon is not 360, then the polygon is
not a quadrilateral.

Get Ready!  To prepare for Lesson 6-9, do Exercises 48 and 49. See Lesson 3-7.
48. Find the equation for the line that contains the origin and (4, 5).
49. Find the equation for the line that contains (p, q) and has slope ab.

412 Chapter 6  Polygons and Quadrilaterals

Concept Byte Quadrilaterals in CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Quadrilaterals
Use With Lesson 6-9 GM-ACFOS.D.9.12. GM-CaOke.4fo.1rm2 aMl gaekoemfoertmricalcognesotmruecttriiocns
wcointhstaruvcatiroientsy wofitthooalvaanrdietmyeotfhtoodosl .a.n.d
Technology MmePth5ods . . .
MP 5

Construct MATHEMATICAL
• Use geometry software to construct a quadrilateral ABCD.
• Construct the midpoint of each side of ABCD. PRACTICES
• Construct segments joining the midpoints, in order, to form quadrilateral EFGH.
DG C
Investigate
• Measure the lengths of the sides of EFGH and their slopes. H F
• Measure the angles of EFGH. A
E
What kind of quadrilateral does EFGH appear to be? B

Exercises hsm11gmse_0609a_t06183.ai

1. Manipulate quadrilateral ABCD. DG C
a. Make a conjecture about the quadrilateral with vertices that are the KN J

midpoints of the sides of a quadrilateral. HO MF
b. Does your conjecture hold when ABCD is concave? I
c. Can you manipulate ABCD so that your conjecture doesn’t hold? L P
A E B
2. Extend  Draw the diagonals of ABCD.
a. Describe EFGH when the diagonals are perpendicular.
b. Describe EFGH when the diagonals are congruent.
c. Describe EFGH when the diagonals are both perpendicular and congruent.

3. Construct the midpoints of EFGH and use them to construct quadrilateral IJKL.
Construct the midpoints of IJKL and use them to construct quadrilateral MNOP.
For MNOP and EFGH, compare the ratios of the lengths of the sides, perimeters,
and areas. How are the sides of MNOP and EFGH related?

4. Writing  In Exercise 1, you made a conjecture as to the type of quadrilateral
EFGH appears to be. Prove your conjecture. Include in your proof the Triangle
Midsegment Theorem, “If a segment joins the midpoints of two sides of a triangle,
then the segment is parallel to the third side and half its length.”

5. Describe the quadrilateral formed by joining the midpoints, in order, of the sides

of each of the following. Justify each response.

a. parallelogram b. rectangle c. rhombus hsdm. 1sq1ugamrese_0609a_t06187.ai
e. trapezoid f. isosceles trapezoid g. kite

Concept Byte  Quadrilaterals in Quadrilaterals 413

6-9 Proofs Using MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Coordinate Geometry
MG-AGFPSE..9B1.24. GU-sGePcEo.o2r.d4in  aUtsees ctooprrdoinveatseismtpoleprove
gsiemopmleetgriecotmheotrriecmthseaolrgeembrsaaiclgaellby.raically.

MP 1, MP 2, MP 3, MP 7

Objective To prove theorems using figures in the coordinate plane

Better draw a The coordinates of three vertices of a rectangle are ( − 2a, 0), (2a, 0),
diagram! and (2a, 2b). A diagonal joins one of these points with the fourth
vertex. What are the coordinates of the midpoint of the diagonal?
Justify your answer.

MATHEMATICAL In the Solve It, the coordinates of the point include variables. In this lesson, you will use
coordinates with variables to write a coordinate proof.
PRACTICES

Essential Understanding  You can prove geometric relationships using variable
coordinates for figures in the coordinate plane.

Proof Problem 1 Writing a Coordinate Proof

What formulas do Use coordinate geometry to prove that the midpoint of the hypotenuse of a
you need? right triangle is equidistant from the three vertices.
You need to find the
distance to a midpoint, Given: △OEF is a right triangle. y
so use the midpoint and E(0, 2b)
distance formulas.
M is the midpoint of EF . M

Prove: EM = FM = OM x

Coordinate Proof: O F(2a, 0)

( ) 2a + 0, 0 + 2b
By the Midpoint Formula, M = 2 2 = (a, b).

By the Distance Formula,

OM = 2a2 + b2 hsm11gmse_0609_t06327

FM = 2(2a - a)2 + (0 - b)2 EM = 2(0 - a)2 + (2b - b)2

= 2a2 + b2 = 2a2 + b2

Since EM = FM = OM, the midpoint of the hypotenuse is equidistant
from the vertices of the right triangle.

Got It? 1. Reasoning  What is the advantage of using coordinates O(0, 0),
E(0, 2b), and F(2a, 0) rather than O(0, 0), E(0, b), and F(a, 0)?

414 Chapter 6  Polygons and Quadrilaterals