46Properties of Rhombuses, *Rectangles, and Squares

MAFS.912.G-C0.3.11 Prove theorems about

parallelograms... rectangles areparallelograms with congruent diagonals. AlsoIVIAFS.912.G-SRT.2.5

MP1.MP3

, r

robjectives To define and classify special types of parallelograms

To use properties of diagonals of rhombuses and rectangles

Getting Ready!

Can you make agood argumentto justify yourobservations?

Fold a piece of notebook paper in half. Fold it in halfagain in the other direction. Draw a diagonal line fromone vertex to the other. Cut through the folded paperalong that line. Unfold the paper. What do you noticeabout the sides and about the diagonals of the figureyou formed?

MATHEMATICAL

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

will study in this lesson.

A LessonAyipf Vocabulary

• rhombus

rectanglesquare

Essential Understanding The parallelograms in the Take Note box below havebasic properties about their sides and angles that help identify them. The diagonals ofthese parallelograms also have certain properties.

Key Concept Special Parallelograms

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

four right angles.

Diagram

J L

n r

j L

n r

c PowerGeometry.com I Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares 375

How do you decidewhether ABCD is a

rhombus, rectangle,or square?Use the definitions of

rhombus, rectangle, andsquare along with themarkings on the figure.

The Venn diagram at the right

shows the relationships amongspecial parallelograms.

Rhombuses Squares Rectangles

Problem 1 Classifying Special Parallelograms

Is CJABCD a rhombus, a rectangle, or a square? Explain.

CD ABCD is a rectangle. Opposite angles of a parallelogram

are congruent so m/LD is 90. By the Same-Side Interior

Angles Theorem, m/-A = 90 and m/LC = 90. Since

ejABCD has four right angles, it is a rectangle. You cannot

conclude that ABCD is a square because you do not know

its side lengths.

Got It? 1. Is cjEFGH a rhombus, a rectangle, or asquare? Explain.

Theorem 6-13

Theorem

If a parallelogram is a

rhombus, then its diagonals are

perpendicular.

If...

ABCD is a rhombus

A D

Then ...

AC ± BD

Theorem 6-14

Theorem

If a parallelogram is a

rhombus, then each

diagonal bisects a pair

of opposite angles.

If...

ABCD is a rhombus

D

Then ...

Z1 s^2

Z.3 = YL4

^5 = /-6

Z7 s Z.8

You will prove Theorem 6-14 in Exercise 45.

376 Chapters Polygons and Quadrilaterals

How are the

numbered anglesformed?

The angles are formedby diagonals, Use whatyou know about thediagonals of a rhombusto find the anglemeasures.

Proof Proof of Theorem 6-13

Given: ABCD is a rhombus.

Prove; The diagonals ofABCD are perpendicular.

Statements Reasons

1) >1 and C are equidistant 1) All sides of a rhombus

from B and D) B and D are =.

are equidistant from A

and C.

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) AC lW 3) Through two points,there is one unique

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

What are the measures of the numbered angles in rhombus ABCDl

m/L 1 = 90 The diagonals of a rhombus are l.

Alternate Interior Angles TheoremmL2 = 58

Each diagonal of a rhombus bisects apair of opposite angles.mZ-3 = 58

mZ.1 4- mlB + mZ.4 = 180 Triangle Angle-Sum Theorem

90 + 58 + mZ.4 = 180 Substitute.

148 + = 180 Simplify.

mZ.4 = 32 Subtract 148 from each side.

Got It? 2. What are the measures of the numbered angles in^ rhombus PQRSl

C PowerGeomeTry.com I Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares 377

The diagonals of a rectangle also have a special property.

How can you find thelength of a diagonal?Since RSBF is a rectangleand its diagonals arecongruent, use theexpressions to write anequation.

Theorem 6" 15

Theorem

If a parallelogram is

a rectangle, then its

diagonals are congruent.

If...

ABCD is a rectangleThen...

AC=BD

D

J L

n rB C

You will prove Theorem 6-15 in Exercise 41.

Problem 3 Finding Diagonal Length

Multiple Choice In rectangle RSBF, SF = 2x + 15 and RB = 5x — 12.

What is the length of a diagonal?

CA^l CI^9 CO 18

You know that the diagonals ofa rectangle are congruent, sotheir lengths are equal.

SF = RB

Cd:>33

Set the algebraic expressions forSf and RB equal to each otherand find the value of x.

Substitute 9 for x in the

expression for RB.

2x + 15 = 5x - 12

15 = 3x - 12

27 = 3x

9 = x

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 arethe lengths of the diagonals of rectangle LMNOl

b. Reasoning What type oftriangle is APMW? Explain.

M N

0

f-

378 Chapter 6 Polygons and Quadrilaterals

Lesson Check

Do you know HOW?

Is each parallelogram a rhombus, rectangle, orsquare? Explain.

1. . ^ „ 2.

130°

3. What are the measures of the

numbered angles in the rhombus?

50°

MATHEMATICAL

Do you UNDERSTAND? PRACTICES

5. Vocabulary Which special parallelograms are

equiangular? Which special parallelograms are

equilateral?

6. Error Analysis Your class Dneeds to find the value of

X for which CJDEFG is a

rectangle. A classmate'swork is shown below. r

(9x - 6)°

/ ̂

r

What is the error? Explain. (2x + 8)°

4. Algebra JKLM is a rectangle. If /L = 4x — 12 andMK = X, what is the value of x? What is the length of

each diagonal?

MATHEMATICAL

PRACTICESPractice and Problem-Solving Exercises

Practice Decide whether the parallelogram is a rhombus, a rectangle, or asquare. Explain.

8.7.

^ See Problem 1.

m

Find the measures of the numbered angles in each rhombus. ^ See Problem 2.

13.

2

\4

(s Lesson 6'4 Properties of Rhombuses, Rectangles, and Squares 379

Apply

Find the measures of the numbered angles in each rhombus.

15. ZP1 16.

Algebra LMNP is a rectangle. Find the value of a: and the length of eachdiagonal.

18. LAf=A:andMi' = 2x-4 19. LN=5x-8andMP==2x+ 1

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

22. IiV=7x-2andMP = 4x + 3 23. IW= 3x + 5andMP = 9x- 10

Determine the most precise name for each quadrilateral.

24. u 25. ■; 26.

^ See Problem 3.

List the quadrilaterals that have the given property. Choose amongparallelogram, rhombus, rectangle, and square.

29. Opposite sides are =.28. All sides are s,

30. Opposite sides are ||.32. All A are right A.

34. Diagonals bisect each other.

36. Diagonals are _L.

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

31. Opposite ^ are =.

33. Consecutive A are supplementary.

35. Diagonals are =.

37. Each diagonal bisects opposite A.

38. rhombus 39. square 2x-7

y-1

4x + 3

j L

n r

2y-5

3y- 9

^ 40. Think About a Plan Write a proof.Pfoof Gjyen; Rectangle PLAN

Prove: ALTP = ANTA

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

380 Chapter b Polygons and Quadrilaterals

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

Given: ABCD is a rectangle.

Prove: AC = 80 g

ABCD is a o. e. ?

Opposite sides ofa o are =.

D

ABCD is

a rectangle.

a. ?

BC^BC

c. ?

f.

SAS

AC^BD

h. ?

AABC and LDCB

are right A.

d. ?

AABC = ADCB

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

42. = 2x + 5,

SVV=5x-20

R

43. m^l = 3y-6

W

1 6zJ?

AA. BD = 4x-y+l

B

45. Prove Theorem 6-14.

Given: ABCD is a rhombus.

Prove: AC bisects ABAD and ABCD.

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 therhombus at the right.

48. Open-Ended On graph paper, draw a parallelogram that is neithera rectangle nor a rhombus. H 2r - A 6

/C 4b-

r+ 1

6r J

6-3

(2x + 6)'

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

49. AC = 2(;c-3)andBD = x+5 50. AC = 2(5fl + 1)andBD = 2(fl + 1)

51. AC = yandBD = 3y-4 52. AC = yandBD = 4 - c

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

Challenge Algebra Find the value of x in the rhombus.

53.

(7x2 _ 10)(6x2 _ 3^)0 54.

(2x^ - 25x)

(3x^ + 60)'

SAT/Aa

Short

> Response

Standardized Test Prep

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

C£> Opposite sides are parallel. CO Adjacent sides are perpendicular.

CT> It is a parallelogram. CO All sides are congruent.

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

isosceles triangles, as shown. What is mZ.1?

CD 18 CD 72

CO 36 CD 108

57. Which term best describes AD in AABC?

CO altitude CO median

CD angle bisector CD perpendicular bisector g

58. Write the first step of an indirect proof that APQR is not a right triangle.

Mixed Review

Can you conclude that the quadrilateral is a parallelogram? Explain.

59. 5 60. 6 61. B

^ See Lesson 6-3.

.C

D

In APQR, points S, T, and U are midpoints. Complete each statement.

62. TQ = ? S3.PQ = _]_ 64. TU= ?

65. ̂ II ? 66. TU 67. PQ •

^ See Lesson 5-1.

3-

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

68. Draw a rhombus that is not a square.

69. Draw a rectangle that is not a square.

^ See Lesson 6-4.

382 Chapter 6 Polygons and Quadrilaterals