CorrectionKey=NL-B;CA-B 24 . 3 DO NOT EDIT--Changes …€¦ · Essential Question: What are the properties of rectangles, rhombuses, and squares? ... application of the Law of Detachment

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Explore Exploring Sides, Angles, and Diagonals of a Rectangle

A rectangle is a quadrilateral with four right angles. The figure shows rectangle ABCD.

Investigate properties of rectangles.

A Use a tile or pattern block and the following method to draw three different rectangles on a separate sheet of paper.

B Use a ruler to measure the sides and diagonals of each rectangle. Keep track of the measurements and compare your results to other students.

Reflect

1. Why does this method produce a rectangle? What must you assume about the tile?

2. Discussion Is every rectangle also a parallelogram? Make a conjecture based upon your measurements and explain your thinking.

3. Use your measurements to make two conjectures about the diagonals of a rectangle.

Conjecture:

Conjecture:

Resource Locker

A B

D C

A B

D C

You must assume that the corners of the tile are right angles. Therefore, each stage of

the drawing produces a line that meets the previous line at a right angle. The completed

quadrilateral has four right angles, so it is a rectangle.

Yes; in every case, opposite sides are congruent. All rectangles are parallelograms because

a quadrilateral in which both pairs of opposite sides are congruent is a parallelogram.

The diagonals of a rectangle bisect each other.

The diagonals of a rectangle are congruent.

Module 24 1217 Lesson 3

24.3 Properties of Rectangles, Rhombuses, and Squares

Essential Question: What are the properties of rectangles, rhombuses, and squares?

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Common Core Math StandardsThe student is expected to:

G-CO.11

Prove theorems about parallelograms. Also G-SRT.5

Mathematical Practices

MP.6 Precision

Language ObjectiveExplain to a partner how to classify different types of quadrilaterals as rectangles, rhombuses, or squares.

HARDBOUND SEPAGE 987

BEGINS HERE

HARDCOVER PAGES 987994

Turn to these pages to find this lesson in the hardcover student edition.

Properties of Rectangles, Rhombuses, and Squares

ENGAGE Essential Question: What are the properties of rectangles, rhombuses, and squares?A rectangle is a parallelogram with congruent

diagonals; a rhombus is a parallelogram with

perpendicular diagonals, each of which bisects a

pair of opposite angles; a square is both a rectangle

and a rhombus and has the properties of each.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photo, drawing attention to the geometrical shapes that form the design of the flag. Then preview the Lesson Performance Task.

1217

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Explore Constructing Perpendicular Bisectors

and Perpendicular Lines

A rectangle is a quadrilateral with four right angles.

The figure shows rectangle ABCD.

Investigate properties of rectangles.

Use a tile or pattern block and the following method to draw three different rectangles on a

separate sheet of paper.

Use a ruler to measure the sides and diagonals of each rectangle. Keep track of the

measurements and compare your results to other students.

Reflect

1. Why does this method produce a rectangle? What must you assume about the tile?

2. Discussion Is every rectangle also a parallelogram? Make a conjecture based upon your measurements

and explain your thinking.

3. Use your measurements to make two conjectures about the diagonals of a rectangle.

Conjecture:

Conjecture:

Resource

Locker

G-CO.11 Prove theorems about parallelograms. Also G-SRT.5

AB

DC

A B

D C

You must assume that the corners of the tile are right angles. Therefore, each stage of

the drawing produces a line that meets the previous line at a right angle. The completed

quadrilateral has four right angles, so it is a rectangle.

Yes; in every case, opposite sides are congruent. All rectangles are parallelograms because

a quadrilateral in which both pairs of opposite sides are congruent is a parallelogram.

The diagonals of a rectangle bisect each other.The diagonals of a rectangle are congruent.

Module 24

1217

Lesson 3

24 . 3 Properties of Rectangles,

Rhombuses, and Squares

Essential Question: What are the properties of rectangles, rhombuses, and squares?

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1217 Lesson 24 . 3

L E S S O N 24 . 3

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Explain 1 Proving Diagonals of a Rectangle are CongruentYou can use the definition of a rectangle to prove the following theorems.

Properties of Rectangles

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

Example 1 Use a rectangle to prove the Properties of Rectangles Theorems.

Given: ABCD is a rectangle.

Prove: ABCD is a parallelogram; _ AC ≅ _ BD .

Statements Reasons

1. ABCD is a rectangle. 1. Given

2. ∠A and ∠C are right angles. 2. Definition of

3. ∠A ≅ ∠C 3. All right angles are congruent.

4. ∠B and ∠D are right angles. 4.

5. ∠B ≅ ∠D 5.

6. ABCD is a parallelogram. 6.

7. ― AD ≅

― CB 7. If a quadrilateral is a parallelogram, then its opposite sides are

congruent.

8. ― DC ≅

― DC 8.

9. ∠D and ∠C are right angles. 9. Definition of rectangle

10. ∠D ≅ ∠C 10. All right angles are congruent.

11. 11.

12. 12.

Reflect

4. Discussion A student says you can also prove the diagonals are congruent in Example 1 by using the SSS Triangle Congruence Theorem to show that △ADC ≅ △BCD. Do you agree? Explain.

Your Turn

Find each measure.

5. AD = 7.5 cm and DC = 10 cm. Find DB.

6. AB = 17 cm and BC = 12.75 cm. Find DB.

A A

D D D

B B

C CC

A

D

B

C

Reflexive Property of Congruence

△ADC ≅ △BCD

― AC ≅ ― BD

SAS Triangle Congruence Theorem

CPCTC

rectangle

Definition of rectangle

All right angles are congruent.If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

No; in order to use the SSS Triangle Congruence Theorem, you would have to know that

_ AC ≅ _ BD , which is what you are trying to prove.

By the Pythagorean Theorem, A C 2 = 7. 5 2 + 1 0 2 , so AC = 12.5 cm. Diagonals of a rectangle are congruent, so DB = AC = 12.5 cm.

Opposite sides of the rectangle are congruent, so DC = AB = 17 cm. By the Pythagorean Theorem, D B 2 = 1 7 2 + 12.7 5 2 , so DB = 21.25 cm.


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Learning ProgressionsIn this lesson, students extend their earlier work with parallelograms to explore the properties of three special quadrilaterals: the rectangle, a quadrilateral with four congruent (right) angles; the rhombus, a quadrilateral with four congruent sides; and the square, a quadrilateral with four congruent (right) angles and four congruent sides. The proofs demonstrate that each of these quadrilaterals is a parallelogram; as such, they inherit all of the properties of parallelograms. Proving these theorems is an application of the Law of Detachment (if p → q is true and p is true, then q is true), which students will use as they further their study of mathematics.


BEGINS HERE EXPLORE Exploring Sides, Angles, and Diagonals of a Rectangle

INTEGRATE TECHNOLOGYStudents have the option of doing the rectangle activity either in the book or online.

QUESTIONING STRATEGIESIs a rectangle a parallelogram? How do you know? Yes; since the opposite sides are

congruent, the Opposite Sides Criterion applies.

EXPLAIN 1 Proving Diagonals of a Rectangle Are Congruent

QUESTIONING STRATEGIESWhat two things do you have to prove in order to prove the theorem? Prove the figure

is a parallelogram and then prove triangles that

contain the diagonals of the rectangle are

congruent.

Does it matter which pair of triangles you prove congruent for this proof? Explain. Yes;

you need to choose a pair of triangles that contain

the diagonals of the rectangle as sides.

AVOID COMMON ERRORSStudents may try to prove that a rectangle is a parallelogram by stating that the opposite sides are congruent and using the Opposite Sides Criterion for a parallelogram. However, the given information states only that the figure is a rectangle (that is, it has four right angles). The fact that a rectangle has opposite sides that are congruent is a consequence of the fact that a rectangle is a parallelogram. In order to avoid circular reasoning, congruent opposite sides cannot be used as part of this proof.

PROFESSIONAL DEVELOPMENT

Properties of Rectangles, Rhombuses, and Squares 1218


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Explain 2 Proving Diagonals of a Rhombus are PerpendicularA rhombus is a quadrilateral with four congruent sides. The figure shows rhombus JKLM.

Properties of Rhombuses

If a quadrilateral is a rhombus, then it is a parallelogram.If a parallelogram is a rhombus, then its diagonals are perpendicular.If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

Example 2 Prove that the diagonals of a rhombus are perpendicular.

Given: JKLM is a rhombus.

Prove: ― JL ⊥ ― MK

Since JKLM is a rhombus, ― JM ≅ . Because JKLM is also a parallelogram, ― MN ≅ ― KN because

. By the Reflexive Property of Congruence, ― JN ≅ ― JN ,

so △JNM ≅ △JNK by the . So, by CPCTC.

By the Linear Pair Theorem, ∠JNM and ∠JNK are supplementary. This means that m∠JNM + m∠JNK = .

Since the angles are congruent, m∠JNM = m∠JNK so by , m∠JNM + m∠JNK = 180° or

2m∠JNK = 180°. Therefore, m∠JNK = and ⊥ ― MK .

Reflect

7. What can you say about the image of J in the proof after a reflection across ― MK ? Why?

8. What property about the diagonals of a rhombus is the same as a property of all parallelograms? What special property do the diagonals of a rhombus have?

Your Turn

9. Prove that if a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. Given: JKLM is a rhombus.Prove: ― MK bisects ∠JML and ∠JKL; ― JL bisects ∠MJK and ∠MLK.

M K

L

J

M K

L

JN

M K

L

JN

_ JK

diagonals of a parallelogram bisect each other

SSS Triangle Congruence Theorem

substitution

∠JNM ≅ ∠JNK

180°

90° _ JL

That its image is L because _ MK is the perpendicular bisector of

_ JL .

The diagonals of rhombuses bisect each other, which is a property of all parallelograms.

The diagonals of rhombuses are perpendicular, which is a property unique to rhombuses.

Since JKLM is a rhombus, _ JM ≅ _ LM and

_ JK ≅

_ LK . _ MK ≅ _ MK by the Reflexive Property of

Congruence. So, △MJK ≅ △MLK by the SSS Triangle Congruence Theorem. Therefore, ∠JMK ≅ ∠LMK and ∠JKM ≅ ∠LKM by CPCTC, so

_ MK bisects ∠JML and ∠JKL. A similar

argument shows that _ JL bisects ∠MJK and ∠MLK.





BEGINS HERE

COLLABORATIVE LEARNING

Whole Class ActivityAs a class, come up with a list of the properties common to some of the quadrilaterals studied in the lesson, such as “Opposite sides congruent” and “Diagonals perpendicular.” Use the list to construct a table summarizing which quadrilateral (parallelogram, rectangle, rhombus, square) has each property. Save the table as a class review resource.

EXPLAIN 2 Proving Diagonals of a Rhombus Are Perpendicular

QUESTIONING STRATEGIESWhy do the diagonals of a rhombus bisect each other? Because a rhombus is a

parallelogram, the diagonals bisect each other.

To prove the diagonals of a rhombus are perpendicular, do you need to show that each

of the four angles formed by the intersecting diagonals is a right angle? Why or why not? No; you

need to show only that one angle is a right angle. If

one angle is a right angle, it’s easy to see that the

others must also be right angles by the Vertical

Angles Theorem and the Linear Pair Theorem.

CONNECT VOCABULARY Have students complete a quadrilateral chart, with pictures and explanations for each type of quadrilateral they have encountered in this unit (rhombus, square, rectangle, other parallelograms, and so on).

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Explain 3 Using Properties of Rhombuses to Find Measures

Example 3 Use rhombus VWXY to find each measure.

Find XY.

All sides of a rhombus are congruent, so _ VW ≅ _ WX and VW = WX.

Substitute values for VW and WX. 6m - 12 = 4m + 4

Solve for m. m = 8

Sustitute the value of m to find VW. VW = 6 (8) - 12 = 36

Because all sides of the rhombus are congruent, then _ VW ≅ _ XY , and XY = 36.

Find ∠YVW.

The diagonals of a rhombus are , so ∠WZX is a right angle and

m∠WZX = .

Since m∠WZX = (3 n 2 - 0.75) °, then .

Solve for n. 3 n 2 - 0.75 = 90

n =

Substitute the value of n to find m∠WVZ.

m∠WVZ =

Since _ VX bisects ∠YVW, then

Substitute 53.5° for m∠WVZ. m∠YVW = 2 (53.5°) = 107°

Your Turn

Use the rhombus VWXY from Example 3 to find each measure.

10. Find m∠VYX. 11. Find m∠XYZ.

Y X

W

Z

(9n + 4)°

(3n2 - 0.75)°

6m - 12

4m + 4

V

(3 n 2 - 0.75°) = 90°

∠YVZ ≅ ∠WVZ

5.5

53.5°

90°

perpendicular

From Part B, m∠YVW = 107° 107° + m∠VYX = 180° m∠VYX = 73°

m∠XYZ = 1 __ 2 (m∠VYX)

m∠XYZ = 1 _ 2 (73°) m∠XYZ = 36. 5°


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BEGINS HERE

DIFFERENTIATE INSTRUCTION

ModelingDivide students into groups. Have each group cut out models of a rectangle, a rhombus, and a square from construction paper. Then have students use a protractor and a ruler to verify the properties of rectangles and rhombuses from the theorems in this lesson.

EXPLAIN 3 Using Properties of Rhombuses to Find Measures

QUESTIONING STRATEGIESWhat type of angles do the diagonals of a rhombus form? right angles

AVOID COMMON ERRORSStudents may be confused when asked to solve equations to find segments of special quadrilaterals. Remind students that the sides and angles marked in the figure are not necessarily the ones asked for in the problem. Have students, when they read a problem, copy the figure and circle the length or measure they are asked to find.



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Explain 4 Investigating the Properties of a SquareA square is a quadrilateral with four sides congruent and four right angles.

Example 4 Explain why each conditional statement is true.

If a quadrilateral is a square, then it is a parallelogram.

By definition, a square is a quadrilateral with four congruent sides. Any quadrilateral with both pairs of opposite sides congruent is a parallelogram, so a square is a parallelogram.

If a quadrilateral is a square, then it is a rectangle.

By definition, a square is a quadrilateral with four .

By definition, a rectangle is also a quadrilateral with four . Therefore, a square is a rectangle.

Your Turn

12. Explain why this conditional statement is true: If a quadrilateral is a square, then it is a rhombus.

13. Look at Part A. Use a different way to explain why this conditional statement is true: If a quadrilateral is a square, then it is a parallelogram.

Elaborate

14. Discussion The Venn diagram shows how quadrilaterals, parallelograms, rectangles, rhombuses, and squares are related to each other. From this lesson, what do you notice about the definitions and theorems regarding these figures?

15. Essential Question Check-In What are the properties of rectangles and rhombuses? How does a square relate to rectangles and rhombuses?

Rectangle

Parallelogram

SquareRhombus

Quadrilateral

right angles

right angles

By definition, a rhombus is a quadrilateral with four congruent sides. Since a square is also a quadrilateral that has four congruent sides, then a square is a rhombus.

Possible answer: All four angles of a square are right angles. All right angles are congruent. Any quadrilateral with both pairs of opposite angles congruent is a parallelogram, so a square is a parallelogram.

A rectangle, a rhombus, and a square are all

defined as quadrilaterals, but they can only be shown to be parallelograms by theorems

that prove their properties.

A rectangle is a parallelogram with four right angles and congruent diagonals. A rhombus

is a parallelogram with four congruent sides, diagonals that are perpendicular, and each of

its diagonals bisects a pair of opposite angles. A square is both a rectangle and a rhombus,

so it has the properties of both.





BEGINS HERE

Rectangle Rhombus

Diagonalsare congruent.

Is a parallelogram

Diagonals areperpendicular.

Diagonals bisecta pair of opposite

angles.

LANGUAGE SUPPORT

Connect VocabularyStudents may have difficulty distinguishing the special quadrilaterals in this lesson. Have them write the definitions on a poster, and look through magazines or books to find pictures of rectangles, rhombuses, and squares that they can add to the poster. Have them group the pictures based on the type of figure, show the pictures to the class, and then describe the identifying characteristics of each type of figure.

EXPLAIN 4 Investigating the Properties of a Square

QUESTIONING STRATEGIESWhy is a square also a rhombus? A square is a

quadrilateral with 4 congruent sides, so a

square is a rhombus.

AVOID COMMON ERRORSWhen working with shapes they are familiar with, like squares, some students have trouble distinguishing between properties that belong to the shape by definition and properties that must be proven. When discussing these definitions, point out which properties are included and which are not.

ELABORATE QUESTIONING STRATEGIES

How do the definitions of the special quadrilaterals in this lesson differ from the

theorems in this lesson? The definitions only

identify the figures as quadrilaterals. Proving the

properties as theorems shows they are also

parallelograms.

SUMMARIZE THE LESSONHave students make a graphic organizer or chart to summarize properties of rectangles

and rhombuses. Sample:

1221 Lesson 24 . 3


• Online Homework• Hints and Help• Extra Practice

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1. Complete the paragraph proof of the Properties of Rectangles Theorems.Given: ABCD is a rectangle.Prove: ABCD is a parallelogram;

_ AC ≅ _

BD .

Proof that ABCD is a : Since ABCD is a rectangle, ∠A and

∠C are right angles. So ∠A ≅ ∠C because .

By similar reasoning, ∠B ≅ ∠D. Therefore, ABCD is a parallelogram because

Proof that the diagonals are congruent: Since ABCD is a parallelogram,

― AD ≅ ― BC because .

Also, by the Reflexive Property of Congruence. By the definition of a

rectangle, ∠D and ∠C are right angles, and so

because all right angles are . Therefore, △ADC ≅ △BCD by the

and ≅ by CPCTC.

Find the lengths using rectangle ABCD.

2. AB = 21; AD = 28. What is the value of AC + BD?

3. BC = 40; CD = 30. What is the value of BD - AC?

4. An artist connects stained glass pieces with lead strips. In this rectangular window, the strips are cut so that FH = 34 in. Find JG. Explain.

Evaluate: Homework and Practice

A B

CD

A

B C

D

E

F G

H

J

parallelogram

all right angles are congruent

it is a quadrilateral that has congruent opposite angles.

opposite sides of a parallelogram are congruent


congruent

_ DC ≅ _ DC

B D 2 = 2 1 2 + 2 8 2 , so BD = 35. Then, AC = 35, so AC + BD = 70.

_ AC

_ BD

∠D ≅ ∠C

The diagonals of the rectangle are congruent, so BD = AC. Then, BD - AC = 0.

_ EG ≅

_ FH because the diagonals of a rectangle are congruent. Since FH measures 34 inches,

EG also measures 34 inches. Since diagonals of a parallelogram bisect each other,

JG = 1 _ 2 EG. So, JG = 1 _ 2 (34) = 17 in.




IN1_MNLESE389762_U9M24L3.indd 1222 2/26/16 4:18 AMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

1–3 2 Skills/Concepts MP.2 Reasoning

4–6 2 Skills/Concepts MP.4 Modeling

7–19 2 Skills/Concepts MP.2 Reasoning

20 3 Strategic Thinking MP.3 Logic



BEGINS HERE

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreExploring Properties of Rectangles

Exercise 1

Example 1Proving Diagonals of a Rectangle Are Congruent

Exercises 2–6

Example 2Proving Diagonals of a Rhombus Are Perpendicular

Exercise 8

Example 3Using Properties of Rhombuses to Find Measures

Exercises 9–12

Example 4Investigating the Properties of a Square

Exercise 19

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Some students may benefit from a hands-on approach for exploring special quadrilaterals. Have students make a physical model of a parallelogram with paper strips and brads. Ask them to adjust the model until it is a rectangle, as shown in an exercise, and then measure its diagonals. Then have them adjust the model to form a rhombus and measure the angles made by the intersecting diagonals.



The rectangular gate has diagonal braces. Find each length.

5. Find HJ. 6. Find HK.

7. Find the measure of each numbered angle in the rectangle.

8. Complete the two-column proof that the diagonals of a rhombus are perpendicular.

Given: JKLM is a rhombus.

Prove: ― JL ⊥ ― MK

Statements Reasons

1. _

JM ≅ _

JK 1. Definition of rhombus

2. _

MN ≅ _

KN 2.

3. _

JN ≅ _

JN 3. Reflexive Property of Congruence

4. 4. SSS Triangle Congruence Theorem

5. ∠JNM ≅ ∠JNK 5.

6. ∠JNM and ∠JNK are supplementary. 6.

7. 7. Definition of supplementary

8. ∠JNM = ∠JNK 8. Definition of congruence

9. + ∠JNK = 180° 9. Substitution Property of Equality

10. 2m∠JNK = 180° 10. Addition

11. m∠JNK = 90° 11. Division Property of Equality

12. 12. Definition of perpendicular lines

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H

G

L

J

K48 in.

30.8 in.

ge07se_c06l04003aAB

5

61°

1

4 3

2

M K

L

JN

HK = JG

= 2 (JL)

= 2 (30.8 in.)

= 61.6 in.

_ HJ ≅ _ KG

HJ = KG

= 48 in.

∠3 and ∠5 are right angles, so they each measure 90°.∠2 ≅ to the give angle, so m∠2 = 61°.m∠1 = 90° - 61° = 29°; ∠4 ≅ ∠1, so m∠4 = m∠1 = 29°.

So, m∠1 = 29°, m∠2 = 61°, m∠3 = 90°, m∠4 = 29°, and m∠5 = 90°.

Diagonals of a parallelogram bisect each other.

Linear Pair Theorem

CPCTC

△JNM ≅ △JNK

_ JL ⊥ _ MK

m∠JNM + m∠JNK = 180°

m∠JNK


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IN1_MNLESE389762_U9M24L3 1223 4/25/14 8:44 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices



INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 You may want to review the properties of parallelograms before assigning the exercises. Then summarize the properties of rectangles and rhombuses and point out that these figures are also parallelograms. Explain that a square has all the properties of a parallelogram, a rectangle, and a rhombus.

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ABCD is a rhombus. Find each measure.

9. Find AB.

10. Find m∠ABC.

Find the measure of each numbered angle in the rhombus.

11. 12.

13. Select the word that best describes when each of the following statements are true. Select the correct answer for each lettered part.A. A rectangle is a parallelogram. always sometimes neverB. A parallelogram is a rhombus. always sometimes neverC. A square is a rhombus. always sometimes neverD. A rhombus is a square. always sometimes neverE. A rhombus is a rectangle. always sometimes never

B C

D

F

A

(4y - 1)°12y°7x + 2

4x + 15

527°

1

432

570° 1

432

Possible answer:

4x + 15 = 7x + 2

13 = 3x

4 1 _ 3 = x

_ AB ≅ _ CD , so AB = CD = 7x + 2 = 7 (4 1 _ 3 ) + 2 = 32 1 _ 3

Possible answer:

_ AC ⊥ ̄ BD , so m∠AFB = 90°; 12y = 90, so y = 7.5.

m∠ABC = 180° - 2 (4y -1) °

= 180° - 2 (4 ⋅ 7.5 -1) ° = 122°

∠2 ≅ ∠3 ≅ ∠5 ≅ to the given angle, so they each measure 27°.m∠1 + m∠2 + 27° = 180° m∠1 + 27° + 27° = 180° m∠1 = 126° ∠4 ≅ ∠1, so m∠4 = 126°So, m∠1 = 126°, m∠2 = 27°, m∠3 = 27°, m∠4 = 126°, and m∠5 = 27°.

∠4 ≅ to the given angle, so m∠4 = 70°.∠1 ≅ ∠2 ≅ ∠3 ≅ ∠5

m∠1 + m∠2 + 70° = 180° m∠1 + m∠1 + 70° = 180° m∠1 = 55°So, m∠1 = 55°, m∠2 = 55°, m∠3 = 55°, m∠4 = 70° and m∠5 = 55°.




IN1_MNLESE389762_U9M24L3 1224 9/30/14 12:49 PM


BEGINS HEREAVOID COMMON ERRORSStudents might expect to use only the current lesson’s properties and theorems when writing proofs. Point out that proofs often build on previous knowledge and thus use concepts learned in earlier lessons.v



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14. Use properties of special parallelograms to complete the proof.Given: EFGH is a rectangle. J is the midpoint of

_ EH .

Prove: △FJG is isosceles.

Statements Reasons

1. EFGH is a rectangle. J is the midpoint of

_ EH .

1. Given

2. ∠E and ∠H are right angles. 2. Definition of rectangle

3. ∠E ≅ ∠H 3.

4. EFGH is a parallelogram. 4.

5. 5.

6. 6.

7. 7.

8. 8.

9. 9.

15. Explain the Error Find and explain the error in this paragraph proof. Then describe a way to correct the proof.Given: JKLM is a rhombus.Prove: JKLM is a parallelogram. Proof: It is given that JLKM is a rhombus. So, by the definition of a rhombus, ― JK ≅ ― LM , and ― KL ≅ ― MJ . If a quadrilateral is a parallelogram, then its opposite sides are congruent. So JKLM is a parallelogram.

F G

E HJ

K L

J M

All right angles are congruent.

A rectangle is a parallelogram.

In a parallelogram, opposite sides are congruent.

Definition of midpoint


CPCTC

Definition of isosceles triangle△FJE is isosceles.

_ FJ ≅ _ GJ

_ EJ ≅

_ HJ

△FJE ≅ △GJH

_ EF ≅ _ GH

You cannot use a theorem that assumes the quadrilateral is a parallelogram to justify the final statement because you do not know that JKLM is a parallelogram. That is what you are trying to prove. Instead, use the converse, which states that if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. So therefore, JKLM is a parallelogram.




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The opening of a soccer goal is shaped like a rectangle.

16. Draw a rectangle to represent a soccer goal. Label the rectangle ABCD to show that the distance between the goalposts,

_ BC , is three times the distance from the top of

the goalpost to the ground. If the perimeter of ABCD is 64 feet, what is the length of

_ BC ?

17. In your rectangle from Evaluate 16, suppose the distance from B to D is (y + 10) feet, and the distance from A to C is (2y - 5.3) feet. What is the approximate length of

_ AC ?

18. PQRS is a rhombus, with PQ = (7b - 5) meters and QR = (2b - 0.5) meters. If S is the midpoint of

_ RT , what is the length of

_ RT ?

T

S

R

Q

P

Possible drawing:

Perimeter = AB + BC + CD + DA

64 = x + 3x + x + 3x

64 = 8x

8 = x

BC = 3x = 3 (8) = 24 feet

The diagonals of a rectangle are congruent, so _ AC ≅

_ BD .

AC = BD

2y - 5.3 = y + 10

y = 15.3

So, AC = 2 (15.3) - 5.3 = 30.6 - 5.3 = 25.3 feet.

Possible answer:

By the midpoint, _ TS ≅

_ RS .

By the definition of a rhombus, _ PQ ≅ _ QR ≅

_ RS ≅

_ SP .

PQ = QR

7b - 5 = 2b − 0.5

b = 0.9

RT = TS + RS = 2 (RS) = 2 (QR)

= 2 (2b − 0.5) = 2 (2 (0.9) − 0.5) = 2 (1.8 − 0.5) = 2 (1.3) = 2.6

So, the length of _ RT is 2.6 meters.

B 3x

xA

C

D






BEGINS HEREPEERTOPEER DISCUSSIONAsk students to work with a partner to write conditional statements in the form “If a quadrilateral is a [figure], then it is a [figure].” An example would be “If a quadrilateral is a rectangle, then it is a parallelogram.” Also have them write conditional statements in the form “If a parallelogram is a [figure], then it is a [figure],” for example, “If a parallelogram is a square, then it is a rhombus.” Have them write as many of these statements as possible and then compare their statements with other student pairs.



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19. Communicate Mathematical Ideas List the properties that a square “inherits” because it is each of the following quadrilaterals.

a. a parallelogram

b. a rectangle

c. a rhombus

H.O.T. Focus on Higher Order Thinking

Justify Reasoning For the given figure, describe any rotations or reflections that would carry the figure onto itself. Explain.

20. A rhombus that is not a square

21. A rectangle that is not a square

22. A square

23. Analyze Relationships Look at your answers for Exercises 20–22. How does your answer to Exercise 22 relate to your answers to Exercises 20 and 21? Explain.

• Both pairs of opposite sides are parallel.

• Both pairs of opposite sides are congruent.

• Both pairs of opposite angles are congruent.

• One angle is supplementary to both of its consecutive angles.

• The diagonals bisect each other.

• The diagonals are congruent.

• The diagonals are perpendicular.

• Each diagonal bisects a pair of opposite angles.

180° rotation around its center; reflectional symmetry across a line that contains opposite vertices (two lines)

180° rotation around its center; reflectional symmetry across a line that contains the midpoints of opposite sides (two lines)

90° rotation around its center; reflectional symmetry across a line that contains opposite vertices (two lines), or a line that contains the midpoints of opposite sides (two lines)

A square has the reflectional properties of both a rhombus that is not a square and a rectangle that is not a square. Because squares have all angles congruent and all sides congruent, as opposed to only all sides congruent (rhombuses) or only all angles congruent (rectangles), a square has 90° rotational symmetry.




JOURNALHave students describe the properties of a rectangle and a rhombus, then have them draw and label examples of each type of figure.

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The portion of the Arkansas state flag that is not red is a rhombus. On one flag, the diagonals of the rhombus measure 24 inches and 36 inches. Find the area of the rhombus. Justify your reasoning.

Lesson Performance Task

12 in.

18 in.

□

432 in . 2

Possible answer: Draw the diagonals of the rhombus. Since they are

perpendicular, the four triangles into which you have divided the rhombus

are right triangles. The area of each is 1 _ 2 bh = 1 _ 2 (18) (12) = 108 i n 2 . There are

four such triangles with a total area of 4 ∙ 108 = 432 i n 2 .




EXTENSION ACTIVITY

The design of the Arkansas flag is composed of a number of geometrical shapes. Ask students to write and solve at least four problems involving the names, perimeters, areas, and/or other information relating to the shapes on the flag. Students should continue to use 24 inches and 36 inches as the lengths of the diagonals of the larger rhombus.

INTEGRATE MATHEMATICAL PRACTICESFocus on PatternsMP.8 Have students look at the rhombus on the Arkansas flag with diagonals measuring 24 inches and 36 inches. Pose these questions:

• Suppose you cut the rhombus along its diagonals to make four shapes, and then rearranged the shapes to form another type of quadrilateral that you studied in this lesson. Name the quadrilateral and its dimensions. a rectangle measuring 18

in. × 24 in. or 12 in. × 36 in.

• Find the area of the new quadrilateral. Then propose a method for finding the area of a rhombus if you know the lengths of its diagonals. Divide the product of the diagonals

by 2.

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 What is the relationship between the combined areas of the four red triangles on an Arkansas flag, and the area of the rest of the flag? Answer without referring to the actual dimensions of the flag. They are equal. Sample answer: The four

red triangles can be rearranged to form the

rhombus that makes up the rest of the flag. So, the

two are equal in area.

Scoring Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain.0 points: Student does not demonstrate understanding of the problem.



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