8-5...activity assess i can… use the properties of rhombuses, rectangles, and squares to solve...

Activity Assess

I CAN… use the properties of rhombuses, rectangles, and squares to solve problems.

Properties of SpecialParallelograms

8-5 EXPLORE & REASON

Consider these three figures.

Figure 1 Figure 3Figure 2

A. What questions would you ask to determine whether each figure is a parallelogram?

B. Communicate Precisely What questions would you ask to determine whether Figure 1 is a rectangle? What additional questions would you ask to determine whether Figure 2 is a square?

C. If all three figures are parallelograms, what is the most descriptive name for Figure 3? How do you know?

STUDY TIPRecall that a rhombus is a parallelogram, so it has all the properties of parallelograms.

EXAMPLE 1 Find the Diagonals of a Rhombus

A. Parallelogram ABCD is a rhombus. What are the measures of ∠1, ∠2, ∠3, and ∠4 ?

4 3

1 2

D

CA E

B

By the Converse of the Perpendicular Bisector Theorem, B and D are on the perpendicular bisector of ‾ AC , so ‾ AC ⟂ ‾ BD .

All four angles formed by the intersection of the diagonals are right angles, so the measure of ∠1, ∠2, ∠3, and ∠4 is 90.

B. Parallelogram JKLM is a rhombus. How are ∠1, ∠2, ∠3, and ∠4 related?

By SSS, △JKL ≅ △JML , so ∠1 ≅ ∠2 and ∠3 ≅ ∠4.

The diagonals of a rhombus bisect the angles at each vertex.

Try It! 1. a. What is WY ? b. What is m∠RPS ?

3

5

Z

YW

X

P

Q

R

S 70°

All four sides of a rhombus are congruent.

What properties of rhombuses, rectangles, and squares differentiate them from other parallelograms?

ESSENTIAL QUESTION

CONCEPTUAL UNDERSTANDING

1 2

3 4

J

L

K M

_

JL ≅ _

JL

LESSON 8-5 Properties of Special Parallelograms 391

PearsonRealize.com

Activity Assess

Try It! 2. Each quadrilateral is a rhombus.

a. What is m∠MNO? b. What is QT?

O

NM

L62°

S

R

Q

P T

2y + 33y − 1

5y − 4

COMMON ERRORYou may incorrectly state that m∠ADE = m∠DAE . Remember that consecutive angles are not necessarily congruent.

EXAMPLE 2 Find Lengths and Angle Measures in a Rhombus

A. Quadrilateral ABCD is a rhombus. What is m∠ADE ?

m∠DAE + m∠AED + m∠ADE = 180

53 + 90 + m∠ADE = 180

m∠ADE = 37

B. Quadrilateral GHJK is a rhombus. What is GH?

Step 1 Find x.

2x + 3 = 4x − 7

2x = 10

x = 5

Step 2 Use the value of x to find GH.

HJ = 3(5) + 1 = 16

GH = HJ

GH = 16

If a parallelogram is a rhombus, then its diagonals are perpendicular bisectors of each other.

PROOF: SEE EXERCISE 14.

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

PROOF: SEE EXERCISE 17.

If...

Then... ‾ WY and ‾ XZ are perpendicular bisectors of each other.

If...

Then... ∠1 ≅ ∠2, ∠3 ≅ ∠4, ∠5 ≅ ∠6, and ∠7 ≅ ∠8.

ZY

WX

3 4

51

78

26

THEOREM 8-16

THEOREM 8-17

CEA

B

D

53°

‾ AC bisects ∠BAD , so m∠DAC = 53 .

J

K

G

H3x + 1

2x + 3 4x − 7

‾ AC ⟂ ‾ BD , so m∠AED = 90.

392 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com

Activity Assess

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

PROOF: SEE EXAMPLE 3.

If...

Then... ‾ AC ≅ ‾ BD

A B

D C

THEOREM 8-18

Try It! 3. A carpenter needs to check the gate his apprentice built to be sure it is rectangular. The diagonals measure 52 inches and 53 inches. Is the gate rectangular? Explain.

EXAMPLE 3 Prove Diagonals of a Rectangle Are Congruent

Write a proof for Theorem 8-18.

Given: PQRS is a rectangle.

P Q

S R

Prove: ‾ PR ≅ ‾ QS

Plan: To show that the diagonals are congruent, find a pair of congruent triangles that each diagonal is a part of. Both △PSR and △QRS appear to be congruent. Think about how to use properties of rectangles to show they are congruent. Draw each triangle separately and label the congruent sides.

P

S R

Q

S R

Proof:

Statements Reasons

1) PQRS is a rectangle. 1) Given

2) PQRS is a parallelogram. 2) Def. of rectangle 3) ‾ PS ≅ ‾ QR 3) Opposite sides of a

parallelogram are congruent. 4) ∠PSR and ∠QRS are right angles. 4) Def. of rectangle 5) ∠PSR ≅ ∠QRS 5) All right angles are congruent. 6) ‾ SR ≅ ‾ RS 6) Reflexive Prop. of Equality 7) △PSR ≅ △QRS 7) SAS Triangle Congruence Thm. 8) ‾ PR ≅ ‾ QS 8) CPCTC

PROOF

STUDY TIPWhen you see triangles in a diagram for a proof, you can often use congruent triangles and CPCTC to complete the proof.


Activity Assess

Try It! 4. A rectangle with area 1,600 m 2 is 4 times as long as it is wide. What is the sum of the diagonals?

Try It! 5. Square ABCD has diagonals ‾ AC and ‾ BD . What is m∠ABD? Explain.

EXAMPLE 4 Find Diagonal Lengths of a Rectangle

Paul is training his horse to run the course at a pace of 4 meters per second or faster. Paul rides his horse from D to C to E to B in 1 minute 30 seconds. The figure ABCD is a rectangle. Did he make his goal?

Use the Pythagorean Theorem to find BD. Then use properties of rectangles to find each segment length and the total distance. Finally, determine his speed.

(BD ) 2 = 8 0 2 + 19 2 2

(BD ) 2 = 43,264

BD = 208

Use the properties of rectangles to find the total distance.

CE = EB = 104

DC + CE + EB = 192 + 104 + 104 = 400

Determine the pace.

400 ÷ 90 ≈ 4.4

Paul’s horse ran at a pace of about 4.4 m/s, so he made his goal.

Formulate

Compute

Interpret

EXAMPLE 5 Diagonals and Angle Measures of a Square

Figure WXYZ is a square. If WY + XZ = 92 , what is the area of △WPZ ?

Since the figure is also a rhombus, ‾ WY ⟂ ‾ XZ and WP and ZP are the base and height of △WPZ.

Step 1 Find the lengths of the diagonals.

WY + XZ = 92

WY = XZ = 46

Step 2 Find WP and ZP.

WP = 1 __ 2 (WY) = 23

ZP = 1 __ 2 (XZ) = 23

Step 3 Find the area of △WPZ .

area(△WPZ) = 1 __ 2 (23)(23) = 264.5

The area of △WPZ is 264.5 square units.

APPLICATION

IMG740828

Apply the Pythagorean Theorem.

WXYZ is a rectangle, so ‾ WY ≅ _

XZ .

Diagonals are congruent and bisect each other.

WXYZ is a parallelogram, so ‾ WY and

_ XZ bisect each other.

P

X

Y

W

ZUSE STRUCTUREConsider the four triangles formed by the diagonals of a square. What observations do you make about these triangles?

A

E

B

CD

80 m

192m


D C

A B

P R

Q

S

W X

Z Y

Concept Summary Assess

CONCEPT SUMMARY Properties of Special Parallelograms

Do You UNDERSTAND?

1. ESSENTIAL QUESTION What properties of rhombuses, rectangles, and squares differentiate them from other parallelograms?

2. Error Analysis Figure QRST is a rectangle. Ramona wants to show that the four interior triangles are congruent. What is Ramona’s error?

✗

Diagonals of a rectangle are congruent andbisect each other, so RP ~= TP ~= QP ~= SP.Because the diagonals are perpendicularbisectors, RPS, SPT, TPQ, and QPRare right angles. Therefore, by SAS,

∆RPS ~= ∆SPT ~= ∆TPQ ~= ∆PQR.

3. Construct Arguments Is any quadrilateral with four congruent sides a rhombus? Explain.

Do You KNOW HOW?

Find each length and angle measure for rhombus DEFG. Round to the nearest tenth.

4. DF

5. m∠DFG

6. EG

Find each length for rectangle MNPQ. Round to the nearest tenth.

7. MP

8. MQ

Find each length and angle measure for square WXYZ.

9. m∠YPZ

10. m∠XWP

11. XZ

12. What is the value of x?

R

P

S

Q T

F

E

G

D62°

55.6

N P

M Q5 4

P

X

Y

W

Z7

Rectangle

WORDS

Rhombus Square

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

If a parallelogram is a rhombus, then the diagonals are perpendicular and bisect each pair of opposite angles.

If a parallelogram is a square, the properties of both a rectangle and a rhombus apply.

‾ AC ≅ ‾ BD SYMBOLS ‾ PR ⟂ ‾ QS ‾ WY ≅ ‾ XZ

‾ WY ⟂ ‾ XZ

DIAGRAMS

(8x − 6)°

(4x + 5)°


Z

PYW

X

PRACTICE & PROBLEM SOLVING

UNDERSTAND PRACTICE

Additional Exercises Available Online

Practice Tutorial

For Exercises 18–20, find each angle measure for rhombus ABCD. SEE EXAMPLES 1 AND 2

18. m∠ACD

19. m∠ABC

20. m∠BEA

For Exercises 21–23, find each length for rhombus PQRS. Round to the nearest tenth. SEE EXAMPLES 1 AND 2

21. TR

22. QS

23. PS

For Exercises 24–27, find each length and angle measure for rectangle GHJK. Round to the nearest tenth. SEE EXAMPLES 3 AND 4

24. m∠GHK

25. m∠HLJ

26. GJ

27. HL

For Exercises 28–30, find each length and value for square QRST. Round to the nearest tenth. SEE EXAMPLE 5

28. SV

29. RT

30. perimeter of △RVS

31. If ABCD is a square, what is GC?

13. Construct Arguments Write a proof of Theorem 8-16.

Given: WXYZ is a rhombus.

Prove: ‾ WY and ‾ XZ are perpendicular bisectors of each other.

14. Error Analysis Figure ABCD is a rhombus. What is Malcolm’s error?

✗

Since ABCD is a rhombus,AB ~= CD. Since the diagonalsof a rhombus bisect eachother, AE ~= BE ~= CE ~= DE.So, by SSS, ∆ABE ~= ∆CDE.

C

D

B

AE

15. Mathematical Connections The area of rectangle WXYZ is 115.5 in . 2 . What is the perimeter of △XYZ ? Explain your work.

X

W

Y

P

QZ

4 in.

16. Construct Arguments Write a proof of Theorem 8-17.

Given: ABCD is a rhombus.

Prove: ∠1 ≅ ∠2, ∠3 ≅ ∠4, ∠5 ≅ 6, ∠7 ≅ ∠8

17. Higher Order Thinking A square is cut apart and reassembled into a rectangle as shown. Which figure has a greater perimeter? Explain.

443

31

1

22

4

56

3

78

12

D

CA E

B

A

BC

D

E

(4x − 3)°

(7x − 6)°

2x + 2

2x − 1

4x − 7

S

RP T

Q

K

J

G

H

L

52°

7

10

V

S

T

R

Q

4

D

B

CGFEA 10 16


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APPLY ASSESSMENT PRACTICE

Mixed Review Available Online

Practice Tutorial


32. Model With Mathematics Jordan wants a collapsible puppy pen that gives his puppy at least 35 square feet of area and at least 10 feet of diagonal length. Should Jordan buy the pen shown? Explain.

6 ft

6 ft 6 ft

6 ft

33. Make Sense and Persevere Luis is using different types of wood to make a rectangular inlay top for a chest with the pattern shown.

A

E

D

B

C

40°

35 in. 25 in.

G

F

a. What angle should he cut for ∠CDG? Explain.

b. If he makes the table top correctly, what will the length of the completed top be?

34. Look for Relationships A carpenter is building a support for a stage. What should be the measures of ∠1, ∠2, ∠3 , and ∠4 ? Explain your answers.

8 in.

13 in.

17 in.

8 in.

13 in.

17 in.

1

2

3

4

34°

61°

134°

35. Which statements are true about all rectangles? Select all that apply.

Ⓐ Diagonals bisect each other.

Ⓑ Adjacent sides are perpendicular.

Ⓒ Diagonals are perpendicular.

Ⓓ Consecutive angles are supplementary.

36. SAT/ACT Which expression gives m∠DBC ?

D

CA

B

(3x)°

Ⓐ (180 − 3x ___ 2 )

° Ⓒ ( 180 − 3x ________

2 )

°

Ⓑ (180 − 3x) ° Ⓓ ( 3x ___ 2 − 180)

°

37. Performance Task At a carnival, the goal is to toss a disc into one of three zones to win a prize. Zone 1 is a square, zone 2 is a rhombus, and zone 3 is a rectangle. Some measurements have been provided.

A D H

F J K

LM

GE

Zone 1 Zone 2 Zone 3

CB

EG = 3.7 ftAC = 7 ftFH = 8 ftJL = 11 ftKL = 10.5 ftm∠EFH = 25°

Part A What are the lengths of the sides of each zone?

Part B What are the angle measures of each zone?

Part C What is the area of each zone?


8-5...activity assess i can… use the properties of rhombuses, rectangles, and squares to solve...

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