A

B C

D

E

Activity Assess

Conditions of Special

Parallelograms

8-6

I CAN… identify rhombuses, rectangles, and squares by the characteristics of their diagonals.

EXAMPLE 1 Use Diagonals to Identify Rhombuses

Information about diagonals can help to classify a parallelogram. In parallelogram ABCD, ‾ AC is perpendicular to ‾ BD . What else can you conclude about the parallelogram?

A B

CD

E

The four triangles are congruent by SAS, so ‾ AB ≅ ‾ CB ≅ ‾ CD ≅ ‾ AD .

Since ABCD is a parallelogram with four congruent sides, ABCD is a rhombus.

MODEL & DISCUSS

The sides of the lantern are identical quadrilaterals.

A. Construct Arguments How could you check to see whether a side is a parallelogram? Justify your answer.

B. Does the side appear to be rectangular? How could you check?

C. Do you think that diagonals of a quadrilateral can be used to determine whether the quadrilateral is a rectangle? Explain.

Try It! 1. If ∠JHK and ∠JGK are complementary, what else can you conclude about GHJK? Explain.

STUDY TIPParallelograms have several properties, and some properties may not help you solve a particular problem. Here, the fact that diagonals bisect each other allows the use of SAS.

Any angle at E either forms a linear pair or is a vertical angle with ∠AEB, so all four angles are right angles.

LG J

K

H

The diagonals of a parallelogram bisect each other, so ‾ AE ≅

_ CE and

‾ DE ≅ _

BE .

A B

CD

E

CONCEPTUAL UNDERSTANDING

Which properties of the diagonals of a parallelogram help you to classify a parallelogram?

ESSENTIAL QUESTION

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Activity Assess

STUDY TIPDrawing diagonals in parallelograms can help you see additional information that is useful in solving problems.

EXAMPLE 2 Prove Theorem 8-20

Write a proof of Theorem 8-20.

Given: Parallelogram FGHJ with ∠1 ≅ ∠2 and ∠3 ≅ ∠4

Prove: FGHJ is a rhombus.

Proof:

HF

J

G

431

2

HF

J

G

431

2

HF

J

G

431

2

By ASA, △FHJ ≅ △FHG . By the Alternate By the Converse of Thus,

_ FJ ≅ ‾ FG . Interior Angles the Isosceles Triangle

Theorem, ∠1 ≅ ∠4 , so Theorem, ‾ FG ≅ ‾ HG ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4 . and

_ FJ ≅ ‾ HJ .

Using the Transitive Property of Congruence, ‾ FG ≅ ‾ HG ≅ _

FJ ≅ ‾ HJ . Since FGHJ is a parallelogram with congruent sides, it is a rhombus.

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

PROOF: SEE EXERCISE 9.

If a diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus.

PROOF: SEE EXAMPLE 2.

If...

Then... ‾ JK ≅ ‾ KL ≅ ‾ LM ≅ ‾ MJ

M

KJ

L

If...

Then... ‾ AB ≅ ‾ BC ≅ ‾ CD ≅ ‾ DA

A

B

D

C

THEOREM 8-19 Converse of Theorem 8-16

THEOREM 8-20 Converse of Theorem 8-17

HF

J

G

431

2

PROOF

Try It! 2. Refer to the figure FGHJ in Example 2. Use properties of parallelograms to show that if ∠1 ≅ ∠2 and ∠3 ≅ ∠4 , then the four angles are congruent.

LESSON 8-6 Conditions of Special Parallelograms 399

8 ft

8 ft

10 ft

10 ft

Activity Assess

EXAMPLE 3 Use Diagonals to Identify Rectangles

Ashton measures the diagonals for his deck frame and finds that they are congruent. Will the deck be rectangular?

Since opposite sides are congruent, the supports form a parallelogram. To show that the structure is rectangular, show that the angles are right angles.

C

B

D

A

In a parallelogram, consecutive angles are supplementary. Angles that are congruent and supplementary are right angles. Similarly, ∠DAB and ∠CBA are also right angles.

The frame forms a parallelogram with four right angles, which is a rectangle.

CONSTRUCT ARGUMENTSThere are often multiple ways to prove something. How could you use properties of parallelograms to show the figure is a rhombus without the congruent angles shown?

Opposite sides and the diagonals are congruent, so △ ACD ≅ △BDC by SSS. Therefore, ∠ADC ≅ ∠BCD .

Try It! 3. If the diagonals of any quadrilateral are congruent, is the quadrilateral a rectangle? Justify your answer.

CONTINUED ON THE NEXT PAGE

EXAMPLE 4 Identify Special Parallelograms

Can you conclude whether each parallelogram is a rhombus, a square, or a rectangle? Explain.

A. Parallelogram ABCD

D

CA

B

Diagonal ‾ BD bisects ∠ABC and ∠ADC, so parallelogram ABCD is a rhombus.

THEOREM 8-21 Converse of Theorem 8-18

By SAS, △ABD ≅ △CBD .

∠ADB ≅ ∠CDB by CPCTC.

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

PROOF: SEE EXERCISE 11.

If...

Then... ∠XWZ, ∠WZY, ∠XYZ, and ∠WXY are right angles

X

W

Y

Z

XZ ≅ WY

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Activity Assess

Try It! 4. Is each parallelogram a rhombus, a square, or a rectangle? Explain.

a. b.

EXAMPLE 4 CONTINUED

EXAMPLE 5 Use Properties of Special Parallelograms

Quadrilateral STUV is a rhombus. What are the values of x and y?

If a parallelogram is a rhombus then each diagonal bisects opposite angles. So, ‾ TV bisects ∠ SVU and ∠ STU .

Solve for x. Solve for y.

m∠SVT = m∠UVT m∠STV = m∠UTV

4x + 3 = 5x −10 65 − y = 5y + 5

−x = −13 −6y = −60

x = 13 y = 10

MAKE SENSE AND PERSEVEREConsider the information given in the diagram. How can you determine whether

_ TV bisects

the angles?

Try It! 5. In parallelogram ABCD, AC = 3w − 1 and BD = 2(w + 6) . What must be true for ABCD to be a rectangle?

B. Parallelogram PQRS

S

P R

Q

Since the parallelogram is a rhombus and a rectangle, it is a square.

Diagonals are perpendicular, so PQRS is a rhombus.

Diagonals are congruent, so PQRS is a rectangle.

H

G

J

L

K

V

W

U

TX

U

T

V

S

(5x − 10)°

(65 − y)°

(4x + 3)°

(5y + 5)°

_

TV bisects ∠SVU and ∠STU.

LESSON 8-6 Conditions of Special Parallelograms 401

Activity Assess

EXAMPLE 6 Apply Properties of Special Parallelograms

A group of friends set up a kickball field with bases 60 ft apart. How can they verify that the field is a square?

Opposite sides are congruent, so the field is a parallelogram.

All sides are congruent, so the parallelogram is a rhombus.

60 ft

60 ft60 ft

60 ft

Home Plate

The field is a rhombus. To show that the rhombus is a square, show that it is also a rectangle.

A parallelogram is a rectangle if the diagonals are congruent.

Home Plate

Third Base First Base

Second Base

The group of friends can verify the field is a square if they find that the distances from first base to third base and from second base to home plate are equal.

Try It! 6. Is MNPQ a rhombus? Explain.

APPLICATION

P

N

R

Q

M

58°58°

32°

Measure the distances from first to third base and from home plate to second base.

COMMON ERRORThe order in which the quadrilateral is identified matters. Be sure to first show that the quadrilateral is a parallelogram before applying the theorems to identify the quadrilateral as a rhombus or a rectangle.

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Concept Summary Assess

CONCEPT SUMMARY Conditions of Special Parallelograms

Do You UNDERSTAND?

1. ESSENTIAL QUESTION Which properties of the diagonals of a parallelogram help you to classify a parallelogram?

2. Error Analysis Sage was asked to classify DEFG. What was Sage’s error?

✗

Since DF = EG, DEFG is a rectangle.

Since EG DF, DEFG is also a rhombus.

Therefore, DEFG is a square.

3. Construct Arguments Write a biconditional statement about the diagonals of rectangles. What theorems justify your statement?

4. Use Appropriate Tools Make a concept map showing the relationships among quadrilaterals, parallelograms, trapezoids, isosceles trapezoids, kites, rectangles, squares, and rhombuses.

Do You KNOW HOW?

For Exercises 5–8, is the parallelogram a rhombus, a square, or a rectangle?

5.

55

55

6.

45°45°

45°45°

7. 8.

9. What value of x will make the parallelogram a rhombus?

10. If m∠1 = 36 and m∠2 = 54, is PQRS a rhombus, a square, a rectangle, or none of these? Explain.

P S

Q R

U1 2

36°

x°

A parallelogram is a rectangle if

• diagonals are congruent

A parallelogram is a rhombus if

• diagonals are perpendicular

• a diagonal bisects angles

A parallelogram is a square if

• diagonals are perpendicular and congruent

• a diagonal bisects angles and diagonals are congruent

RHOMBUS SQUARERECTANGLE

E

G

FD

DF = EG

LESSON 8-6 Conditions of Special Parallelograms 403

PRACTICE & PROBLEM SOLVING

UNDERSTAND PRACTICE

Additional Exercises Available Online

Practice Tutorial

For Exercises 17 and 18, determine whether each figure is a rhombus. Explain your answer.

SEE EXAMPLES 1 AND 2

17. 100°

40°40°

18.

19. What is the perimeter of parallelogram WXYZ? SEE EXAMPLE 3

X Y

W Z13

13 13

13

24

For Exercises 20 and 21, determine the name that best describes each figure: parallelogram, rectangle, square, or rhombus. SEE EXAMPLE 4

20. 21.

For Exercises 22–24, give the condition required for each figure to be the specified shape.

SEE EXAMPLES 5 AND 6

22. rectangle

3x − 1 x + 13

23. rhombus

(6x + 6)°

24. rhombus

(3x + 12)° (5x − 24)°

11. Construct Arguments Write a proof for Theorem 8-19 using the following diagram.

CE

D

B

A

12. Error Analysis Becky is asked to classify PQRS. What is her error?

✗PR bisects opposite angles SPQ and QRS, soPQRS must be a rhombus.

13. Construct Arguments Write a proof for Theorem 8-21 using the following diagram.

G

FK

H

J

14. Construct Arguments Write a proof to show that if ABCD is a parallelogram and ∠ABE ≅ ∠BAE , then ABCD is a rectangle.

A

E

B

D C

15. Mathematical Connections If WXYZ is a rhombus with W(−1, 3) and Y(9, 11), what must be an equation of

⟷ XZ in order for WXYZ to be a

rhombus? Explain how you found your answer.

16. Higher Order Thinking The longer diagonal of a rhombus is three times the length of the shorter diagonal. If the shorter diagonal is x, what expression gives the perimeter of the rhombus?

P

S

Q

R

50°50°

48°48°

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PRACTICE & PROBLEM SOLVING

APPLY ASSESSMENT PRACTICE

Mixed Review Available Online

Practice Tutorial

25. Look for Relationships Melissa charges $1.50 per square meter for laying sod. She says she can compute the amount to charge for the pentagonal lawn by evaluating 1.50(1 2 2 + 0.25(1 2 2 )). Do you agree? Explain.

12 m

26. Make Sense and Persevere Jeffery is making a wall design with tape. How much tape does he need to put the design shown on his wall? Explain how you used the information in the diagram to find your answer.

30 in.

40 in.

30 in.

23 in.

=

=

= =

27. Use Structure After knitting a blanket, Monisha washes and stretches it out to the correct size and shape. Opposite sides line up with the edges of a rectangular table. She plans to sew a ribbon around the edge of the blanket. How much ribbon will she need?

58 in.58 in.

42 in.

42 in.

58 in.

28. Are the terms below valid classifications for the figure? Select Yes or No.

Yes No

❑

❑

❑

❑❑

❑

❑

❑

❑ ❑

Square

Rhombus

Parallelogram

Rectangle

Trapezoid

29. SAT/ACT Parallelogram ABCD has diagonals with lengths AC = 7x + 6 and BD = 9x − 2 . For which value of x is ABCD a rectangle?

Ⓐ 2 Ⓑ 4 Ⓒ 7 Ⓓ 9 Ⓔ 34

30. Performance Task Zachary is using the two segments shown as diagonals of quadrilaterals he is making for a decal design for the cover of his smart phone.

5 cm 5 cm

Part A Make a table showing at least four types of different quadrilaterals that Zachary can make using the segments as diagonals. For each type of quadrilateral, draw a diagram showing an example. Label angle measures where the diagonals intersect, and label segment lengths of the diagonals.

Part B Are some types of quadrilaterals not possible using these diagonals? Explain.

Part C Which has the greater area, a square or a rectangle? Explain.

LESSON 8-6 Conditions of Special Parallelograms 405