9.4 Conditions for Rectangle, Rhombus, Square Date: Mixed Practice: Algebra and proofs

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Fill in the blanks to complete each theorem. 1. If one pair of consecutive sides of a parallelogram are congruent, then the

parallelogram is a _____________________.

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

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

4. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a _____________________.

5. If one angle of a parallelogram is a right angle, then the parallelogram is a _____________________.

Use the figure for Problems 6–7. Determine whether each conclusion is valid. If not, tell what additional information is needed to make it valid.

6. Given: AC and BD bisect each other. AC BD#

Conclusion: ABCD is a square.

7. Given: ,AC BD AB BCA #

Conclusion: ABCD is a rhombus.

Complete Problems 8–11 to show that the conclusion is valid. Given: , ,JK ML JM KL# # and .JK KL# �M is a right angle.

Conclusion: JKLM is a square.

8.Because JK ML# and ,JM KL# JKLM is a _____________________.

9.Because JKLM is a parallelogram and �M is a right angle, JKLM is a ____________________.

10.Because JKLM is a parallelogram and ,JK KL# JKLM is a _____________________.

11. Because JKLM is a _____________________ and a _____________________, JKLM is a square.

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Evaluate: Homework and Practice

1. Suppose Anna draws two line segments, _ AB and

_ CD that intersect at point E. She draws them in such a way that

_ AB ≅ _ CD , _ AB ⊥ _ CD , and

_ AB and _ CD bisect each

other. What is the best name to describe ACBD? Explain.

2. Write a two-column proof that if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Given: EFGH is a parallelogram; _ EG ≅ _ HF .

Prove: EFGH is a rectangle.

Statements Reasons

1. 1.

Determine whether each quadrilateral must be a rectangle. Explain.

3.

Given: BD = AC

4.

F

E

G

H

C

D

B

A

Square; because the diagonals of the quadrilateral bisect each other, it is a

parallelogram; because the diagonals are congruent, it is a rectangle and

because the diagonals are perpendicular, it is a rhombus. A figure that is

both a rectangle and a rhombus must be a square.

Given

2. If a quadrilateral is a parallelogram, then its opposite sides are congruent.

3. Reflexive Property of Congruence

4. SSS Triangle Congruence Theorem

5. CPCTC

6. Consecutive angles of a parallelogram are supplementary.

EFGH is a parallelogram; _ EG ≅

_ HF .

6. ∠FEH and ∠GHE are supplementary.

4. △EFH ≅ △HGE

2. _ EF ≅

_ GH

5. ∠FEH ≅ ∠GHE

7. Congruent supplementary angles are right angles.

8. EFGH is a rectangle.

7. m∠FEH = 90°

3. _ EH ≅

_ EH

8. Definition of rectangle

No information about the angles is

known, so it cannot be determined if it

is a rectangle.

No information is known about its sides or

angles, so it may not be a parallelogram. So,

it cannot be determined if it is a rectangle.

Module 9 466 Lesson 4

GE_MNLESE385795_U3M09L4.indd 466 2/26/16 1:32 AMExercise Depth of Knowledge (D.O.K.)COMMON

CORE Mathematical Practices

1–10 2 Skills/Concepts MP.2 Reasoning

11–16 2 Skills/Concepts MP.5 Using Tools

17–18 2 Skills/Concepts MP.4 Modeling

19 2 Skills/Concepts MP.2 Reasoning

20 3 Strategic Thinking MP.2 Reasoning

21 3 Strategic Thinking MP.3 Logic

22 3 Strategic Thinking MP.3 Logic

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreProperties of Rectangles, Rhombuses, and Squares

Exercise 1

Example 1Proving that Congruent Diagonals Is a Condition for Rectangles

Exercises 2–4

Example 2Proving Conditions for Rhombuses

Exercises 5–7

Example 3Applying Conditions for Special Parallelograms

Exercises 8–16

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Some students may not realize how important each word is in a definition or theorem. To explain one of the theorems in this lesson, ask students to focus on exactly what they know about a given parallelogram (or quadrilateral) in order to make a conclusion about how to further classify the parallelogram. Tell them to make sure that the statement they are trying to prove contains no more and no less information than is needed to proceed deductively to the conclusion.

Conditions for Rectangles, Rhombuses, and Squares 466

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Each quadrilateral is a parallelogram. Determine whether each parallelogram is a rhombus or not.

5.

6.

Give one characteristic about each figure that would make the conclusion valid.

7. Conclusion: JKLM is a rhombus.

8. Conclusion: PQRS is a square.

Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.

9. Given: _ EG and

_ FH bisect each other. _ EG ⟘

_ FH

Conclusion: EFGH is a rhombus.

10. _ FH bisects ∠EFG and ∠EHG.

Conclusion: EFGH is a rhombus.

Find the value of x that makes each parallelogram the given type.

11. square 12. rhombus

K

J

L

M

N

Q

P

R

S

T

E

G

H F

14 - x

2x + 5

(13x + 5.5)°

Rhombus; a parallelogram with

perpendicular diagonals is a rhombus.

Rhombus; a parallelogram with a pair of

consecutive sides congruent is a rhombus.

You need to know that JKLM is a

parallelogram.

Possible answer: You need to know that

∠QPS is a right angle.

The conclusion is valid. The conclusion is not valid. You need to

know that EFGH is a parallelogram.

3 = x

x = 6.5

Module 9 467 Lesson 4

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GE_MNLESE385795_U3M09L4 467 6/27/14 5:27 PM

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Before doing the exercises, you may want to review the conditions for rectangles, rhombuses, and squares. In particular, if a parallelogram

has one right angle, it is a rectangle.has congruent diagonals, it is a rectangle.has congruent consecutive sides, it is a rhombus.has perpendicular diagonals, it is a rhombus.is a rectangle and a rhombus, it is a square.

467 Lesson 9 . 4

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oughton Mifflin H

arcourt Publishing Company

In Exercises 13–16, determine which quadrilaterals match the figure: parallelogram, rhombus, rectangle, or square? List all that apply.

13. Given: _ XY ≅

_ ZW , _ XY ‖

_ ZW , _ WY ≅

_ XZ , _ WY ⊥

_ XZ 14. Given: _ XY ≅

_ ZW , _ XW ≅

_ ZY _ WY ≅

_ ZX

15. Given: ∠WXY ≅ ∠YZW, ∠XWZ ≅ ∠ZYX, ∠XWY ≅ ∠YWZ, ∠XYW ≅ ∠ZYW

16. Given: m∠WXY = 130°, m∠XWZ = 50°, m∠WZY = 130°

17. Represent Real-World Problems A framer uses a clamp to hold together pieces of a picture frame. The pieces are cut so that

_ PQ ≅ _ RS and

_ QR ≅ _ SP . The clamp

is adjusted so that PZ, QZ, RZ, and SZ are all equal lengths. Why must the frame be a rectangle?

18. Represent Real-World Problems A city garden club is planting a square garden. They drive pegs into the ground at each corner and tie strings between each pair. The pegs are spaced so that ― WX ≅ ― XY ≅ ― YZ ≅ ― ZW . How can the garden club use the diagonal strings to verify that the garden is a square?

19. A quadrilateral is formed by connecting the midpoints of a rectangle. Which of the following could be the resulting figure? Select all that apply.

parallelogram rectangle

rhombus square

X

W

Y

Z

X

W

Y

Z

X

W

Y

Z

X

W

Y

Z

Q R

P SZ

ge07sec06l05004a ABeckmann

X

W Y

Z

V

parallelogram, rhombus, rectangle, square parallelogram, rectangle

parallelogram, rhombus parallelogram

Since both pairs of opposite sides are congruent, PQRS is a parallelogram. Since PZ, QZ, RZ, and SZ are all equal lengths, PZ + RZ = QZ + SZ. So

_ QS ≅ _ PR . Since the diagonals are congruent, PQRS is a rectangle.

Because both pairs of opposite sides of the quadrilateral garden are congruent, the garden is a parallelogram. All four sides are congruent, so it is a rhombus. The club members can measure the lengths of the diagonals to see if they are equal. Then, the parallelogram is a rectangle. If the garden is a rhombus and a rectangle, then it is a square.

Module 9 468 Lesson 4

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GE_MNLESE385795_U3M09L4.indd 468 3/4/16 5:27 AM

AVOID COMMON ERRORSStudents may be confused about how to use the theorems in this lesson. Explain how some of the theorems in the lesson can be used as alternate definitions. For example, some people define a rectangle as a parallelogram with one right angle. In this case, the remaining properties and the definition as a quadrilateral with four right angles follow.

Conditions for Rectangles, Rhombuses, and Squares 468

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