topic 6 polygons and quadrilaterals

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Topic 6 Polygons and Quadrilaterals

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English/Spanish Vocabulary Audio Online:

English Spanish

equiangular polygon, p. 249 polígono equiángulo

equilateral polygon, p. 249 polígono equilátero

isosceles trapezoid, p. 281 trapecio isósceles

kite, p. 282 cometa

midsegment of a trapezoid, p. 281 segmento medio de un trapecio

parallelogram, p. 255 paralelogramo

rectangle, p. 269 rectángulo

regular polygon, p. 249 polígono regular

rhombus, p. 269 rombo

square, p. 269 cuadrado

trapezoid, p. 281 trapecio

VOCABULARY

6-1 The Polygon Angle-Sum Theorems

6-2 Properties of Parallelograms

6-3 Proving That a Quadrilateral

Is a Parallelogram

6-4 Properties of Rhombuses,

Rectangles, and Squares

6-5 Conditions for Rhombuses,

Rectangles, and Squares

6-6 Trapezoids and Kites

TOPIC OVERVIEW

246 Topic 6 Polygons and Quadrilaterals

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If You Need Help . . .Vocabulary onlineYou’ll find definitions of math terms in both English and Spanish. All of the terms have audio support.

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The Mystery Sides

Have you every looked closely at honeycombs? What shape are they? How do you know? Most often the cells in the honeycombs look like hexagons, but they might also look like circles. Scientists now believe that the bees make circular cells that become hexagonal due to the bees’ body heat and natural physical forces.

What are some strategies you use to identify shapes? Think about this as you watch the 3-Act Math video.

Scan page to see a video for this 3-Act Math Task.

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Technology Lab Exterior Angles of Polygons

teks (5)(A), (1)(E)Use With Lesson 6-1

Use geometry software. Construct a polygon similar to the one at the right. Extend each side as shown. Mark a point on each ray so that you can measure the exterior angles.

Use your figure to explore properties of a polygon.

• Measure each exterior angle.

• Calculate the sum of the measures of the exterior angles.

• Manipulate the polygon. Observe the sum of the measures of the exterior angles of the new polygon.

hsm11gmse_0601a_t06178Exercises 1. Write a conjecture about the sum of the measures of the exterior angles (one at

each vertex) of a convex polygon. Test your conjecture with another polygon.

2. The figures below show a polygon that is decreasing in size until it finally becomes a point. Describe how you could use this to justify your conjecture in Exercise 1.

3. The figure at the right shows a square that has been copied several times. Notice that you can use the square to completely cover, or tile, a plane, without gaps or overlaps.

a. Using geometry software, make several copies of other regular polygons with 3, 5, 6, and 8 sides. Regular polygons have sides of equal length and angles of equal measure.

b. Which of the polygons you made can tile a plane? c. Measure one exterior angle of each polygon (including the square). d. Write a conjecture about the relationship between the measure of an exterior

angle and your ability to tile a plane with the polygon. Test your conjecture with another regular polygon.

hsm11gmse_0601a_t06179

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43

2

1


543 2

1


543 2

1


�

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248 Technology Lab Exterior Angles of Polygons

An equilateral polygon is a polygon with all sides congruent.

An equiangular polygon is a polygon with all angles congruent.

A regular polygon is a polygon that is both equilateral and equiangular.

hsm11gmse_0601_t06299 hsm11gmse_0601_t06300 hsm11gmse_0601_t06301

Key Concept Classifying Polygons Based on Sides and Angles

The sum of the interior angle measures of a polygon depends on the number of sides the polygon has.

ESSENTIAL UNDERSTANDING

TEKS (5)(A) Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools.

TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Additional TEKS (1)(E), (1)(F)

TEKS FOCUS

•Equiangular polygon – An equiangular polygon is a polygon with all angles congruent.

•Equilateral polygon – An equilateral polygon is a polygon with all sides congruent.

•Regular polygon – A regular polygon is a polygon that is both equilateral and equiangular.

•Number sense – the understanding of what numbers mean and how they are related

VOCABULARY

The sum of the measures of the interior angles of an n-gon is (n - 2)180.

The measure of each interior angle of a regular n-gon is (n - 2)180

n .

Theorem 6-1 Polygon Angle-Sum Theorem

Corollary to the Polygon Angle-Sum Theorem

You will prove the Corollary to the Polygon Angle-Sum Theorem in Exercise 16.

For a proof of Theorem 6-1, see the Reference section on page 683.


249PearsonTEXAS.com

The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

For the pentagon, m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360.

Theorem 6-2 Polygon Exterior Angle-Sum Theorem

1

23

4

5

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You will prove Theorem 6-2 in Exercise 9.

Problem 1

Investigating Interior Angles of Polygons

A Choose from among a variety of tools (such as a ruler, a compass, or geometry software) to investigate the sums of the measures of the interior angles of different polygons. Explain your choice.

Geometry software is a good way to identify the measures of the interior angles of polygons. You can quickly make many different polygons and use the software to find the measures of their angles.

B Use geometry software to make several triangles, quadrilaterals, pentagons, and hexagons. Then complete the table.

C Use the data in the table in part B to make a conjecture about the sum of the measures of the interior angles of a polygon.

Notice that the numbers in the table are all multiples of 180. Look at the patterns:

Triangle 1 # 180 = 180 Pentagon 3 # 180 = 540

Quadrilateral 2 # 180 = 360 Hexagon 4 # 180 = 720

Conjecture: If you subtract 2 from the number of sides and multiply by 180, you will get the sum of the measures of the interior angles of any polygon.

TEKS Process Standard (1)(C)

Triangle 1

Triangle 2

Triangle 3

Quadrilateral 1

Quadrilateral 2

Quadrilateral 3

PolygonSum of Interior Angle

Measures

180

180

180

360

360

360

Pentagon 1

Pentagon 2

Pentagon 3

Hexagon 1

Hexagon 2

Hexagon 3

PolygonSum of Interior Angle

Measures

540

540

540

720

720

720

How can recording data in a table help you make a conjecture?Recording data in a table is an organized way to present and analyze information. You can look for patterns in the data and make a conjecture.

250 Lesson 6-1 The Polygon Angle-Sum Theorems

How does the word regular help you answer the question?The word regular tells you that each angle has the same measure.

Problem 3

Problem 2

Finding a Polygon Angle Sum

What is the sum of the interior angle measures of a heptagon?

Sum = (n - 2)180 Polygon Angle-Sum Theorem

= (7 - 2)180 Substitute 7 for n.

= 5 # 180 Simplify.

= 900

The sum of the interior angle measures of a heptagon is 900.

Using the Polygon Angle-Sum Theorem STEM

Biology The common housefly, Musca domestica, has eyes that consist of approximately 4000 facets. Each facet is a regular hexagon. What is the measure of each interior angle in one hexagonal facet?

Measure of an angle = (n - 2)180n Corollary to the Polygon Angle-Sum Theorem

= (6 - 2)1806 Substitute 6 for n.

= 4 # 1806 Simplify.

= 120

The measure of each interior angle in one hexagonal facet is 120.

How many sides does a heptagon have?A heptagon has 7 sides.

251PearsonTEXAS.com

Problem 5

Problem 4

Using the Polygon Angle-Sum Theorem

What is mjY in pentagon TODAY?

Use the Polygon Angle-Sum Theorem for n = 5.

m∠T + m∠O + m∠D + m∠A + m∠Y = (5 - 2)180

110 + 90 + 120 + 150 + m∠Y = 3 # 180 Substitute.

470 + m∠Y = 540 Simplify.

m∠Y = 70 Subtract 470 from each side.

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Y

AD

O110�

150�120�

How does the diagram help you?You know the number of sides and four of the five angle measures.

Investigating Exterior Angles of Polygons

A Choose from a variety of tools (such as a ruler, a protractor, or a graphing calculator) to investigate exterior angles of polygons. Explain your choice.

A protractor is a useful tool for investigating exterior angles of polygons because you use protractors to measure angles.

B Draw an exterior angle at each vertex of three different polygons. Investigate patterns and write a conjecture about the exterior angles.

Step 1 Draw three different polygons. Then draw the exterior angles at each vertex of the polygons as shown.

Step 2 Use the protractor to measure the exterior angles of each polygon. Observe any patterns. Write a conjecture about the exterior angles of polygons.

Notice that for each polygon the sum of the measures of the exterior angles is 360.

Triangle: 135 + 120 + 105 = 360

Quadrilateral: 79 + 63 + 130 + 88 = 360

Pentagon: 90 + 90 + 58 + 58 + 64 = 360

Conjecture: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

105°

120°135°

63°

79°

88° 130°

90°

90°58°

64°

58°

What polygons can you draw to investigate patterns? If you draw a triangle, a quadrilateral, and a pentagon, you can investigate patterns for different numbers of exterior angles in each polygon.


Problem 6

Finding an Exterior Angle Measure

What is mj1 in the regular octagon at the right?

By the Polygon Exterior Angle-Sum Theorem, the sum of the exterior angle measures is 360. Since the octagon is regular, the interior angles are congruent. So their supplements, the exterior angles, are also congruent.

m∠1 = 3608 Divide 360 by 8, the number of sides in an octagon.

= 45 Simplify.

TEKS Process Standard (1)(F)

hsm11gmse_0601_t06303

1

2

3 45

7

6

8What kind of angle is j1?Looking at the diagram, you know that ∠1 is an exterior angle.

PRACTICE and APPLICATION EXERCISES

ONLINE

HO

M E W O RK

For additional support whencompleting your homework, go to PearsonTEXAS.com.

Find the measure of one interior angle in each regular polygon.

1. 2. 3.

4. Sketch an equilateral polygon that is not equiangular.

5. A triangle has two congruent interior angles and an exterior angle that measures 100. Find two possible sets of interior angle measures for the triangle.

Analyze Mathematical Relationships (1)(F) Find the value of each variable.

6. 7. 8.

9. a. A polygon has n sides. An interior angle of the polygon and an adjacent exterior angle form a straight angle. What is the sum of the measures of the n straight angles? Of the n interior angles?

b. Using your answers in part (a), what is the sum of the measures of the n exterior angles? What theorem does this prove?

10. a. Use geometry software or other tool to explore the relationships among the interior angles of quadrilaterals. Draw several quadrilaterals with parallel opposite sides. Measure the interior angles.

b. Make two conjectures about the interior angles of this type of quadrilateral.

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110�87 �

z �

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x �

z �

w � y �(z � 13)�

(z � 10)�

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3x �

2x �

x �4x �

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Scan page for a Virtual Nerd™ tutorial video.

253PearsonTEXAS.com

11. Explain Mathematical Ideas (1)(G) Your friend says she has another way to find the sum of the interior angle measures of a polygon. She picks a point inside the polygon, draws a segment to each vertex, and counts the number of triangles. She multiplies the total by 180, and then subtracts 360 from the product. Does her method work? Explain.

12. The measure of an interior angle of a regular polygon is three times the measure of an exterior angle of the same polygon. What is the name of the polygon?

Apply Mathematics (1)(A) The gift package at the right contains fruit and cheese. The fruit is in a container that has the shape of a regular octagon. The fruit container fits in a square box. A triangular cheese wedge fills each corner of the box.

13. Find the measure of each interior angle of a cheese wedge.

14. Display Mathematical Ideas (1)(G) Show how to rearrange the four pieces of cheese to make a regular polygon. What is the measure of each interior angle of the polygon?

15. a. Select Tools to Solve Problems (1)(C) Choose from a variety of tools (such as a ruler, a compass, or geometry software) to investigate the exterior angles of regular polygons. Explain your choice. Draw three regular polygons, each with a different number of sides. Then draw the exterior angles at each vertex of the polygons.

b. Make two conjectures about the exterior angles of regular polygons.

16. a. In the Corollary to the Polygon Angle-Sum Theorem, explain why the measure of an interior angle of a regular n-gon is given by the formulas 180(n - 2)

n and 180 - 360n .

b. Use the second formula to explain what happens to the measures of the interior angles of regular n-gons as n becomes a large number. Explain also what happens to the polygons.

hsm 11gm se_0601_t06061.a i

TEXAS Test Practice

17. The car at each vertex of a Ferris wheel holds a maximum of five people. The sum of the interior angle measures of the Ferris wheel is 7740. What is the maximum number of people the Ferris wheel can hold?

18. The Public Garden is located between two parallel streets: Maple Street and Oak Street. The garden faces Maple Street and is bordered by rows of shrubs that intersect Oak Street at point B. What is m∠ABC, the angle formed by the shrubs?

19. △ABC ≅ △DEF . If m∠A = 3x + 4, m∠C = 2x, and m∠E = 4x + 5, what is m∠B?


B

A C

Oak Street

Maple Street

Shrubs Shru

bs

PublicGarden

64�

37�


Term Description Diagram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

You can abbreviate parallelogram with the symbol ▱.


In a quadrilateral, opposite sides do not share a vertex and opposite angles do not share a side.


B C

DA

AB and CDare oppositesides.

�A and �Care oppositeangles.

Angles of a polygon that share a side are consecutive angles. In the diagram, ∠A and ∠B are consecutive angles because they share side AB.


A

D C

B�B and �Care alsoconsecutiveangles.

Key Concept Parallelograms and Their Parts

Parallelograms have special properties regarding their sides, angles, and diagonals.


TEKS (6)(E) Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems.

TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

Additional TEKS (1)(G)

TEKS FOCUS

•Consecutive angles – Consecutive angles of a polygon share a common side.

•Opposite angles – Opposite angles of a quadrilateral are two angles that do not share a side.

•Opposite sides – Opposite sides of a quadrilateral are two sides that do not share a vertex.

•Parallelogram – A parallelogram is a quadrilateral with two pairs of parallel sides.

•Analyze – closely examine objects, ideas, or relationships to learn more about their nature

VOCABULARY


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

If . . .ABCD is a ▱

Then . . .AB ≅ CD and BC ≅ DA


TheoremIf a quadrilateral is a parallelogram, then its consecutive angles are supplementary.


Then . . .


B C

DA

m∠A + m∠B = 180 m∠B + m∠C = 180 m∠C + m∠D = 180 m∠D + m∠A = 180


TheoremIf a quadrilateral is a parallelogram, then its opposite angles are congruent.

If . . .ABCD is a ▱.

Then . . .∠A ≅ ∠C and ∠B ≅ ∠D

For a proof of Theorem 6-5, see Problem 2.

TheoremIf a quadrilateral is a parallelogram, then its diagonals bisect each other.


Then . . .AE ≅ CE and BE ≅ DE


TheoremIf three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

If . . .<AB

>}<CD

>}<EF> and AC ≅ CE

Then . . .BD ≅ DF


Theorem 6-3


B C

DA


B C

DA


B C

DA


B C

DA


B C

DA


B C

DA


A E

B C

D

A

C D

FE

B


A

C D

FE

B


Theorem 6-4

Theorem 6-5

Theorem 6-6

Theorem 6-7

256 Lesson 6-2 Properties of Parallelograms

Problem 2

Problem 1

Using Consecutive Angles

Multiple Choice What is mjP in ▱PQRS?

26 116

64 126

m∠P + m∠S = 180 Consecutive angles of a ▱ are supplementary.

m∠P + 64 = 180 Substitute.

m∠P = 116 Subtract 64 from each side.

The correct answer is C.

Using Properties of Parallelograms in a Proof

Given: ▱ABCD

Prove: ∠A ≅ ∠C and ∠B ≅ ∠D

Proof

TEKS Process Standard (1)(G)


B

A D

C


Def. of consecutive ⦞

Supplements of thesame ∠ are ≅.

Supplements of thesame ∠ are ≅.

Consecutive ⦞are supplementary.

Def. of consecutive ⦞ Def. of consecutive ⦞

Given

ABCD is a ▱.

∠B and ∠C aresupplementary.


∠C and ∠D aresupplementary.

∠A and ∠B areconsecutive ⦞.


∠A and ∠B aresupplementary.

∠A ≅ ∠C ∠B ≅ ∠D

∠B and ∠C areconsecutive ⦞.

∠C and ∠D areconsecutive ⦞.

S

P

R

Q

64

What information from the diagram helps you get started?From the diagram, you know m∠PSR and that ∠P and ∠PSR are consecutive angles. So you can write an equation and solve for m∠P.

Why is a flow proof useful here?A flow proof allows you to see how the pairing of two statements leads to a conclusion.

257PearsonTEXAS.com

Problem 4

Problem 3

Using Algebra to Find Lengths

Solve a system of linear equations to find the values of x and y in ▱KLMN . What are KM and LN?

L

K

M

N

Py � 10 y � 2

2x � 8x

hsm11gmse_0602_t06481.aiThe diagonals of a parallelogram bisect each other.

Set up a system of linear equations by substituting the algebraic expressions for each segment length.

KP ≅ M PLP ≅ N P

① y + 10 = 2x − 8 ② x = y + 2

y + 10 = 2(y + 2) − 8 y + 10 = 2y + 4 − 8 y + 10 = 2y − 4

10 = y − 4 14 = y

x = 14 + 2 = 16

KM = 2(KP) LN = 2(LP) = 2(y + 10) = 2(x) = 2(14 + 10) = 2(16) = 48 = 32

Substitute ( y + 2) for x in equation ①. Then solve for y.

Substitute 14 for y in equation ②. Then solve for x.

Use the values of x and y to find KM and LN.

Using Parallel Lines and Transversals

In the figure at the right, <AE

> } <BF

> } <CG

> } <DH

>,

AB = BC = CD = 2, and EF = 2.25. What is EH?

EF = FG = GH Since } lines divide AD into equal parts, they also divide EH into equal parts.

EH = EF + FG + GH Segment Addition Postulate

EH = 2.25 + 2.25 + 2.25 = 6.75 Substitute.



A

EF

GH

B C DWhat information do you need?You know the length of EF . To find EH, you need the lengths of FG and GH.


PRACTICE and APPLICATION EXERCISESON

LINE

HO

M E W O RK


1. What are the values of x and y in the parallelogram?

2. The perimeter of ▱ABCD is 92 cm. AD is 7 cm more than twice AB. Find the lengths of all four sides of ▱ABCD.

In the figure, PQ = QR = RS. Find each length.

3. ZU 4. XZ

5. TU 6. XV

7. YX 8. YV

9. WX 10. WV

11. Justify Mathematical Arguments (1)(G) Complete this two-column proof of Theorem 6-6.

Given: ▱ABCD

Prove: AC and BD bisect each other at E.

Statements Reasons

1) ABCD is a parallelogram. 1) Given

2) AB } DC 2) a. ?

3) ∠1 ≅ ∠4; ∠2 ≅ ∠3 3) b. ?

4) AB ≅ DC 4) c. ?

5) d. ? 5) ASA

6) AE ≅ CE; BE ≅ DE 6) e. ?

7) f. ? 7) Definition of bisector

Find the values of x and y in ▱PQRS.

12. PT = 2x, TR = y + 4, QT = x + 2, TS = y

13. PT = x + 2, TR = y, QT = 2x, TS = y + 3

14. PT = y, TR = x + 3, QT = 2y, TS = 3x - 1

Use the diagram at the right for each proof.

15. Given: ▱RSTW and ▱XYTZ

Prove: ∠R ≅ ∠X

16. Given: ▱RSTW and ▱XYTZ

Prove: XY } RS

Find the measures of the numbered angles for each parallelogram.

17. 18. 19.


3y � 3x �

y �


W S U

ZY R

Q

PV

X

T

2.25

3

Proof


1

2

3

4

AE

B C

D


Q

P

T

S

R

Proof


S

R W

X

Y

Z

T

Proof


12

3

110�38�


1 2 3

28�81�


12

3

85�

48�


259PearsonTEXAS.com

20. Apply Mathematics (1)(A) A pantograph is an expandable device, shown at the right. Pantographs are used in the television industry in positioning lighting and other equipment. In the photo, points D, E, F, and G are the vertices of a parallelogram. ▱DEFG is one of many parallelograms that change shape as the pantograph extends and retracts.

a. If DE = 2.5 ft, what is FG? b. If m∠E = 129, what is m∠G?

c. What happens to m∠D as m∠E increases or decreases? Explain.

21. Prove Theorem 6-4.

Given: ▱ABCD

Prove: ∠A is supplementary to ∠B. ∠A is supplementary to ∠D.

22. Explain Mathematical Ideas (1)(G) Is there an SSSS congruence theorem for parallelograms? Explain.

23. Prove Theorem 6-7. Use the diagram at the right.

Given: <AB

> } <CD

> } <EF>, AC ≅ CE

Prove: BD ≅ DF

24. Explain Mathematical Ideas (1)(G) Explain how to separate a blank card into three strips that are the same height by using lined paper, a straightedge, and Theorem 6-7.

Proof


B

A D

C

E

D

G

F

E

D

G

F

Proof

E

C

54

16

3

2

A B

GD

FH


TEXAS Test Practice

25. PQRS is a parallelogram with m∠Q = 4x and m∠R = x + 10. Which statement explains why you can use the equation 4x + (x + 10) = 180 to solve for x?

A. The measures of the interior angles of a quadrilateral have a sum of 360.

B. Opposite sides of a parallelogram are congruent.

C. Opposite angles of a parallelogram are congruent.

D. Consecutive angles of a parallelogram are supplementary.

26. In the figure of DEFG at the right, DE } GF . Which statement must be true?

F. m∠D + m∠E = 180 H. DE ≅ GF

G. m∠D + m∠G = 180 J. DG ≅ EF

27. An obtuse triangle has side lengths of 5 cm, 9 cm, and 12 cm. What is the length of the side opposite the obtuse angle?

A. 5 cm B. 9 cm C. 12 cm D. not enough information

28. Find the measure of one exterior angle of a regular hexagon. Explain your method.

hsm11gmse_0602_t12794

P Q

S R


D E

FG


You can decide whether a quadrilateral is a parallelogram if its sides, angles, and diagonals have certain properties.





TEKS FOCUS


VOCABULARY

Theorem If . . . Then . . .

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

AB ≅ CD BC ≅ DA

ABCD is a ▱


TheoremIf an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

If . . . m∠A + m∠B = 180 m∠A + m∠D = 180

Then . . .ABCD is a ▱


Theorem 6-8


B C

DA


B C

DA


B C

DA


B C

DA

TheoremIf both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

If . . .∠A ≅ ∠C ∠B ≅ ∠D

Then . . .ABCD is a ▱


B C

DA


B C

DA


Theorem 6-10

Theorem 6-9

6-3 Proving That a Quadrilateral Is a Parallelogram

261PearsonTEXAS.com


If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

AE ≅ CE BE ≅ DE

ABCD is a ▱

Theorem 6-11


A E

B C

D


B C

DA


If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

BC ≅ DA BC } DA

ABCD is a ▱

Theorem 6-12


B

A D

C


B C

DA

Method Source Diagram

Prove that both pairs of opposite sides are parallel. Definition of parallelogram

Prove that both pairs of opposite sides are congruent. Theorem 6-8

Prove that an angle is supplementary to both of its consecutive angles.

Theorem 6-9

hsm11gmse_0603_t12047

75�

75�

105�

Prove that both pairs of opposite angles are congruent.

Theorem 6-10

Prove that the diagonals bisect each other. Theorem 6-11

Prove that one pair of opposite sides is congruent and parallel.

Theorem 6-12

Concept Summary Proving That a Quadrilateral Is a Parallelogram








262 Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram

Problem 2

Problem 1

Proving Theorem 6-8

Given: AB ≅ CD, BC ≅ DA

Prove: ABCD is a parallelogram.

Statements Reasons

1) Draw BD. 1) Construction

2) AB ≅ CD and BC ≅ DA 2) Given

3) BD ≅ BD 3) Reflexive Property of Congruence

4) △ABD ≅ △CDB 4) SSS

5) ∠ADB ≅ ∠CBD and ∠CDB ≅ ∠ABD

5) Corresponding parts of congruent triangles are congruent.

6) AB } DC and BC } AD 6) Converse of the Alternate Interior Angles Theorem

7) ABCD is a parallelogram. 7) Definition of parallelogram

Proof


B C

DA

Proving Theorem 6-10

Given: ∠A ≅ ∠C, ∠B ≅ ∠D


Statements Reasons

1) ∠A ≅ ∠C, ∠B ≅ ∠D 1) Given

2) x + y + x + y = 360 2) The sum of the measures of the angles of a quadrilateral is 360.

3) 2(x + y) = 360 3) Distributive Property

4) x + y = 180 4) Division Property of Equality

5) ∠A and ∠B are supplementary. ∠A and ∠D are supplementary.

5) Definition of supplementary angles

6) AD } BC, AB } DC 6) Converse of the Same-Side Interior Angles Postulate

7) ABCD is a parallelogram. 7) Definition of parallelogram

Proof



A

B C

D

x � y �

Why do you start by drawing BD?In many proofs about parallelograms, it is convenient to have a pair of triangles that you can show to be congruent. Draw a diagonal to form two triangles.

How do you get started with the proof?Since the goal is to show that opposite sides are parallel, you can label the angle measures as in the diagram and show that same-side interior angles are supplementary.

263PearsonTEXAS.com

Problem 3


Given: AC and BD bisect each other at E.


Proof


B

A D

C

E

Vertical ⦞ are ≅. Def. of segment bisector

Given

∠AEB ≅ ∠CED

△AEB ≅ △CED

Vertical ⦞ are ≅.

∠BEC ≅ ∠DEA

SAS

∠BAE ≅ ∠DCE

Corresp. parts of ≅ are ≅. Corresp. parts of ≅ are ≅.

∠ECB ≅ ∠EAD

△BEC ≅ △DEA

SAS

AC and BD bisect each other at E.

If alternate interior ⦞ ≅,then lines are �.

AB � CD

If alternate interior ⦞ ≅,then lines are �.

BC � AD

AE ≅ CEBE ≅ DE

Def. of parallelogram

ABCD is a parallelogram.

Finding Values for Parallelograms

A For what value of y must PQRS be a parallelogram?

Step 1 Find x.

3x - 5 = 2x + 1 If opp. sides are ≅, then the quad. is a ▱.

x - 5 = 1 Subtract 2x from each side.

x = 6 Add 5 to each side.

Step 2 Find y.

y = x + 2 If opp. sides are ≅, then the quad. is a ▱.

= 6 + 2 Substitute 6 for x.

= 8 Simplify.

For PQRS to be a parallelogram, the value of y must be 8.


QP

S R

yx � 2

3x � 5

2x � 1

8

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

. . . . . . .

Problem 4

continued on next page ▶

Which theorem should you use?The diagram gives you information about sides. Use Theorem 6-8 because it uses sides to conclude that a quadrilateral is a parallelogram.

How can you get started?Notice that in the diagram there are several pairs of triangles. Use the given information to prove pairs of triangles congruent. Then use their corresponding parts to show that ABCD is a parallelogram.


Problem 5

continuedProblem 4

B For what values of w and z must ABCD be a parallelogram?

Step 1 Find w.

5w - 30 = 3w + 10 If opp. angles are ≅, then the quad. is a ▱.

2w - 30 = 10 Subtract 3w from each side.

2w = 40 Add 30 to each side.

w = 20 Divide each side by 2.

Step 2 Find z.

8z - 10 = 7z + 5 If opp. angles are ≅, then the quad. is a ▱.

z - 10 = 5 Subtract 7z from each side.

z = 15 Add 10 to each side.

For ABCD to be a parallelogram, the value of w must be 20 and the value of z must be 15.

(7z 1 5)8

(8z 2 10)8

(5w 2 30)8

(3w 1 10)8

A B

D C

Deciding Whether a Quadrilateral Is a Parallelogram

Can you prove that the quadrilateral is a parallelogram based on the given information? If so, write a paragraph proof. If not, explain.


A Given: AB = 5, CD = 5, m∠A = 50, m∠D = 130



A

D

B

C

50�

130�

5

5

Yes. Proof: Because it is given that m∠A = 50 and m∠D = 130, same-side interior angles A and D are supplementary. So AB } CD. It is given that AB = 5 and CD = 5, so AB ≅ CD. Therefore, ABCD is a parallelogram by Theorem 6-12.

B Given: HI ≅ HK, JI ≅ JK

Prove: HIJK is a parallelogram.


H

K J

I

No. By Theorem 6-8, you need to show that both pairs of opposite sides, not consecutive sides, are congruent.


How do you decide if you have enough information?If you can satisfy every condition of a theorem about parallelograms, then you have enough information.

265PearsonTEXAS.com

Problem 6

continuedProblem 5

C Given: m∠N = m∠Q = 39, m∠P = 141

Prove: MNPQ is a parallelogram.

Q

M N

P398 1418

398

Yes. Proof: It is given that m∠N = m∠Q = 39 and m∠P = 141. Since the sum of the angle measures of a quadrilateral is 360, m∠M = 141. Since m∠N = m∠Q and m∠P = m∠M, ∠N ≅ ∠Q and ∠M ≅ ∠P. Therefore, MNPQ is a parallelogram by Theorem 6-10.

D Given: GE = 24, GH = 12, DF = 32, HF = 16

Prove: DEFG is a parallelogram.

D

G

H

F

E

12 16

Yes. Proof: It is given that GE = 24, GH = 12, DF = 32, and HF = 16. By the Segment Addition Postulate, HE = 12 and DH = 16, so the diagonals of the quadrilateral bisect each other. DEFG is a parallelogram by Theorem 6-11.

26 ft26 ft

26 ft26 ft

RQ

P S

RQ

P S

6 ft6 ft6 ft6 ft

Identifying Parallelograms

Vehicle Lifts A truck sits on the platform of a vehicle lift. Two moving arms raise the platform until a mechanic can fit underneath. Why will the truck always remain parallel to the ground as it is lifted? Explain.

The angles of PQRS change as platform QR rises, but its side lengths remain the same. Both pairs of opposite sides are congruent, so PQRS is a parallelogram by Theorem 6-8. By the definition of a parallelogram, PS } QR. Since the base of the lift PS lies along the ground, platform QR, and therefore the truck, will always be parallel to the ground.

As the arms of the lift move, what changes and what stays the same?The angles the arms form with the ground and the platform change, but the lengths of the arms and the platform stay the same.



LINE

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1. Given: AB ≅ CD, DE ≅ FC, EA ≅ BF 2. Given: ∠M ≅ ∠P, ∠MNQ ≅ ∠PQN,

Prove: ABCD is a parallelogram. ∠MQN ≅ ∠PNQ

Prove: MNPQ is a parallelogram.

NM

Q P

3. Given: M is the midpoint of HK and JL. 4. Given: ∠A and ∠C are right angles,

Prove: HJKL is a parallelogram. AD ≅ CB


B

C

D

A

Analyze Mathematical Relationships (1)(F) For what values of x and y must ABCD be a parallelogram?

5.


B

A D

C3x �

3y �

(y � 78)�

(4x � 21)�

6. 7.

8. 9.


3x � 6 y � 4

A B

CD

2y � 2

3y � 9

10.

11. Display Mathematical Ideas (1)(G) Sketch two noncongruent parallelograms ABCD and EFGH so that AB ≅ EF and BC ≅ FG.

Can you prove that the quadrilateral is a parallelogram based on the given information? Explain.

12. 13. 14.

Proof Proof

A B

CD

FE

Proof Proof

L

H JM

K


BA

D C

y � 1

2x

42y � 7


BA

D C5x � 8

2x � 7


D

A B

C

(3x � 10)�

5y�

(8x � 5)�


(4x � 33)�

A

B C

D

(2x � 15)�

hsm11gmse_0603_t06154.ai hsm11gmse_0603_t06155.ai hsm11gmse_0603_t06157.ai


267PearsonTEXAS.com

TEXAS Test Practice

18. Which piece of additional information would allow you to prove that PQRS is a parallelogram?

A. PQ ≅ RS C. ∠PTQ ≅ ∠RTS

B. QR ≅ SP D. ∠QPR ≅ ∠SRP

19. In quadrilateral ABCD, m∠A = 3x + 2, m∠B = x - 22, and m∠C = 2x + 52. Which value of x allows you to conclude that ABCD is a parallelogram?

F. 50 H. 28

G. 34 J. -12

20. Quadrilateral JKLM is a parallelogram. Which of the following does NOT guarantee that JNPM is a parallelogram?

A. N is the midpoint of JK and P is the midpoint of ML.

B. JM ≅ NP

C. JM } NP

D. ∠JMP ≅ ∠NPL

21. Write a proof using the diagram.

Given: △NRJ ≅ △CPT, JN } CT

Prove: JNTC is a parallelogram.

QP

S R

T

N

P

K

L

J

M


N

JR

P

C

T

15. Apply Mathematics (1)(A) Quadrilaterals are formed on the side of this fishing tackle box by the adjustable shelves and connecting pieces. Explain why the shelves are always parallel to each other no matter what their position is.

16. Justify Mathematical 17. Prove Theorem 6-9. Arguments (1)(G) Prove Given: ∠A is supplementary to ∠B.

Theorem 6-12. ∠A is supplementary to ∠D. Given: BC } DA, BC ≅ DA Prove: ABCD is a parallelogram. Prove: ABCD is a parallelogram.

A B

CD

Proof Proof


A

B C

D


B

A D

C



TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Additional TEKS (1)(F), (6)(E)

TEKS FOCUS

•Rectangle – A rectangle is a parallelogram with four right angles.

•Rhombus – A rhombus is a parallelogram with four congruent sides.

•Square – A square is a parallelogram with four congruent sides and four right angles.

•Number sense – the understanding of what numbers mean and how they are related

VOCABULARY

The parallelograms in the Take Note box below have basic properties about their sides and angles that help identify them. The diagonals of these parallelograms also have certain properties.


TheoremIf a parallelogram is a rhombus, then its diagonals are perpendicular.

If . . .ABCD is a rhombus

Then . . .AC # BD

Theorem 6-13

hsm11gmse_0604_t06022

A

B C

D

hsm11gmse_0604_t06023

A

B C

D

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.

Key Concept Special Parallelograms

hsm11gmse_0604_t06018 hsm11gmse_0604_t06019

hsm11gmse_0604_t06020

For a proof of Theorem 6-13, see Lesson 7-3.

6-4 Properties of Rhombuses, Rectangles, and Squares

269PearsonTEXAS.com

Problem 1

Classifying Special Parallelograms

Is ▱ABCD a rhombus, a rectangle, or a square? Explain.

▱ABCD is a rectangle. Opposite angles of a parallelogram are congruent, so m∠D is 90. By the Same-Side Interior Angles Theorem, m∠A = 90 and m∠C = 90. Since ▱ABCD has four right angles, it is a rectangle. You cannot conclude that ABCD is a square because you do not know its side lengths.

A B

CD

E

F

G

A B

CD

E

F

G

HH

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

If . . .ABCD is a rhombus

Then . . .

∠1 ≅ ∠2 ∠3 ≅ ∠4 ∠5 ≅ ∠6 ∠7 ≅ ∠8


Theorem 6-14

hsm11gmse_0604_t06022

A

B C

D

hsm11gmse_0604_t06024

A

B C

D

1

78

34

65

2

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

If . . .ABCD is a rectangle

Then . . .

AC ≅ BD


Theorem 6-15

hsm11gmse_0604_t06028

A

C

D

B

hsm11gmse_0604_t06031

A

B C

D

How do you decide whether ABCD is a rhombus, a rectangle, or a square?Use the definitions of rhombus, rectangle, and square along with the markings on the figure.

270 Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares

Problem 3

How are the numbered angles formed?The angles are formed by diagonals. Use what you know about the diagonals of a rhombus to find the angle measures.

Problem 2

Investigating Diagonals of Quadrilaterals

A Choose from a variety of tools (such as a protractor, a ruler, a compass, or a geoboard) to investigate patterns in the diagonals of quadrilaterals. Explain your choice.

A manipulative such as a geoboard makes it easy to make different types of quadrilaterals and their diagonals.

B Make several parallelograms, rectangles, and rhombuses. Then make a conjecture about the diagonals of each type of quadrilateral.

TEKS Process Standards (1)(C)

How can you measure distances on a geoboard?You can use the grid of pegs to indicate horizontal and vertical units.

Finding Angle Measures

What are the measures of the numbered angles in rhombus ABCD?

m∠1 = 90 The diagonals of a rhombus are #.

m∠2 = 58 Alternate Interior Angles Theorem

m∠3 = 58 Each diagonal of a rhombus bisects a

pair of opposite angles.

m∠1 + m∠3 + m∠4 = 180 Triangle Angle-Sum Theorem

90 + 58 + m∠4 = 180 Substitute.

148 + m∠4 = 180 Simplify.

m∠4 = 32 Subtract 148 from each side.


hsm11gmse_0604_t06026

A

B C

D

13

58�

24

Parallelogram

Conjecture: The diagonals of a parallelogram bisect each other.

Rectangle

Conjecture: The diagonals of a rectangle are congruent.

Rhombus

Conjecture: The diagonals of a rhombus are perpendicular.

271PearsonTEXAS.com

Finding Diagonal Length

Multiple Choice In rectangle RSBF, SF = 2x + 15 and RB = 5x − 12. What is the length of a diagonal?

1 9 18 33

hsm11gmse_0604_t06032

S B

FR

Problem 4

How can you find the length of a diagonal?Since RSBF is a rectangle and its diagonals are congruent, use the expressions to write an equation.


ONLINE

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Find the measures of the numbered angles in each rhombus.

1. 2. 3.

LMNP is a rectangle. Find the value of x and the length of each diagonal.

4. LN = x and MP = 2x - 4 5. LN = 5x - 8 and MP = 2x + 1

6. LN = 3x + 1 and MP = 8x - 4 7. LN = 9x - 14 and MP = 7x + 4

8. LN = 7x - 2 and MP = 4x + 3 9. LN = 3x + 5 and MP = 9x - 10


Given: ABCD is a rhombus.

Prove: AC bisects ∠BAD and ∠BCD.

243

135�

hsm 11gm se_0604_t05920.a i

60�1

23

hsm 11gm se_0604_t05921.a i

35�21

3

hsm 11gm se_0604_t05935.a i

ProofA

C

D

B

3 4

2 1


SF = RB

2x + 15 = 5x − 12 15 = 3x − 12 27 = 3x 9 = x RB = 5x − 12

= 5(9) − 12 = 33

The correct answer is D.

You know that the diagonals of a rectangle are congruent, so their lengths are equal.

Set the algebraic expressions for SF and RB equal to each other and find the value of x.

Substitute 9 for x in the expression for RB.



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

11. 12.

13. Justify Mathematical Arguments (1)(G) Complete the flow proof of Theorem 6-15.

Given: ABCD is a rectangle.

Prove: AC ≅ BD

14. Connect Mathematical Ideas (1)(F) Summarize the properties of squares that follow from a square being (a) a parallelogram, (b) a rhombus, and (c) a rectangle.

15. Analyze Mathematical Relationships (1)(F) Find the angle measures and the side lengths of the rhombus at the right.

16. Create Representations to Communicate Mathematical Ideas (1)(E) On graph paper, draw a parallelogram that is neither a rectangle nor a rhombus.

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

17. AC = 2(x - 3) and BD = x + 5 18. AC = 2(5a + 1) and BD = 2(a + 1)

19. AC =3y5 and BD = 3y - 4 20. AC = 3c

9 and BD = 4 - c

A

B C

D

hsm 11gm se_0604_t06239.a i

Proof

ABCD is a ▱.

Opposite sides of a ▱ are ≅.

∠ABC ≅ ∠DCB∠ABC and ∠DCBare right ⦞.

ABCD isa rectangle. SAS

AC ≅ BDBC ≅ BC


b.

c.

d.g.

h.a.

e.

f.

K J

GH

b � 3 (2x � 6)�

r � 1x�

2r � 4

4b � 6r


273PearsonTEXAS.com

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

21. rhombus 22. square

23. Justify Mathematical Arguments (1)(G) Write a proof.

Given: Rectangle PLAN

Prove: △LTP ≅ △NTA

24. a. Select Tools to Solve Problems (1)(C) To investigate the diagonals and the interior angles of rhombuses, choose from the following tools: ruler, paper folding, or graphing calculator. Explain your choice.

b. Make several rhombuses with their diagonals. Observe any patterns. Make a conjecture about the diagonals and the interior angles of rhombuses.

Find the value of x in the rhombus.

25. 26.

4x � 3

3y5x

15

hsm 11gm se_0604_t05942.a i

3y � 9

2x � 7

y � 1 2y � 5

hsm 11gm se_0604_t05943.a i

ProofL

T

AN

P


(7x 2 � 10)�(6x 2 � 3x)�

hsm 11gm se_0604_t05944.a i

(2x 2 � 25x)�

(3x 2 � 60)�

hsm 11gm se_0604_t05945.a iTEXAS Test Practice

27. A part of a design for a quilting pattern consists of a regular pentagon and five isosceles triangles, as shown. What is m∠1?

A. 18 C. 72

B. 36 D. 108

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

F. Opposite sides are parallel. H. Adjacent sides are perpendicular.

G. It is a parallelogram. J. All sides are congruent.

29. Which term best describes AD in △ABC?

A. altitude C. median

B. angle bisector D. perpendicular bisector

30. Write the first step of an indirect proof that △PQR is not a right triangle.

hsm11gmse_0604_t12846

1

DB

A

C

hsm 11gm se_0604_t05947.a i


TheoremIf a quadrilateral is a parallelogram with perpendicular diagonals, then the quadrilateral is a rhombus.

If . . .ABCD is a ▱ and AC # BD

Then . . .ABCD is a rhombus


Theorem 6-16

A D

CB


A

B C

D


TheoremIf a quadrilateral is a parallelogram with a diagonal that bisects a pair of opposite angles, then the quadrilateral is a rhombus.

If . . .ABCD is a ▱, ∠1 ≅ ∠2, and ∠3 ≅ ∠4

Then . . .ABCD is a rhombus


Theorem 6-17

A

B C

D

43

12


A

B C

D


You can determine whether a parallelogram is a rhombus or a rectangle based on the properties of its diagonals.





TEKS FOCUS


VOCABULARY

6-5 Conditions for Rhombuses, Rectangles, and Squares

275PearsonTEXAS.com

TheoremIf a quadrilateral is a parallelogram with perpendicular, congruent diagonals, then the quadrilateral is a square.

If . . .ABCD is a ▱, AC # BD, and AC ≅ BD

Then . . .ABCD is a square

A D

B C


Theorem 6-19

A D

E

B C

Problem 1


Given: ABCD is a parallelogram, AC # BD

Prove: ABCD is a rhombus.

Since ABCD is a parallelogram, AC and BD bisect each other, so BE ≅ DE. Since AC # BD, ∠AED and ∠AEB are congruent right angles. By the Reflexive Property of Congruence, AE ≅ AE. So △AEB ≅ △AED by SAS. Corresponding parts of congruent triangles are congruent, so AB ≅ AD. Since opposite sides of a parallelogram are congruent, AB ≅ DC ≅ BC ≅ AD. By definition, ABCD is a rhombus.

Proof


A

B C

D

E


TheoremIf a quadrilateral is a parallelogram with congruent diagonals, then the quadrilateral is a rectangle.

If . . .ABCD is a ▱, and AC ≅ BD

Then . . .ABCD is a rectangle


Theorem 6-18

A

B C

D

hsm11gmse_0605_t06037.ai hsm11gmse_0604_t06028

A

C

D

B

How can knowing that the quadrilateral is a parallelogram help you prove the theorem? You can use any of the properties of parallelograms to help you.

276 Lesson 6-5 Conditions for Rhombuses, Rectangles, and Squares

Problem 3

Problem 2


Write a two-column proof to prove Theorem 6-19.

Given: ABCD is a parallelogram, AC # DB, and AC ≅ DB

Prove: ABCD is a square.

Statements Reasons

1) ABCD is a parallelogram, AC # DB, and AC ≅ DB

1) Given

2) ABCD is a rectangle. 2) Theorem 6-18

3) ∠DAB, ∠ABC, ∠BCD, and ∠CDA are right angles. 3) Def. of a rectangle

4) ABCD is a rhombus. 4) Theorem 6-16

5) AB ≅ BC ≅ CD ≅ DA 5) Def. of a rhombus

6) ABCD is a square. 6) Def. of a square

Proof A D

E

B C

Identifying Rhombuses, Rectangles, and Squares

Can you conclude that quadrilateral ABCD is a rhombus, a rectangle, or a square? If so, write a paragraph proof. If not, explain.

A Given: Quadrilateral ABCD with AE ≅ BE ≅ CE ≅ DE

Prove: ABCD is a rhombus, a rectangle, or a square.

E

DA

B C

Yes. Proof: It is given that AE ≅ BE ≅ CE ≅ DE in quadrilateral ABCD. By the definition of segment bisector, AC and DB bisect each other. By Theorem 6-11, ABCD is a parallelogram. By the definition of congruent segments, AE = BE = CE = DE. By the Segment Addition Postulate, AC = DB. So AC ≅ DB by the definition of congruent segments. Therefore, by Theorem 6-18, ABCD is a rectangle.

B Given: Quadrilateral ABCD with AE ≅ CE, BE ≅ DE

Prove: ABCD is a rhombus, a rectangle, or a square.

DA

E

B C

No. The diagonals bisect each other, so by Theorem 6-11, quadrilateral ABCD is a parallelogram. The diagonals are not perpendicular, so ABCD is not a rhombus or a square. The diagonals are not congruent, so ABCD is not a rectangle or a square.


How can knowing the figure is a rectangle help you prove it is a square?A rectangle has four 90° angles. If you know the figure is a rectangle, you only need to show all sides are congruent to prove it is a square.

How do you get started? Use the properties of rhombuses, rectangles, and squares and the theorems you learned to help you determine whether each figure is a rhombus, a rectangle, or a square.

277PearsonTEXAS.com

Problem 5

Problem 4

Using Properties of Special Parallelograms

Algebra For what value of x is ▱ABCD a rhombus?


A

B C

D

(4x � 8)�

(6x � 2)�

For ▱ABCD to be a rhombus, its diagonals must bisect a pair of opposite angles.

Set the expressions for m∠ABD and m∠CBD equal to each other.

Solve for x.

m∠ABD = m∠CBD

6x - 2 = 4x + 8

2x - 2 = 8 2x = 10 x = 5

Using Properties of ParallelogramsCommunity Service Builders use properties of diagonals to “square up” rectangular shapes like building frames and playing-field boundaries. Suppose you are on the volunteer building team at the right. You are helping to lay out a rectangular patio for a youth center.

A How can you use properties of diagonals to locate the four corners?

You can use two theorems.

• Theorem 6-11: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

• Theorem 6-18: If a quadrilateral is a parallelogram with congruent diagonals, then the quadrilateral is a rectangle.

Step 1 Cut two pieces of rope that will be the diagonals of the foundation rectangle. Cut them the same length because of Theorem 6-18.

Step 2 Join the two pieces of rope at their midpoints because of Theorem 6-11.

Step 3 Pull the ropes straight and taut. The ends of the ropes will be the corners of a rectangle.

B Can you adapt this method slightly to stake off a square play area? Explain.

Yes, you can if you make the diagonals perpendicular. The result will be a rectangle and a rhombus, so the play area will be square.

If a quadrilateral is both a rectangle and a rhombus, why is it a square? If a quadrilateral is a rectangle, then its diagonals are congruent bisectors. If it is a rhombus, then its diagonals are perpendicular bisectors. So, by Theorem 6-19, the quadrilateral is a square.



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For what value of x is the figure the given special parallelogram?

1. rhombus 2. rectangle 3. rectangle

4. rectangle 5. rhombus 6. rectangle

7. Analyze Mathematical Relationships (1)(F) Decide whether the given information is sufficient to show the quadrilateral is a rectangle. Explain.

a. AE ≅ CE and DE ≅ BE

b. AD ≅ BC, AB ≅ DC, and m∠DAB = 90

c. AB } CD, AD } BC, and AC ≅ DB

d. AE ≅ CE ≅ DE ≅ BE

8. Apply Mathematics (1)(A) You can use a simple device called a turnbuckle to “square up” structures that are parallelograms. For the gate pictured at the right, you tighten or loosen the turnbuckle on the diagonal cable so that the rectangular frame will keep the shape of a parallelogram when it sags. What are two ways you can make sure that the turnbuckle works? Explain.

9. Explain Mathematical Ideas (1)(G) Suppose the diagonals of a parallelogram are both perpendicular and congruent. What type of special quadrilateral is it? Explain your reasoning.

Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain.

10. 11. 12.


(6x � 9)�

(2x � 39)�


8x � 3 4x � 7

4 4

L O

NM

LN � 4x � 7MO � 2x � 13


(5x � 2)�

3x�

hsm11gmse_0605_t05967.ai hsm11gmse_0605_t05968.ai

(8x � 7)�(3x � 6)�

hsm11gmse_0605_t05969

(4x � 12)�

(3x � 4)�

D

CE

B

A

STEM

hsm11gmse_0605_t05961.aihsm11gmse_0605_t05962.ai hsm11gmse_0605_t05963.ai


279PearsonTEXAS.com

Create Representations to Communicate Mathematical Ideas (1)(E) Given two segments with lengths a and b (a ≠ b), what special parallelograms meet the given conditions? Show each sketch.

13. Both diagonals have length a. 14. The two diagonals have lengths a and b.

15. One diagonal has length a, and one side of the quadrilateral has length b.


Given: ABCD is a parallelogram.

AC bisects ∠BAD and ∠BCD.

Prove: ABCD is a rhombus.


Given: ▱ABCD, AC ≅ BD

Prove: ABCD is a rectangle.

Explain Mathematical Ideas (1)(G) Explain how to construct each figure given its diagonals.

18. parallelogram 19. rectangle 20. rhombus

Determine whether the quadrilateral can be a parallelogram. Explain.

21. The diagonals are congruent, but the quadrilateral has no right angles.

22. Each diagonal is 3 cm long, and two opposite sides are 2 cm long.

23. Two opposite angles are right angles, but the quadrilateral is not a rectangle.

24. Justify Mathematical Arguments (1)(G) In Theorem 6-17, replace “a pair of opposite angles” with “one angle.” Write a paragraph that proves this new statement to be true, or give a counterexample to prove it to be false.

Proof

hsm11gmse_0605_t05970

43

2 1

A

B C

D

Proof

hsm11gmse_0605_t05971

A

B

C

D

Proof

TEXAS Test Practice

25. Each diagonal of a quadrilateral bisects a pair of opposite angles of the quadrilateral. What is the most precise name for the quadrilateral?

A. parallelogram B. rhombus C. rectangle D. not enough information

26. Given a triangle with side lengths 7 and 11, which value could NOT be the length of the third side of the triangle?

F. 13 G. 7 H. 5 J. 2

27. What is the sum of the measures of the exterior angles, one at each vertex, in a pentagon?

A. 180 B. 360 C. 540 D. 108

28. The midpoint of PQ is (-1, 4). One endpoint is P(-7, 10). What are the coordinates of endpoint Q? Explain your work.




Additional TEKS (1)(C)

TEKS FOCUS

•Base angles of a trapezoid – The base angles of a trapezoid are the two angles that share a base of the trapezoid.

•Bases of a trapezoid – The bases of a trapezoid are the parallel sides of the trapezoid.

•Isosceles trapezoid – An isosceles trapezoid is a trapezoid with legs that are congruent.

•Kite – A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.

•Legs of a trapezoid – The legs of a trapezoid are the nonparallel sides of the trapezoid.

•Midsegment of a trapezoid – The midsegment of a trapezoid is the segment that joins the midpoints of its legs.

•Trapezoid – A trapezoid is a quadrilateral with exactly one pair of parallel sides.


VOCABULARY

The angles, sides, and diagonals of a trapezoid have certain properties.

The angles, sides, and diagonals of a kite have certain properties.


Term Description

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides of a trapezoid are called bases. The nonparallel sides are called legs. The two angles that share a base of a trapezoid are called base angles. A trapezoid has two pairs of base angles.

Diagram

An isosceles trapezoid is a trapezoid with legs that are congruent.

A midsegment of a trapezoid is the segment that joins the midpoints of its legs.

Key Concept Trapezoids and Their Parts

hsm11gmse_0606_t06314

base

base anglesbase angles

base

leg leg

hsm11gmse_0606_t06315

A

B C

D

hsm11gmse_0606_t06325

AR

M N

T P


281PearsonTEXAS.com

TheoremIf a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent.

If . . .TRAP is an isosceles trapezoid with bases RA and TP

Then . . .∠T ≅ ∠P, ∠R ≅ ∠A


Theorem 6-20

hsm11gmse_0606_t06317

T

R A

P

hsm11gmse_0606_t06317

T

R A

P

TheoremIf a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

If . . .ABCD is an isosceles trapezoid

Then . . .AC ≅ BD


Theorem 6-21

hsm11gmse_0606_t06321

A

B C

D

hsm11gmse_0606_t06324

A

B C

D

TheoremIf a quadrilateral is a trapezoid, then(1) the midsegment is parallel to the bases, and(2) the length of the midsegment is half the sum of the lengths of the bases.

If . . .TRAP is a trapezoid with midsegment MN

Then . . .(1) MN } TP, MN } RA, and

(2) MN = 12(TP + RA)

You will prove Theorem 6-22 in Lesson 7-3.

Theorem 6-22 Trapezoid Midsegment Theorem

hsm11gmse_0606_t06325

AR

M N

T P

Term DescriptionA kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.

Diagram

Key Concept Kites

hsm11gmse_0606_t06333282 Lesson 6-6 Trapezoids and Kites

TheoremIf a quadrilateral is a kite, then its diagonals are perpendicular.

If . . .ABCD is a kite

Then . . .AC # BD


Theorem 6-23

hsm11gmse_0606_t06335

A

B

C

D

hsm11gmse_0606_t06336

A

B

C

D

Concept Summary Relationships Among Quadrilaterals

Quadrilateral

Kite

Parallelogram

RhombusRectangle

Square

Trapezoid

Isoscelestrapezoid

Only 1 pair of parallel sides

No pairs of

parallel sides

2 pairs ofparallel sides

hsm11gmse_0606_t06342.aiProblem 1

Finding Angle Measures in Trapezoids

CDEF is an isosceles trapezoid and mjC = 65. What are mjD, mjE, and mjF ?

m∠C + m∠D = 180 Two angles that form same-side interior angles along one leg are supplementary.

65 + m∠D = 180 Substitute.

m∠D = 115 Subtract 65 from each side.

Since each pair of base angles of an isosceles trapezoid is congruent, m∠C = m∠F = 65 and m∠D = m∠E = 115.


hsm11gmse_0606_t06318

C

D E

F65�

What do you know about the angles of an isosceles trapezoid?You know that each pair of base angles is congruent. Because the bases of a trapezoid are parallel, you also know that two angles that share a leg are supplementary.

283PearsonTEXAS.com

Problem 3

How is an isosceles trapezoid different from other trapezoids? An isosceles trapezoid is a trapezoid whose nonparallel legs are congruent.

What do you notice about the diagram?Each trapezoid is part of an isosceles triangle with base angles that are the acute base angles of the trapezoid.

Problem 2

Finding Angle Measures in Isosceles Trapezoids

Paper Fans The second ring of the paper fan shown at the right consists of 20 congruent isosceles trapezoids that appear to form circles. What are the measures of the base angles of these trapezoids?

Step 1 Find the measure of each angle at the center of the fan. This is the measure of the vertex angle of an isosceles triangle.

m∠1 = 36020 = 18

Step 2 Find the measure of each acute base angle of an isosceles triangle.

18 + x + x = 180 Triangle Angle-Sum Theorem

18 + 2x = 180 Combine like terms.

2x = 162 Subtract 18 from each side.

x = 81 Divide each side by 2.

Step 3 Find the measure of each obtuse base angle of the isosceles trapezoid.

81 + y = 180

Two angles that form same-side interior angles along one leg are supplementary.

y = 99 Subtract 81 from each side.

Each acute base angle measures 81. Each obtuse base angle measures 99.


Investigating the Diagonals of Isosceles Trapezoids

A Choose from a variety of tools (such as a protractor, a ruler, or a compass) to investigate patterns in the diagonals of the three given isosceles trapezoids. Explain your choice.

A ruler is useful for measuring segments.

B Make a conjecture about the diagonals of isosceles trapezoids.

Isosceles Trapezoid ABCD

AC = 3 cm and BD = 3 cm.

So AC = BD and AC ≅ BD.

TEKS Process Standard (1)(C)

A B

D C

284 Lesson 6-6 Trapezoids and Kites

Problem 5

Problem 4

continuedProblem 3

Isosceles Trapezoid EFGH Isosceles Trapezoid JKLM

EG = 2.5 cm and FH = 2.5 cm. JL = 2 cm and KM = 2 cm.

So EG = FH and EG ≅ FH . So JL = KM and JL ≅ KM .

Conjecture: If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

E F

H GKL

M

J

Finding Angle Measures in Kites

Quadrilateral DEFG is a kite. What are mj1, mj2, and mj3?

m∠1 = 90 Diagonals of a kite are #.

90 + m∠2 + 52 = 180 Triangle Angle-Sum Theorem

142 + m∠2 = 180 Simplify.

m∠2 = 38 Subtract 142 from each side.

△DEF ≅ △DGF by SSS. Since corresponding parts of congruent triangles are congruent, m∠3 = m∠GDF = 52.

D

E G

F

1 23

52�


How can you check your answer?Find LM and QR. Then see if QR equals half of the sum of the base lengths.

How are the triangles congruent by SSS?DE ≅ DG and FE ≅ FG because a kite has congruent consecutive sides. DF ≅ DF by the Reflexive Property of Congruence.

Using the Midsegment of a Trapezoid

Algebra QR is the midsegment of trapezoid LMNP. What is x?

QR = 12 (LM + PN) Trapezoid Midsegment

Theorem

x + 2 = 12 [(4x - 10) + 8] Substitute.

x + 2 = 12 (4x - 2) Simplify.

x + 2 = 2x - 1 Distributive Property

3 = x Subtract x and add 1 to each side.

hsm11gmse_0606_t06328

L M

Q R

P N8

4x � 10

x � 2

285PearsonTEXAS.com


ONLINE

HO

M E W O RK


1. Justify Mathematical Arguments (1)(G) The plan suggests a proof of Theorem 6-20. Write a proof that follows the plan.

Given: Isosceles trapezoid ABCD with AB ≅ DC

Prove: ∠B ≅ ∠C and ∠BAD ≅ ∠D

Plan: Begin by drawing AE } DC to form parallelogram AECD so that AE ≅ DC ≅ AB. ∠B ≅ ∠C because ∠B ≅ ∠1 and ∠1 ≅ ∠C. Also, ∠BAD ≅ ∠D because they are supplements of the congruent angles, ∠B and ∠C.

Analyze Mathematical Relationships (1)(F) Find the value(s) of the variable(s) in each isosceles trapezoid or kite.

2. 3. 4.

5. Explain Mathematical Ideas (1)(G) If KLMN is an isosceles trapezoid, is it possible for KM to bisect ∠LMN and ∠LKN ? Explain.

Apply Mathematics (1)(A) The beams of the bridge at the right form quadrilateral ABCD. △AED @ △CDE @ △BEC and mjDCB = 120.

6. Classify the quadrilateral. Explain your reasoning.

7. Find the measures of the other interior angles of the quadrilateral.

8. The perimeter of a kite is 66 cm. The length of one of its sides is 3 cm less than twice the length of another. Find the length of each side of the kite.

9. Prove the converse of Theorem 6-20: If a trapezoid has a pair of congruent base angles, then the trapezoid is isosceles.

Name each type of special quadrilateral that can meet the given condition. Make sketches to support your answers.

10. exactly one pair of congruent sides 11. two pairs of parallel sides

12. four right angles 13. adjacent sides that are congruent

14. perpendicular diagonals 15. congruent diagonals


Given: Isosceles trapezoid ABCD with AB ≅ DC

Prove: AC ≅ DB

Proof

hsm11gmse_0606_t06008

A

B C

D

1E

hsm11gmse_0606_t06001

QS � x � 5RP � 3x � 3

Q R

P S2x�

(x � 6)�(2x � 4)�

hsm11gmse_0606_t06005 hsm11gmse_0606_t06006

y�

(2y � 20)�

(4x � 30)�

(3x � 5)�

STEM

BA

CD

E BA

CD

E

Proof

Proof

hsm11gmse_0606_t06010

A

B C

D


286 Lesson 6-6 Trapezoids and Kites

17. Prove the converse of Theorem 6-21: If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.

18. Given: Isosceles trapezoid TRAP with TR ≅ PA

Prove: ∠RTA ≅ ∠APR

19. Prove that the angles formed by the noncongruent sides of a kite are congruent.

Determine whether each statement is true or false. Justify your response.

20. All squares are rectangles. 21. A trapezoid is a parallelogram.

22. A rhombus can be a kite. 23. Some parallelograms are squares.

24. Every quadrilateral is a parallelogram. 25. All rhombuses are squares.

26. Select Tools to Solve Problems (1)(C) A wallpaper border pattern consists of isosceles trapezoids, each with two diagonals separating it into four triangles as shown.

To investigate the trapezoids, choose from the following tools: protractor, ruler, compass, or graphing calculator. Explain your choice. Then observe any patterns. Make a conjecture about the triangles that are formed by the diagonals.

27. Given: Isosceles trapezoid TRAP with TR ≅ PA; BI is the perpendicular bisector of RA, intersecting RA at B and TP at I.

Prove: BI is the perpendicular bisector of TP.

28. <BN

> is the perpendicular bisector of AC at N. Describe the set of points, D, for

which ABCD is a kite.

For a trapezoid, consider the segment joining the midpoints of the two given segments. How are its length and the lengths of the two parallel sides of the trapezoid related? Justify your answer.

29. the two nonparallel sides 30. the diagonals

Proof

Proof

hsm11gmse_0606_t06009

T

R A

P

Proof

Proof

hsm11gmse_0606_t15810

T

R A

P

hsm11gmse_0606_t06011

A

B

CN

TEXAS Test Practice

31. Which statement is never true?

A. Square ABCD is a rhombus. C. Parallelogram PQRS is a square.

B. Trapezoid GHJK is a parallelogram. D. Square WXYZ is a parallelogram.

32. A quadrilateral has four congruent sides. Which name best describes the figure?

F. trapezoid H. rhombus

G. parallelogram J. kite

33. Given DE is congruent to FG and EF is congruent to GD, prove ∠E ≅ ∠G.

hsm11gmse_0606_t06013

D

E

G

F

287PearsonTEXAS.com

TOPIC VOCABULARY

•baseanglesofatrapezoid, p. 281

•basesofatrapezoid, p. 281

• consecutiveangles, p. 255

•equiangularpolygon, p. 249

•equilateralpolygon, p. 249

• isoscelestrapezoid, p. 281

•kite, p. 282

• legsofatrapezoid, p. 281

•midsegmentofatrapezoid, p. 281

•oppositeangles, p. 255

•oppositesides, p. 255

•parallelogram, p. 255

• rectangle, p. 269

• regularpolygon, p. 249

• rhombus, p. 269

• square, p. 269

• trapezoid, p. 281

Check Your UnderstandingChoose the vocabulary term that correctly completes the sentence.

1. A parallelogram with four congruent sides is a(n) ? .

2. A polygon with all angles congruent is a(n) ? .

3. Angles of a polygon that share a side are ? .

4. A(n) ? is a quadrilateral with exactly one pair of parallel sides.

Topic 6 Review


ExercisesFind the measure of an interior angle and an exterior angle of each regular polygon.

5. hexagon 6. 16-gon 7. pentagon

8. What is the sum of the exterior angles for each polygon in Exercises 5–7?

Find the measure of the missing angle.

9. 10.

hsm11gmse_06cr_t06357

x� 83�89�

119�


z�

122�

79�

ExampleFind the measure of an interior angle of a regular 20‑gon.

Measure =(n - 2)180

n Corollary to the Polygon Angle-Sum Theorem

=(20 - 2)180

20 Substitute.

= 18 # 18020 Simplify.

= 162

The measure of an interior angle is 162.

Quick ReviewThe sum of the measures of the interior angles of an n-gon is (n - 2)180. The measure of one interior angle of a regular

n-gon is (n - 2)180

n . The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

288 Topic 6 Review

6-3 Proving That a Quadrilateral Is a Parallelogram

ExercisesDetermine whether the quadrilateral must be a parallelogram.

17. 18.

Find the values of the variables for which ABCD must be a parallelogram.

19. 20. hsm11gmse_06cr_t06370hsm11gmse_06cr_t06372


A

B C

D

(4y � 4)�

(2x � 6)�4x�

(3y � 20)�


4x � 2

3y �

3 3y � 1

3x

A

B C

D


ExercisesFind the measures of the numbered angles for each parallelogram.

11. 12.

13. 14.

Find the values of x and y in ▱ABCD.

15. AB = 2y, BC = y + 3, CD = 5x - 1, DA = 2x + 4

16. AB = 2y + 1, BC = y + 1, CD = 7x - 3, DA = 3x


38�

99�

23

1


79�

21

3


63�

37� 2

13


21 3

ExampleFind the measures of the numbered angles in the parallelogram.

Since consecutive angles are supplementary, m∠1 = 180 - 56, or 124. Since opposite angles are congruent, m∠2 = 56 and m∠3 = 124.


56�

2 31

ExampleMust the quadrilateral be a parallelogram?

Yes, both pairs of opposite angles are congruent.


Quick ReviewOpposite sides and opposite angles of a parallelogram are congruent. Consecutive angles in a parallelogram are supplementary. The diagonals of a parallelogram bisect each other. If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Quick ReviewA quadrilateral is a parallelogram if any one of the following is true.

• Both pairs of opposite sides are parallel.

• Both pairs of opposite sides are congruent.

• Consecutive angles are supplementary.

• Both pairs of opposite angles are congruent.

• The diagonals bisect each other.

• One pair of opposite sides is both congruent and parallel.

289PearsonTEXAS.com

6-5 Conditions for Rhombuses, Rectangles, and Squares

ExercisesCan you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain.

29. 30.

For what value of x is the figure the given parallelogram? Justify your answer.

31. 32. hsm11gmse_06cr_t06383.aihsm11gmse_06cr_t06384.ai

hsm11gmse_06cr_t06386.ai

Rhombus

(5x � 30)� (3x � 6)�


Rectangle

2x � 1 x � 32 2

6-4 Properties of Rhombuses, Rectangles, and Squares

ExercisesFind the measures of the numbered angles in each special parallelogram.

21. 22.

Determine whether each statement is always, sometimes, or never true.

23. A rhombus is a square.

24. A square is a rectangle.

25. A rhombus is a rectangle.

26. The diagonals of a parallelogram are perpendicular.

27. The diagonals of a parallelogram are congruent.

28. Opposite angles of a parallelogram are congruent.


2

32�3

1


256�

31

ExampleWhat are the measures of the numbered angles in the rhombus?

m∠1 = 60 Each diagonal of a rhombus bisects a pair of opposite angles.

m∠2 = 90 The diagonals of a rhombus are #.

60 + m∠2 + m∠3 = 180 Triangle Angle-Sum Thm.

60 + 90 + m∠3 = 180 Substitute.

m∠3 = 30 Simplify.

2

60�

31


Quick ReviewA 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.

The diagonals of a rhombus are perpendicular. Each diagonal bisects a pair of opposite angles.

The diagonals of a rectangle are congruent.

ExampleCan you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain.

Yes, the diagonals are perpendicular, so the parallelogram is a rhombus.


Quick ReviewIf a quadrilateral is a parallelogram with a diagonal that bisects two angles of the parallelogram, then the quadrilateral is a rhombus. If a quadrilateral is a parallelogram with perpendicular diagonals, then the quadrilateral is a rhombus. If a quadrilateral is a parallelogram with congruent diagonals, then the quadrilateral is a rectangle.

290 Topic 6 Review


ExercisesFind the measures of the numbered angles in each isosceles trapezoid.

33. 34.

Find the measures of the numbered angles in each kite.

35. 36.

37. A trapezoid has base lengths of (6x - 1) units and 3 units. Its midsegment has a length of (5x - 3) units. What is the value of x?


45� 3

1 2


80�3

12


65�

12


134�

2

38�ExampleABCD is an isosceles trapezoid. What is m∠C?

Since BC } AD, ∠C and ∠D are same-side interior angles.

m∠C + m∠D = 180

m∠C + 60 = 180 Substitute.

m∠C = 120 Subtract 60 from each side.


60�

B C

A D

Same-side interior angles are supplementary.

Quick ReviewThe parallel sides of a trapezoid are its bases, and the nonparallel sides are its legs. Two angles that share a base of a trapezoid are base angles of the trapezoid. The midsegment of a trapezoid joins the midpoints of its legs.

The base angles of an isosceles trapezoid are congruent. The diagonals of an isosceles trapezoid are congruent.

The diagonals of a kite are perpendicular.

291PearsonTEXAS.com

Topic 6 TEKS Cumulative Practice

Multiple ChoiceRead each question. Then write the letter of the correct answer on your paper.

1. Which list could represent the lengths of the sides of a triangle?

A. 7 cm, 10 cm, 25 cm

B. 4 in., 6 in., 10 in.

C. 1 ft, 2 ft, 4 ft

D. 3 m, 5 m, 7 m

2. Which quadrilateral CANNOT contain four right angles?

F. square H. trapezoid

G. rhombus J. rectangle

3. What is the circumcenter of △ABC with vertices A(-7, 0), B(-3, 8), and C(-3, 0)?

A. (-7, -3) C. (-4, 3)

B. (-5, 4) D. (-3, 4)

4. ABCD is a rhombus. To prove that the diagonals of a rhombus are perpendicular, which pair of angles below must you prove congruent by using corresponding parts of congruent triangles?

F. ∠AEB and ∠DEC

G. ∠AEB and ∠AED

H. ∠BEC and ∠AED

J. ∠DAB and ∠ABC

5. FGHJ is a quadrilateral. If at least one pair of opposite angles in quadrilateral FGHJ is congruent, which statement is false?

A. Quadrilateral FGHJ is a trapezoid.

B. Quadrilateral FGHJ is a rhombus.

C. Quadrilateral FGHJ is a kite.

D. Quadrilateral FGHJ is a parallelogram.

6. For which value of x are lines g and h parallel?

F. 12 H. 18

G. 15 J. 25

7. In △GHJ, GH ≅ HJ . Using the indirect proof method, you attempt to derive a contradiction by proving that ∠G and ∠J are right angles. Which theorem will contradict this claim?

A. Triangle Angle-Sum Theorem

B. Side-Angle-Side Theorem

C. Converse of the Isosceles Triangle Theorem

D. Angle-Angle-Side Theorem

8. Which quadrilateral must have congruent diagonals?

F. kite

G. rectangle

H. parallelogram

J. rhombus

hsm11gmse_06cu_t06102.ai

A

D

E

C

B

hsm11gmse_06cu_t06104 .ai

g

h

(2x � 10)�(5x � 5)�

292 Topic 6 TEKS Cumulative Practice

9. What values of x and y make the quadrilateral below a parallelogram?

A. x = 2, y = 1 C. x = 1, y = 2

B. x = 3, y = 5 D. x = 2, y = 97

10. Which is the most valid conclusion based on the statements below?

If a triangle is equilateral, then it is isosceles. △ABC is not equilateral.

F. △ABC is not isosceles.

G. △ABC is isosceles.

H. △ABC may or may not be isosceles.

J. △ABC is equilateral.

Gridded Response 11. What is m∠1 in the figure below?

12. ∠ABE and ∠CBD are vertical angles, and both are complementary with ∠FGH . If m∠ABE = (3x - 1), and m∠FGH = 4x, what is m∠CBD?

13. What is the value of x in the kite below?

14. The outer walls of the Pentagon in Arlington, Virginia, are formed by two regular pentagons, as shown at the right. What is the value of x?

Constructed Response 15. What are the possible values for n to make ABC a valid

triangle? Show your work.

16. The pattern of a soccer ball contains regular hexagons and regular pentagons. The figure at the right shows what a section of the pattern would look like on a flat surface. Use the fact that there are 360° in a circle to explain why there are gaps between the hexagons.

Does the information help you prove that ABCD is a parallelogram? Explain.

17. AC bisects BD.

18. AB ≅ DC , AB } DC

19. AB ≅ DC , BC ≅ AD

20. ∠DAB ≅ ∠BCD , ∠ABC ≅ ∠CDA

21. CD has endpoints C(5, 7) and D(10, -5). What are the coordinates of the midpoint of CD? What is CD? Show your work.


6y � 4

5x � 6

y � 1

3x � 2


1

31�

38�

69�

22� x�



x�


5n � 4

2n

n �

1

C

BA


hsm11gmse_06ct_t05844.ai

A B

D C

293PearsonTEXAS.com

topic 6 polygons and quadrilaterals

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