solutions manual - prek...

the isosceles right triangle, and the diagonal is thehypotenuse.

9. Always true by the Parallelogram Opposite AnglesConjecture

10. Sometimes true. It is true only if the parallelogramis a rectangle. Consecutive angles of a parallelogramare always supplementary, but are only congruent ifthey are right angles.

11. 20. By the Rectangle Diagonals Conjecture, the diag-onals of a rectangle bisect each other and arecongruent. By the “bisect each other” part, KC �CR � 10, so KR � 20, and by the “are congruent”part, WE � KR, so WE � 20.

12. 37°. By the Parallelogram Consecutive AnglesConjecture, �P and �PAR are supplementary, so48° � (y � 95°) � 180°. Then y � 95° � 132°,so y � 37°.

13. x � 45°, y � 90°. A diagonal divides a square intotwo isosceles right triangles, so x � 45°. Thediagonals of a square are perpendicular (SquareDiagonals Conjecture), so y � 90°.

14. DIAM is not a rhombus because it is not equilateraland opposite sides are not parallel. You can useslopes to determine that opposite sides of thisquadrilateral are not parallel, so it is not evena parallelogram.

15. BOXY is a rectangle because its adjacent sides areperpendicular.

16. Yes. TILE is a rhombus, and every rhombus isa parallelogram.

17. Possible construction: You know by the Square DiagonalsConjecture that the diagonalsof a square are perpendicularand congruent. Duplicate LV�.Construct its perpendicularbisector. From the intersectionpoint of the diagonals, set the radius of yourcompass at �

12�(LV ). Draw a circle with this center

and radius. Label the two points where the circleintersects the perpendicular bisector of LV� as Eand O. Connect the points L, O, V, and E to formthe square.

E V

L O

false

true

Find an equation for the line passing through

H(8, 3) with slope ��29�.

�xy �

�38� � ��

29� � �

�92�

9(y � 3) � �2(x � 8)

9y � 27 � �2x � 16

y � ��29�x � �

493�

IMPROVING YOUR REASONING SKILLS

Beginner Puzzle: green left, red up, red right, reddown

Intermediate Puzzle: orange down, orange right,red up, red right, red down

Advanced Puzzle: orange down, orange left, red up,red right, yellow down, yellow left, red down

LESSON 5.6

EXERCISES

1. Sometimes true. It is only true if the parallelogramis a rectangle.

2. Always true. By the definition of rectangle, all theangles are congruent. By the Quadrilateral SumConjecture and division, each angle measures 90°,so any two angles of a rectangle, including consecu-tive angles, are supplementary.

3. Always true by the Rectangle Diagonals Conjecture

4. Sometimes true; it is only true if the rectangle is asquare.

5. Always true by the Square Diagonals Conjecture

6. Sometimes true. It is true only if the rhombus isequiangular.

7. Always true. All squares fit the definition of rectangle.

8. Always true. All sides of a square are congruent andform right angles, so the sides become the legs of

false

true

false

true

false

true

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DISCOVERING GEOMETRY COURSE SAMPLER 131

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m�CDA. So the quadrilateral is equiangular. Using�1 � �5 and the Converse of AIA, AB� � CD�. Using�3 � �7 and the Converse of AIA, BC� � AD��.Therefore ABCD is an equiangular parallelogram, soit is a rectangle.

21. If the diagonals are congruent and bisect each other,then the room is rectangular (Rectangle DiagonalsConjecture).

22. The platform stays parallel to the floor becauseopposite sides of a rectangle are parallel. (The Paral-lelogram Opposite Sides Conjecture applies becausea rectangle is a parallelogram.)

23. The crosswalks form a parallelogram: The streets areof different widths, so the crosswalks are of differentlengths. The streets would have to cross at rightangles for the crosswalks to form a rectangle. Thestreets would have to cross at right angles and also beof the same width for the crosswalk to form a square.

24. Place one side of the ruler along one side of theangle. Draw a line with the other side of the ruler.Repeat with the other side of the angle. Draw a linefrom the vertex of the angle to the point where thetwo lines meet.

25. Rotate your ruler so that each endpoint of thesegment barely shows on each side of the ruler.Draw the parallel lines on each side of your ruler.Now rotate your ruler the other way and repeat theprocess to get a rhombus. The original segment isone diagonal of the rhombus. The other diagonalwill be the perpendicular bisector of the originalsegment.

26. 2. Given; 4. SSS; 5. CPCTC; 9. Definition of rhombus

27. Yes, it is true for rectangles.

Given: �1 � �2 � �3 � �4

Show: ABCD is a rectangle

Paragraph Proof: By the Quadrilateral SumConjecture, m�1 � m�2 � m�3 � m�4 � 360°.Because all four angles are congruent, each anglemeasures 90°. Because �4 and �5 form a linear

18. One possible construction: Duplicate �B. Bisect it.Mark off distance BK along the bisector. At K dupli-cate angles of measure �

12� m�B on either side of BK�.

Label as A and E the intersections of the rays of �Bwith the rays of the new angles which are not BK�.

Another possible construction: Duplicate �B andthen bisect it. Mark off the length BK on the anglebisector. Then construct the perpendicular bisectorof BK�. Label the points where this perpendicularintersects the sides of �B as E and K. Connect thepoints B, A, K, and E to form the rhombus.

19. Possible construction: Duplicate PS� and construct a perpendi-cular to PS� through S. Openyour compass to radius PEand draw an arc intersectingthe perpendicular line thatyou have constructed. Label the intersection pointof the arc and the perpendicular line as E.Construct arcs of length PI and EI. Label theirintersection as I. Connect the points P, I, E, and Sto form the rectangle.

20. Converse: If the diagonals of a quadrilateral arecongruent and bisect each other, then the quadrilat-eral is a rectangle.

Given: Quadrilateral ABCD with diagonals AC� � BD�. AC� and BD� bisect each other

Show: ABCD is a rectangle

Paragraph Proof: Because the diagonals arecongruent and bisect each other, AE� � BE� �DE� � EC�. Using the Vertical Angles Conjecture,�AEB � �CED and �BEC � �DEA. So �AEB ��CED and �AED � �CEB by SAS. Using theIsosceles Triangle Conjecture and CPCTC, �1 ��2 � �5 � �6, and �3 � �4 � �7 � �8. Eachangle of the quadrilateral is the sum of two angles,one from each set, so for example, m�DAB �m�1 � m�8. By the addition property of equality,m�1 � m�8 � m�2 � m�3 � m�5 � m�4 �m�6 � m�7. So m�DAB � m�ABC � m�BCD �

A

E

B

CD

81 2

3

456

7

I E

P S

A

B

K

E

A

B E

K

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�y �

x �(�

21)

� � �1

�xy �

�12� � �1

y � 1 � �1(x � 2)

y � 1 � �x � 2

y � �x � 1

To find points on this line in addition to the twothat are given, substitute any number for x otherthan 2 and �3 into the equation y � �x � 1 tofind the corresponding value for y. If x � 1, y � 0;if x � 0, y � 1; if x � �1, y � 2; if x � �2,y � 3. Therefore, four additional points on the lineare (1, 0), (0, 1), (�1, 2), and (�2, 3). Any three ofthese points would be sufficient; there are infinitelymany other points on the line.

30. y � �89�x � �

896�, or 8x � 9y � �86. The perpendicular

bisector of the segment with endpoints (�12, 15) and(4, �3) is the line through the midpoint of this segmentand perpendicular to it. First find the midpoint:

��122� 4�, �

15 �

2(�3)��

�

28�, �

122�� (�4, 6)

The slope of this segment is

�4�

�

3(�

�

11

52)

� � ��

1168

� � ��98

�

So, its perpendicular bisector will have slope �89�. Find

an equation of the line with m � �89� through (�4, 6).

�x �y �

(�64)� � �

89�

�xy �

�64� � �

89�

9(y � 6) � 8(x � 4)

9y � 54 � 8x � 32

y � �89�x � �

896�, or 8x � 9y � �86

31. y � ��170�x � �

152�, or 7x � 10y � �24. The median to

AB� goes through C(8, �8) and the midpoint of AB�,which is (�2, �1). Find an equation of the line thatcontains (8, �8) and (�2, �1). First find the slope.

m � ��88��

(�(�

21))

� � ��170�

Now use the slope and either point to find the equa-tion of the line. Here, the point (�2, �1) is used.

�xy �

�((��

12

))� � �

�10

7�

�xy �

�12� � �

�10

7�

10(y � 1) � �7(x � 2)

10y � 10 � �7x � 14

7x � 10y � �24, or y � ��170�x � �

152�

pair, m�4 � m�5 � 180°. Substitute 90° for m�4and solve to get m�5 � 90°. By definition ofcongruent angles, �5 � �3, and they are alternateinterior angles, so AD� � BC� by the Converse of theParallel Lines Conjecture. Similarly, �1 and �5 arecongruent corresponding angles, so AB� � CD�� by theConverse of the Parallel Lines Conjecture. Thus,ABCD is a parallelogram by the definition of paral-lelogram. Because it is an equiangular parallelo-gram, ABCD is a rectangle.

28. a � 54°, b � 36°, c � 72°, d � 108°, e � 36°,f � 144°, g � 18°, h � 48°, j � 48°, and k � 84°.First, a � 54° (CA Conjecture). Now look at theright triangle in which the measure of one of theacute angles is b. The other acute angle in thistriangle is the vertical angle of the angle of measurea, so its measure is 54°, and b � 90° � 54° � 36°.Next, 2c � b � 180°, so 2c � 144°, and c � 72°.The angle with measure d forms a linear pair withthe alternate interior angle of the angle withmeasure c, so d � 180° � 72° � 108°. In thetriangle that contains the angle with measure e,both of the other angle measures have been foundto be 72°, so this is an isosceles triangle with vertexangle of measure e. Therefore, e � 2 � 72° � 180°,so e � 36°. (Or, by the CA Conjecture, e � b � 36°.)Now look at the quadrilateral that contains theangle with measure f. This is a parallelogrambecause both pairs of opposite sides are parallel(from the given pairs of parallel lines). Notice thatthe angle that forms a linear pair with the angle ofmeasure e is the angle opposite the angle ofmeasure f in the parallelogram. Therefore, by theParallelogram Opposite Angles Conjecture and theLinear Pair Conjecture, f � 180° � e � 144°. Nowlook at the isosceles triangle in which g is themeasure of a base angle. In this triangle, the vertexangle measures 144° (Vertical Angles Conjecture), so2g � 144° � 180°; then 2g � 36°, and g � 18°.Along the upper horizontal line, there are threemarked angles, all of measure h, and one unmarkedangle. The unmarked angle is the alternate interiorangle of the supplement of the angle of measure f,so its measure is 180° � 144° � 36°. Then 3h �36° � 180°, so 3h � 144°, and h � 48°. Next, j isan alternate interior angle of one of the angles ofmeasure h, so j � 48°. Finally, k � j � 48° � 180°(Triangle Sum Conjecture), so k � 84°.

29. Possible answers: (1, 0); (0, 1); (�1, 2); (�2, 3).

Find an equation for the line passing through the

points (2, �1) and (�3, 4), and then find points on

that line. First find the slope: m � �4��

3(��1

2)

� � ��55� �

�1. Now use the slope and one of the points on the

line to find an equation for the line. Here, (2, �1)

is used.

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Because both pairs of opposite sides are parallel,WATR is a parallelogram.

4. Flowchart Proof

5. parallelogram

6. Flowchart Proof

Paragraph Proof: Look at two overlappingtriangles: �GOY and �IYO. IY� � GO�� becauseopposite sides of a rectangle are congruent.(Opposite sides of a parallelogram are congruent,and a rectangle is a parallelogram.) �GOY � �IYOby the definition of a rectangle. (All angles of arectangle are congruent.) Therefore, �GOY ��IYO by SAS, and YG� � OI� by CPCTC.

7. Flowchart Proof

Paragraph Proof: RE� � AB� is given, andBR� � EA� and AR� � EB� because opposite sides ofa parallelogram are congruent (ParallelogramOpposite Sides Conjecture). Therefore,

Given Opposite sides ofparallelogram arecongruent.

Opposite sides ofparallelogram arecongruent.

SSS

5

CPCTC

6

42

Given

�� AR � EB 3 �� BR � EA

1 BEAR is a parallelogram.

�� RE � AB

Definition of rectangle

7 BEAR is a rectangle

�EBR � �ARB � �RAE � �BEA

�EBR � �ARB � �RAE � �BEA

Opposite sides ofrectangle are congruent.

Definition of rectangle

SAS

Same segment

4

CPCTC

5

31 2 �GOY � �IYO �� YO � YO �� IY � GO

�� YG � IO

�GOY � �IYO

Given

1 �� SP � OA

Converse of AIA Conjecture

7 �� PA � SO

Definition of parallelogram

8 SOAP is a parallelogram

Given

2 �� SP � OA

SAS

5�SOP � �APO

Same segment

4 �� OP � OP

AIA Conjecture

3�1 � �2

CPCTC

6�3 � �4

32. Velocity � 1.8 mi/h. Angle of path �106.1° clockwise from the north.

IMPROVING YOUR VISUAL THINKING SKILLS

The matching pairs are (A1, B3), (B1, D3), (A2, C2),(B2, C4), (A3, B4), (C1, C3), (A4, D2), and (D1, D4).

LESSON 5.7

EXERCISES

1. Completion of flowchart proof: 3. OA� � SK�,definition of parallelogram; 5. AIA; 6. SA� � SA�,Same segment; 7. �SOA � �AKS by ASA

Paragraph Proof: By the definition of aparallelogram, SO� � KA� and OA� � SK�. Because �3and �4 are alternate interior angles formed whenthe parallel lines SO�� and KA�� are cut by thetransversal SA��, �3 � �4 by the AIA Conjecture.Similarly, �1 and �2 are alternate interior anglesformed when parallel lines OA�� and SK�� are cut bythe same transversal, so �1 � �2 by the AIAConjecture. SA� is a shared side of the two triangles;every segment is congruent to itself. Therefore,�SOA � �AKS by ASA.

2. Completion of flowchart proof: 4. Given; 5. �THA,Conjecture proved in Exercise 1; 6. �HBA � �ATHby CPCTC

Paragraph Proof: Diagonal BT� dividesparallelogram BATH into two congruent trianglesbecause either diagonal of a parallelogram dividesthe parallelogram into two congruent triangles.(This is the conjecture that you proved inExercise 1.) Therefore, �BAT � �THB, and�BAT � �THB by CPCTC. Also, HT�� dividesparallelogram BATH into two congruent trianglesfor the same reason, so �BAH � �THA, and�HBA � �ATH by CPCTC.

3. Completion of flowchart proof: 1. WA�� RT�,Given; 2. WR�� AT�, Given; 3. WT�� WT��, Samesegment; 4. �WRT � �TAW by SSS; 5. �1 � �2by CPCTC; 6. RT� � WA�� by Converse of theParallel Lines Conjecture; 7. �4 � �3 by CPCTC;8. RW�� TA� by Converse of the Parallel LinesConjecture; 9. Definition of parallelogram

Paragraph Proof: Because WA�� RT� andWR�� AT� (both given) and also WT�� WT��,�WRT � �TAW by SSS. Then �1 � �2 byCPCTC, so RT� � WA�� by the Converse of the ParallelLines Conjecture. Similarly, �3 � �4, so RW�� TA�.

2 mi/h

1.5 mi/h

60°

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76 LESSON 5.1 • Worksheet Discovering Geometry Teaching and Worksheet Masters


Quadrilateral Sum Conjecture

Write a proof of the Quadrilateral Sum Conjecture.

Conjecture: The sum of the measures of the four interiorangles of any quadrilateral is 360°.

Given: Quadrilateral QUAD with an auxiliary line,diagonal DU��, and angles labeled as shown

Show: m�Q � m�U � m�A � m�D � 360°.

Proof

D

d e

a

vu

qQ

U

A

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Discovering Geometry Teaching and Worksheet Masters LESSON 5.1 • Transparency 77©2008 Key Curriculum Press

Exercise 12

ep

da

fg

bcn

m

k j

h

� 1 � �

2

116°

36°

98° 11

6°13

8°12

2°

106°

60°

87°

77°

� 2� 1

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Exercise 9

Giv

en:

Kit

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EN

Yw

ith

BE�

�B

Y�

and

EN�

�Y

N�

Sho

w:

BN�

bise

cts

�B

BN�

bise

cts

�N

Flo

wch

art

Pro

of

N

Y E

B2 1

4 3

��

an

d �

�

6

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init

ion

of

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e bi

sect

or�

� C

ongr

uen

cesh

ortc

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4�

BE

N �

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5

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Exercises 9 and 10

9.

10

.

� Vb

� Vc

� Vh

� Vw

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Discovering Geometry Assessment Resources LESSONS 5.1, 5.2 65©2008 Key Curriculum Press

Complete each statement.

1. The sum of the measures of the n interior angles of an n-gon

is ________________.

2. The number of triangles formed in a decagon when all the diagonals

from one vertex are drawn is ________________.

3. The sum of the measures of the exterior angles of a 25-gon

is ________________.

4. The measure of one angle in a regular octagon is ________________.

5. If the measure of one exterior angle of a regular polygon is 24°, then

the polygon has ________________ sides.

Find each lettered angle measure.

6. a � _____

b � _____

c � _____

d � _____

7. m � _____

n � _____

p � _____

r � _____

s � _____

t � _____

ps

m

r

t n

72�

85�

dc

ba

Chapter 5 • Quiz 1 Form A

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68 LESSONS 5.3, 5.4 Discovering Geometry Assessment Resources


For Problems 1–6, find the lettered measures in each figure. The figures inProblems 1 and 2 are kites.

1. w � _____ x � _____ 2. w � _____ x � _____

y � _____ z � _____ y � _____

The figures in Problems 3 and 4 are isosceles trapezoids.

3. x � _____ 4. Perimeter � 111 cm

y � _____ x � _____

5. x � _____ 6. The figure is a trapezoid.

y � _____ q � _____

z � _____

22

16

q

2313

17 x

z

y

60� 40�

x

23 cm

30 cm

y

x

119�

y

x

w

75�

9 cm

148�

zxy

72�

4 cm

w

51�

Chapter 5 • Quiz 2 Form B

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Discovering Geometry Assessment Resources CHAPTER 5 71©2008 Key Curriculum Press

Part AComplete each statement. Do not use square as an answer.

1. In an isosceles trapezoid, the base angles are ________________.

2. Each interior angle of a regular decagon measures ________________.

3. The vertex angles of a kite are ________________ by the diagonal.

4. The consecutive angles of a parallelogram are ________________.

5. The length of a midsegment between two sides of a triangle

is ________________ the length of the third side.

6. The sum of the measures of the interior angles of a hexagon

is ________________.

7. The nonvertex angles of a kite are ________________.

8. A convex quadrilateral with congruent diagonals is a(n) ________________ or a(n) ________________.

9. An equilateral quadrilateral is a ________________.

10. The measure of an exterior angle of a regular octagon is________________.

Part BFind each lettered angle measure.

1. a � _____ 2. b � _____

3. c � _____ 4. d � _____

5. e � _____ 6. f � _____

7. g � _____ 8. h � _____

Part CFind each lettered measure.

1. Perimeter � 64 2. a � _____

a � _____ b � _____

x � _____ x � _____

y � _____ y � _____ x

ya

b

38

31

47�x

y

a

96�

24�

10

d

f

be

a

h

g

c

40�

88�

76�

58�

70�

(continued)

Chapter 5 • Test Form A

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72 CHAPTER 5 Discovering Geometry Assessment Resources


Chapter 5 • Test (continued) Form A

Name Period Date

3. a � _____

w � _____

x � _____

y � _____

4. A regular hexagonal mirror frame is to be built from stripsof 2-inch-wide pine lattice. At what angles a and b shouldthe lattice be cut?

a � _____

b � _____

Part DUse the segments and angle at right to construct each figure. Use either a compass and a straightedge or patty paper.

1. Rhombus WAVY using �W and usingsegment z as the diagonal WV��. (You don’tneed to use segments x and y.)

2. Kite LMNO using segments y and z asdiagonals and segment x as a side.(You don’t need to use �W.)

Part EWrite a paragraph proof or a flowchart proof for the conjecture:The diagonals of a rectangle are congruent.

Given: Rectangle ABCD with diagonals AC�� and BD��

Show: AC�� BD��

B C

A D

W

x

y

z

b ba a

xa

wy

60� 55�

46

31

(continued)

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Mixed Review

1. In the figure at right, AB� � CD�� and AD�� BC�. Is AB��parallel to CD��? Explain.

For Problems 2 and 3, consider the statement: If two sides and an angle ofone triangle are congruent to two sides and an angle of another triangle,then the triangles are congruent.

2. Is the statement true? If not, give a counterexample or explain why itis not true.

3. Write the converse of the statement. Is the converse true? If not, give acounterexample or explain why it is not true.

4. Find x.

In Problems 5–9, use the figure at right to name an example ofeach of these:

5. An inscribed triangle

6. An isosceles triangle

7. A concave polygon

8. An acute triangle

9. A central angle

For Problems 10–12, m�K � 36°.

10. What is the measure of the complement to �K ?

11. What is the measure of the supplement to �K ?

12. What is the reflex measure of �K ?

ACircle O

OB

D

C

x

45�

D

A C

B

Chapter 5 • Test (continued) Form A

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Chapter 5 • Constructive Assessment Options

Choose one or more of these items to replace part of the chapter test.Let students know that they will receive from 0 to 5 points for eachitem, depending on the correctness and completeness of their answer.

1. (Lesson 5.1)On her Chapter 5 test, Ms. Donovan asked her students to find themeasure of each interior angle of a regular 15-gon. Here are a few ofthe answers students gave:

2340° 156° 180° 24° 168°

a. Tell which of the answers is correct and explain why it is correct.

b. Choose two of the incorrect answers and explain the error thestudent may have made in finding the answer.

2. (Lessons 5.1–5.2)In Lesson 5.1, you found formulas for the sum of the interior anglemeasures of a regular polygon and for the measure of each interiorangle. Now you will consider the “outside” angles of a regular polygon.

In the regular hexagon below, one “outside” angle is marked. Themeasure of this angle is the number of degrees in the rotationindicated by the arrow.

Find a formula for the sum of the measures of the “outside” angles of aregular n-gon and a formula for the measure of one “outside” angle.Show and explain all your work and simplify the formulas as much aspossible.


(continued)

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78 CHAPTER 5 Discovering Geometry Assessment Resources


Chapter 5 • Constructive Assessment Options (continued)

3. (Lessons 5.1–5.3)A classical semicircular arch is really half of a regular polygonbuilt with blocks whose faces are congruent isosceles trapezoids.For example, the inner arch in the diagram below is half of aregular 18-gon.

The angle measures of the isosceles trapezoids forming the arch dependon how many blocks are used to build the arch.

a. If a semicircular arch has only three blocks, what are the anglemeasures of the isosceles trapezoids? If an arch has five blocks,what are the angle measures? Show all your work.

b. Given the number of blocks, b, in the arch, find formulas for themeasures of the trapezoid base angles. You should find twoformulas—one for the “outer” base angles (those whose vertices areon the outside of the arch) and one for the “inner” base angles.Explain the reasoning you used to find the formulas.

4. (Lessons 5.3, 5.5, and 5.6)In the game Find the Oddball, a player looks at four objects anddetermines which one doesn’t belong with the others. For example,consider the four objects below.

A. B. C. D.

You might say that object B doesn’t belong because it has four sides,while the other shapes each have three. Or, you might say that object Cdoesn’t belong because it is the only shape with a right angle. You maybe able to find and explain other oddballs.

Create three different groups of four quadrilaterals, each with at leastone shape that can be considered an oddball. For each group youcreate, identify every possible oddball and explain why it doesn’tbelong with the other objects.

KeystoneVoussoir

Abutment

Rise

Span

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Chapter 5 • Constructive Assessment Options (continued)

5. (Lessons 5.3, 5.5, and 5.6)Consider the following points: A(1, 4), B(2, 12), C(9, 8).

a. Graph the points. Add a fourth point D so that points A, B, C, andD are the vertices of a particular type of quadrilateral. Name thequadrilateral and use algebra to verify one of the properties of thattype of quadrilateral. Show all your work.

b. Find the coordinates of the point where the diagonals ofquadrilateral ABCD intersect. Show all your work.

6. (Lesson 5.4)The modern art section of the Museum of Geometric Art is a largerectangular room. The museum directors want to build a wall in thecenter of the room to create more room for displaying art. The wallwill be built so that it is parallel to two of the opposite sides and itsends are equally distant from the other two sides.

Once the center wall is in place, a path will be painted on the floor around it. The path will be created byconnecting the midsegments of the triangles and trapezoidsformed by connecting the ends of the center wall to thecorners of the room. (See the diagram.)

a. If the room measures 80 ft by 100 ft and the wall is 70 ftlong, how long will the path be? Does your answerdepend on which sides the wall is parallel to? Explain.

b. Now generalize your results. If the room measures a feetby b feet and the center wall is x feet long, how long willthe path around the wall be?

Path

Center wall


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32 CHAPTER 5 Discovering Geometry Practice Your Skills


Lesson 5.1 • Polygon Sum Conjecture

Name Period Date

In Exercises 1 and 2, find each lettered angle measure.

1. a � _____, b � _____, c � _____, 2. a � _____, b � _____, c � _____,

d � _____, e � _____ d � _____, e � _____, f � _____

3. One exterior angle of a regular polygon measures 10°. What isthe measure of each interior angle? How many sides does thepolygon have?

4. The sum of the measures of the interior angles of a regular polygon is2340°. How many sides does the polygon have?

5. ABCD is a square. ABE is an equilateral 6. ABCDE is a regular pentagon. ABFGtriangle. is a square.

x � _____ x � _____

7. Use a protractor to draw pentagon ABCDE with m�A � 85°,m�B � 125°, m�C � 110°, and m�D � 70°. What is m�E?Measure it, and check your work by calculating.

D

FC

BA

x

EG

x

E

A B

D C

85�

44�a

c

b

d

f

e

a

e d

cb

97�

26�

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Discovering Geometry Practice Your Skills CHAPTER 5 35©2008 Key Curriculum Press

Lesson 5.4 • Properties of Midsegments

Name Period Date

In Exercises 1–3, each figure shows a midsegment.

1. a � _____, b � _____, 2. x � _____, y � _____, 3. x � _____, y � _____,

c � _____ z � _____ z � _____

4. X, Y, and Z are midpoints. Perimeter �PQR � 132, RQ � 55, and PZ � 20.

Perimeter �XYZ � _____

PQ � _____

ZX � _____

5. MN�� is the midsegment. Find the 6. Explain how to find the width of the lake fromcoordinates of M and N. Find the A to B using a tape measure, but without slopes of AB� and MN��. using a boat or getting your feet wet.

7. M, N, and O are midpoints. What type of quadrilateral is AMNO? How do you know? Give a flowchart proofshowing that �ONC � �MBN.

8. Give a paragraph or flowchart proof.

Given: �PQR with PD � DF � FH � HRand QE � EG � GI � IR

Show: HI� � FG� � DE� � PQ�

P

D E

F G

H I

R

Q

BM

O N

C

A

LakeA B

x

M

N

yC (20, 10)

B (9, –6)

A (4, 2)

Z Y

XP Q

R

x

z

y29

11

41

13

x

y

z16

21

14b

a c 37�

54�

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(continued)

Polygon Sum ConjectureL E S S O N

5.1CONDENSED

Discovering Geometry Condensed Lessons CHAPTER 5 65©2008 Key Curriculum Press

In this lesson you will

● Discover a formula for finding the sum of the angle measures for any polygon

● Use deductive reasoning to explain why the polygon sum formula works

Triangles come in many different shapes and sizes. However, as you discovered inChapter 4, the sum of the angle measures of any triangle is 180°. In this lessonyou will investigate the sum of the angle measures of other polygons. After youfind a pattern, you’ll write a formula that relates the number of sides of a polygonto the sum of the measures of its angles.

Investigation: Is There a Polygon Sum Formula?Draw three different quadrilaterals. For each quadrilateral, carefully measure thefour angles and then find the sum of the angle measures. If you measurecarefully, you should find that all of your quadrilaterals have the same angle sum.What is the sum? Record it in a table like the one below.

Next draw three different pentagons. Carefully measure the angles in eachpentagon and find the angle sum. Again, you should find that the angle sum isthe same for each pentagon. Record the sum in the table.

Use your findings to complete the conjectures below.

Quadrilateral Sum Conjecture The sum of the measures of the four angles

of any quadrilateral is ________________.

Pentagon Sum Conjecture The sum of the measures of the five angles of

any pentagon is ________________.

Now draw at least two different hexagons and find their angle sum. Record thesum in the table.

The angle sums for heptagons and octagons have been entered in the table foryou, but you can check the sums by drawing and measuring your own polygons.

Look for a pattern in the completed table. Find a general formula for the sum ofthe angle measures of a polygon in terms of the number of sides, n. (Hint: Usewhat you learned in Chapter 2 about finding the formula for a pattern with aconstant difference.) Then complete this conjecture.

Polygon Sum Conjecture The sum of the measures of the n interior angles

of an n-gon is ________________.

Number of sides 3 4 5 6 7 8 . . . n

Sum of angle measures 180° 900° 1080°

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Lesson 5.1 • Polygon Sum Conjecture (continued)

66 CHAPTER 5 Discovering Geometry Condensed Lessons


You can use deductive reasoning to see why your formula works. In each polygonbelow, all the diagonals from one vertex have been drawn, creating triangles.Notice that in each polygon, the number of triangles is 2 less than the numberof sides.

The quadrilateral has been divided into two triangles, each with an angle sum of180°. So, the angle sum for the quadrilateral is 180° � 2, or 360°. The pentagonhas been divided into three triangles, so its angle sum is 180° � 3, or 540°. Theangle sums for the hexagon and the heptagon are 180° � 4 and 180° � 5,respectively. In general, if a polygon has n sides, its angle sum is 180°(n � 2) or,equivalently, 180°n � 360°. This should agree with the formula you found earlier.

You can use the diagram at right to write a paragraph proof ofthe Quadrilateral Sum Conjecture. See if you can fill in the stepsin the proof below.

Paragraph Proof: Show that m�Q � m�U � m�A � m�D � 360°.

q � d � u � 180° and e � a � v � 180° by the ______________ Conjecture.By the Addition Property of Equality, q � d � u � e � a � v � __________.Therefore, the sum of the angle measures of a quadrilateral is 360°.

Here is an example using your new conjectures.

EXAMPLE Find the lettered angle measures.

a. b.

� Solution a. The polygon has seven sides, so the angle sum is 180° � 5, or 900°. Because allthe angles have the same measure, the measure of angle m is 900° � 7, orabout 128.6°.

b. The polygon has five sides, so the angle sum is 180° � 3, or 540°. Therefore,90° � 120° � 110° � 95° � t � 540°. Solving for t gives t � 125°.

t

110�

95�

60�m

D

d e

a

vu

qQ

U

A

4 sides 5 sides3 triangles

6 sides4 triangles

7 sides5 triangles

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(continued)

Properties of MidsegmentsL E S S O N

5.4CONDENSED

Discovering Geometry Condensed Lessons CHAPTER 5 71©2008 Key Curriculum Press

In this lesson you will

● Discover properties of the midsegment of a triangle

● Discover properties of the midsegment of a trapezoid

In Chapter 3, you learned that a midsegment of a triangle is a segment connectingthe midpoints of two sides. In this lesson you will investigate properties ofmidsegments.

Investigation 1: Triangle Midsegment PropertiesFollow Steps 1–3 in your book. Your conclusions should lead to the followingconjecture.

Three Midsegments Conjecture The three midsegments of a triangledivide it into four congruent triangles.

Mark all the congruent angles in your triangle as shownin this example.

Focus on one of the midsegments and the third side of thetriangle (the side the midsegment doesn’t intersect). Lookat the pairs of alternate interior angles and correspondingangles associated with these segments. What conclusioncan you make? Look at the angles associated with each ofthe other midsegments and the corresponding third side.

Now compare the length of each midsegment to the length of the correspondingthird side. How are the lengths related?

State your findings in the form of a conjecture.

Triangle Midsegment Conjecture A midsegment of a triangleis ________________ to the third side and ________________ the lengthof the third side.

The midsegment of a trapezoid is the segment connecting the midpoints of thetwo nonparallel sides.

Investigation 2: Trapezoid Midsegment PropertiesFollow Steps 1–3 in your book. You should find that the trapezoid’sbase angles are congruent to the corresponding angles at themidsegment. What can you conclude about the relationship of themidsegment to the bases?

Now follow Steps 5–7. You should find that the midsegment fits twice ontothe segment representing the sum of the two bases. That is, the length of the

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Lesson 5.4 • Properties of Midsegments (continued)

midsegment is half the sum of the lengths of the two bases. Another way to saythis is: The length of the midsegment is the average of the lengths of the bases.

Use what you have learned about the midsegment of a trapezoid to completethis conjecture.

Trapezoid Midsegment Conjecture The midsegment of a trapezoid

is ________________ to the bases and equal in length to _____________.

Read the text below the investigation on page 277 of your book and study thesoftware construction. Make sure you understand the relationship between theTrapezoid and Triangle Midsegment Conjectures.

Work through the following example yourself before checking the solution.

EXAMPLE Find the lettered measures.

a. b.

� Solution a. By the Triangle Midsegment Conjecture, x � �12�(13 cm) � 6.5 cm.

The Triangle Midsegment Conjecture also tells you that the midsegment isparallel to the third side. Therefore, the corresponding angles are congruent,so m � 72°.

b. By the Trapezoid Midsegment Conjecture, �12�(12 � y) � 9. Solving for y

gives y � 6.

The Trapezoid Midsegment Conjecture also tells you that the midsegmentis parallel to the bases. Therefore, the corresponding angles are congruent,so c � 58°.

By the Trapezoid Consecutive Angles Conjecture, b � 58° � 180°, so b � 122°.

y

b

c

12 cm

9 cm

58�x

m

13 cm

72�

72 CHAPTER 5 Discovering Geometry Condensed Lessons


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Discovering Geometry with The Geometer’s Sketchpad CHAPTER 5 65©2008 Key Curriculum Press

Lesson 5.1 • Polygon Sum Conjecture

You discovered that the angle measures of any triangle always sum to thesame number. Do you think that the same will be true for other polygons?

Investigation: Is There a Polygon Sum Formula?

SketchStep 1 In a new sketch, construct a quadrilateral.

Step 2 Measure all four angles of the quadrilateral.Make sure that the vertex is the second point you select on each angle.

Step 3 Use the calculator to sum all the angle measures. Click on each measurement toenter it into the calculator.

Investigate

1. Drag vertices of the quadrilateral and observe the angle sum. For now,just consider the angle sum in convex quadrilaterals. Record yourobservations as a conjecture (Quadrilateral Sum Conjecture).

Now you’ll extend your conjecture about quadrilaterals to polygons withmore than four sides. Before you get started, you need to decide a fewthings in your group.

2. There are many different types of polygons with more than foursides—pentagons, hexagons, octagons, and so on. Decide whichconvex polygon your group would like to investigate. Make sure thatno other group is investigating the same polygon.

3. Let n represent the number of sides of your polygon. What is n foryour polygon?

4. What is the name of your polygon, if it has one?

5. Before you explore the sum of the angles of your polygon onSketchpad, predict what you will discover. Record your prediction andexplain why you think it is reasonable.

6. Construct your convex polygon and calculate the sum of all its angles.Drag different vertices of your polygon and describe how well yoursketch confirms your prediction from Question 5.

7. Compare your results with results from groups that investigated otherpolygons. After you study the different results, write a conjecture thatgeneralizes the sum of the angles for any convex polygon with n sides(Polygon Sum Conjecture).

A

B

CD

m�DAB � 70.7°m�ABC � 78.9°m�BCD � 94.9°m�CDA � 115.5°

Steps 1 and 2

(continued)

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66 CHAPTER 5 Discovering Geometry with The Geometer’s Sketchpad


Lesson 5.1 • Polygon Sum Conjecture (continued)

EXPLORE MORE

1. Construct a polygon with more than four sides. Use dashedsegments to connect one of the vertices to every other vertex.(Because the vertex is already connected to its neighboring vertices,you don’t need to construct these two segments again.) You havedissected your polygon into triangles. Use this sketch to explainwhy the Polygon Sum Conjecture follows directly from theTriangle Sum Conjecture.

2. Now you can explore the concave polygons you have ignored so far.Your book defines an angle’s measure to be between 0° and 180°.Sketchpad will give angle measures up to and including 180°, but notgreater. By that definition, there is no interior angle at a “caved-in”vertex of a concave polygon. If you consider these points to be verticesof interior angles with measures greater than 180°, does the PolygonSum Conjecture still hold? Use a polygon like the one from ExploreMore 1 to investigate this question.

3. Calculate the measure of an angle of a regular octagon. Use thisinformation to construct a regular octagon by repeatedly rotating asegment by this angle. Save or print your regular octagon.

F

E

D

C

B

A

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Lesson 5.2 • Exterior Angles Demonstration

If you extend each side of a convex polygon in one direction (either allclockwise or all counterclockwise), you create a set of exterior angles.Here you’ll investigate the sum of the measures of one set of exteriorangles in a polygon.

SketchStep 1 Open the sketch Exterior Angles Demo.gsp.

Step 2 Drag each vertex of the pentagon and observe howthe exterior angle measures change.

Step 3 Press Show Sum.

Step 4 Drag each vertex again and observe the effect onthe sum.

Step 5 Make sure the polygon is convex, then press ShrinkPolygon. Observe the effects on the polygon andthe angle measures.

Step 6 To repeat the experiment for another pentagon,press Stretch Back Out, drag one or more verticesto change the polygon, then shrink it again.

Step 7 To try the experiment for a quadrilateral, stretch the polygon backout, press One Less Side, then shrink the polygon.

Step 8 To try the experiment for a triangle, stretch the quadrilateral backout, press Another Side Less, then shrink the polygon.

Investigate1. What is the sum of the measures of one set of exterior angles in any

pentagon, quadrilateral, or triangle?

2. Do you think your answer to Question 1 can be generalized for anypolygon? If so, explain why you think the sum of the measures of oneset of exterior angles is what it is.

3. Explain what happens to exterior angles and their sum when thepolygon is concave.

1

2

34

5

Center of dilation

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Lesson 5.4 • Properties of Midsegments

As you learned in Chapter 3, when you connect the midpoints of two sidesof a triangle, you create a midsegment. The segment connecting themidpoints of the two nonparallel sides of a trapezoid is also called amidsegment. In this activity you will construct midsegments of trianglesand trapezoids and investigate their properties.

Investigation 1: Triangle Midsegment Properties

SketchStep 1 In a new sketch, construct a triangle.

Step 2 Construct the midpoint of each side. (To do this, select each sideand choose Midpoints from the Construct menu.)

Step 3 Construct all three midsegments by connecting the midpoints.

Investigate1. Drag vertices of your original triangle and observe the four small

triangles. How do they compare? Make appropriate measurements tocheck your observations. Then summarize your observations aboutthe four triangles as the Three Midsegments Conjecture.

2. Select a midsegment and change its line width to Thickusing the Display menu. Now find the side of theoriginal triangle that the midsegment is not connectedto and make that side thick as well. Before making anymeasurements, drag a vertex of your original triangleand make predictions about relationships between themidsegment and this third side.

3. Measure the length and slope of both thick segments. Dragvertices of the original triangle and compare the slopes andlengths of these segments. Summarize your observations as theTriangle Midsegment Conjecture.

Investigation 2: Trapezoid Midsegment Properties

SketchStep 1 If you made a tool to construct trapezoids in the previous lesson,

open it. Otherwise, open the sketch Quad Family.gsp. Drag differentparts of the quadrilaterals and determine which one is the trapezoid.Use the ordinary trapezoid, not the isosceles trapezoid. You will usethis quadrilateral for the rest of this investigation, so drag it toanother part of the screen and make it large. If you prefer, you cancopy it and paste it into a new sketch.

A

B

E

D

F

C

m CA � 3.58 cmm ED � 1.79 cmSlope CA � �0.31Slope ED � �0.31

A

B

E

D

F

C

(continued)

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Step 2 The midsegment of a trapezoid is the segment connecting themidpoints of the nonparallel sides. Construct the midsegment ofyour trapezoid.

Step 3 Measure the slopes of the midsegment and the parallel sides.

Investigate

1. Drag vertices of your trapezoid and write a conjecture based onyour observations.

2. Now measure the lengths of the midsegment and the parallel sides.Make a prediction about how the length of the midsegment compareswith the lengths of the parallel sides. Check your prediction usingSketchpad’s calculator. (You may need to use parentheses.) Summarizeyour observations as a second conjecture about the midsegment ofa trapezoid.

3. Now combine your results from your last two conjectures to make a single conjecture about the midsegment of a trapezoid (TrapezoidMidsegment Conjecture).

Steps 2 and 3

Slope CD � 0.10Slope EF � 0.10Slope AB � 0.10m CD � 2.7 cmm EF � 3.1 cmm AB � 3.5 cm

B

C

F

A

D

E

Lesson 5.4 • Properties of Midsegments (continued)

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Using Your Algebra Skills 5 • Equations of Lines Demonstration

In this demonstration you’ll look for relationships between the equation of a line and its slope and y-intercept.

SketchStep 1 Open the sketch Slope Intercept Demo.gsp.

Step 2 Press the Translate and Rotate buttons and observe their effects on the line and on the measurementsand equation.

Step 3 Drag the points y-intercept and drag to rotate line, and observe whatchanges in the sketch. Do not press Make Vertical or Reset untilyou’ve answered the questions.

Investigate

1. Dragging the point y-intercept has the same effect as what action button?

2. What measurement changes when you press Translate?

3. What part of the equation changes when you press Translate?

4. Dragging the point drag to rotate line has the same effect as what button?

5. What measurement changes when you press Rotate?

6. What part of the equation changes when you press Rotate?

7. If a line has slope m and y-intercept b, what is the equation of the line?(This is called the slope-intercept form of the equation.)

8. Make the line horizontal. What is its slope?

9. What is the equation of a horizontal line in slope-intercept form?

SketchStep 4 Press Make Vertical, and observe its effects. Do not press Reset until

you’ve answered the following questions.

Step 5 Press Translate Vertical Line, or drag the point labeled drag to translate line.

Investigate

10. What is the equation of the line when it coincides with the y-axis?

11. What is the slope of a vertical line?

12. Which measurement disappears when the line becomes vertical? Why?

13. What is the equation of a vertical line that has x-intercept c?

14. Explain why the variable y doesn’t appear in the equation of avertical line.

4

2

–2

–5 5

y-intercept

drag to rotate line

y-intercept � 0.00Slope � -0.81equation of line: y � -0.81x

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Lesson 5.6 • Properties of Special Parallelograms

You have discovered some properties that hold true for any parallelogram.Now you will discover some properties that are true only for special typesof parallelograms.

Investigation 1: Do Rhombus Diagonals Have Special Properties?A rhombus is a parallelogram with four congruent sides.

SketchStep 1 Open the sketch Quad Family.gsp. Drag different vertices of

the quadrilaterals and determine which quadrilateral is alwaysa rhombus (but is not always a square). You will use thisquadrilateral for the rest of this investigation, so drag it toanother part of the screen and make it large. If you prefer,you can copy it and paste it into a new sketch.

Step 2 Construct the diagonals of the rhombus.

Step 3 Construct the point of intersection of the diagonals.

Investigate1. Drag the vertices of the rhombus and observe the diagonals. Find

two properties that describe how the diagonals intersect. Makemeasurements to verify your observations. Then summarize yourfindings as the Rhombus Diagonals Conjecture.

2. Now drag different vertices and observe how each diagonal intersectsthe angles of the rhombus. Make some angle measurements to verifyyour observations. Then summarize your findings as the RhombusAngles Conjecture.

Investigation 2: Do Rectangle Diagonals Have Special Properties?A rectangle is an equiangular parallelogram.

SketchStep 1 Look back at the sketch Quad Family.gsp. Drag different vertices of

the quadrilaterals and determine which quadrilateral is always a rectangle (but is not always a square). You will use this quadrilateral for the rest of this investigation, so drag it to another part of the screen and make it large. If you prefer, you can copy it and paste it into a new sketch.

Step 2 Measure at least two of the angles of the rectangle. (You probablywon’t need to measure the others.)

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Lesson 5.6 • Properties of Special Parallelograms (continued)

Investigate1. Drag vertices of the rectangle and write a conjecture about the measure

of its angles.

2. Construct both diagonals of the rectangle. Drag vertices of therectangle and observe the diagonals. Make measurements to verifyyour observations. Then write a conjecture about the diagonals ofa rectangle (Rectangle Diagonals Conjecture).

SketchStep 3 Go back to the sketch Quad Family.gsp. Drag different vertices of

the quadrilaterals and determine which quadrilateral is alwaysa square.

Investigate

3. Test the conjectures you made for rhombuses and rectangles on thesquare. Which of them hold for the square?

4. You may have noticed that you haven’t been given a definition of asquare. Instead, you will write your own.

a. First write a definition of a square that uses the word rhombus.

b. Now write a definition of a square that uses the word rectangle.

5. Using what you know about the diagonals of parallelograms, rectangles,and rhombuses, write a conjecture about the diagonals of a square(Square Diagonals Conjecture).

EXPLORE MORE

In Questions 1–3, construct each parallelogram. Drag vertices to make sureyour construction has enough constraints to keep the polygon you want,but not too many to represent some versions of the polygon.

1. Construct a rhombus.

2. Construct a rectangle.

3. Construct a square.

4. Construct a parallelogram.

a. Bisect a pair of opposite angles of the parallelogram. Make aconjecture about these angle bisectors. Explain why yourconjecture is true.

b. Follow the procedure in part a for a pair of consecutive angles.D

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44 CHAPTER 5 Discovering Geometry More Projects and Explorations


EXPLORATION

Quadrilateral Linkages

How can the wheels of a vehicle turn smoothly, withoutsideways drag, if the steering mechanism regulates only oneaxle? In 1818, German engineer George Lenkenspergerinvented a device that solved this problem. German-Britishprinter Rudolf Ackermann (1764–1834) patented this devicefor him in England. Of course, Lenkensperger andAckermann worked with horse-drawn carriages then.Automobiles were not invented until about 70 years later, butthis invention is still called the Ackermann steering linkage,and you can see it if you look carefully under a car.

In this activity you will investigate the properties of quadrilateral linkageslike the Ackermann steering linkage. You will need Geostrips, or cardboardstrips with paper fasteners, or small wood strips with nuts and bolts tobuild each model. You will also need string and cellophane tape.

Activity: Quadrilaterals at WorkStep 1 Begin by building two linkages. At right is a

parallelogram linkage, and below right is an isoscelestrapezoid linkage.

Step 2 Does the parallelogram linkage stay a parallelogramwhen the linkage is moved around? Does the isoscelestrapezoid linkage stay an isosceles trapezoid when thelinkage is moved? What might be a better name forthis quadrilateral?

Step 3 Set the parallelogram model vertically on your desk by holding thebottom bar AB�. Place the palm of your other hand lightly on thetop bar CD�� and feel the motion of the bar as you move yourhand back and forth. Do this with the isosceles trapezoid linkage,too. What is the difference in “feel” of the two sideways motions?

Step 4 Lay the parallelogram model flat on your desk, and hold or clampthe bottom bar AB� to the desk so that it is fixed. Tug lightly onthe top bar CD�� at point C. What happens? What path (locus ofpoints) can point C take? Can you make point C collinear withpoints A and B? With points A and D? Now do this with theisosceles trapezoid linkage and answer the same questions. (continued)

Center ofturning circle

Inner wheel traces a smallercircle, so it must be turnedat a greater angle..

A B

CD

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Discovering Geometry More Projects and Explorations CHAPTER 5 45©2008 Key Curriculum Press

Step 5 When the bar AB� is fixed, what geometricfigures describe the paths of points C and Dfor the parallelogram linkage? For theisosceles trapezoid linkage? Test yourconjecture by temporarily replacing thepaper fasteners at points C and D withpencil points to draw the paths.

Step 6 The key to smooth turning is to have the wheels always facing at right anglesto the line drawn from the center ofthe wheels to the center of the turningcircle, as shown in the figure in theintroduction. A top view of the linkagein a car’s steering mechanism is shownat right. What type of linkage, parallelogram or trapezoid, does this linkage most resemble? Explain how thislinkage is used to create a smooth-turning vehicle. To help explainthe functioning of this mechanism, you might read about linkagesin a book on mechanical design or auto repair.

Questions

1. Quadrilateral linkages are used for many different mechanisms. Studythe sewing box and rocking horse pictured here. Explain why atrapezoid linkage is used in the rocking horse and why a parallelogramlinkage is used in the sewing box. What is the difference between thedesired motions in these mechanisms?

2. Pop-up cards are another example of parallelogram linkages. Find apop-up card or book and explain how it was made.

3. Some kinds of patio chairs use a quadrilateral linkage like the oneat right. The chains hanging from each arm are attached to thefront and back of each end of the seat. The chains swing freely sothat the seat can swing back and forth. Does the seat always tilt atthe same angle from a horizontal position as it swings back andforth? Model the chair’s chains with two pieces of string taped tothe end of your desk so that they hang down. Tie them to the endsof a pencil to model one end of the seat.

Exploration • Quadrilateral Linkages (continued)

A B

CD

Tie rod

Front axleTop view of steering linkage

Driving straight Turning right

Arm

Chains

Seat

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PROJECT

Symmetry in Snowflakes

You may remember folding and cutting paper snowflakes. In this projectyou’ll fold paper into different polygons and use them to make snowflakes.

Folding a Pentagon

Step 1 Start with a piece of paper that is approximately 9 in. by 12 in.(a piece of notebook paper will work), and begin with paper inhorizontal position.

Step 2 Fold in half with vertical crease EF, bringing left edge AD over toright edge BC.

Step 3 Fold F to A and make a small crease marking the midpoint of FA�(point G). Open the paper back up to Step 2.

Step 4 Fold E to G and make crease HJ.

Step 5 Match HJ� on top of HE� and make crease HK. This crease shouldbisect ∠JHE.

Step 6 Fold the �HFG flap backward, under the paper, using crease HKas your guideline to fold along.

Step 7 Flip over by rotating about point H. Cut along FG�.

Step 8 Keep �HFG. If you open it now, you will get a pentagon. If youwant to make a snowflake, cut along the edges before opening.

(continued)

A B

D C

Step 1

AF

DE

Step 2

AF

DE

G

Step 3

AF

J

H

G

E

Step 4

F

J

HK

G

Step 5

J

H K

G

Step 6

G

F

J

H

K

Step 7

FH

G

Step 8

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GAME

Matho (page 1 of 4)

Matho Board

Matho

MATH


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GAME

Matho (page 2 of 4)

Chapter 5 Review ExercisesFor Exercises 1–13, complete each statement using thebest answer from your Matho Board.

1. A quadrilateral with four congruent sides is called a .

2. The sum of the measures of the interior angles of a pentagon is.

3. The sum of the measures of the exterior angles of a pentagon is.

4. Any angle in a regular octagon measures .

5. The measures of the interior angles of a(n) have asum of 720°.

6. The nonvertex angles of a kite are .

7. The diagonals of a kite are .

8. The consecutive angles between the bases of a trapezoid are.

9. The midsegment of a triangle is times the length ofthe base.

10. The midsegment of a trapezoid is the of the lengthsof the bases.

11. A rhombus is a(n) with perpendicular diagonals.

12. A square is a(n) with congruent sides.

13. If a trapezoid has congruent base angles, it is .

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75�

165�

d e

a

c

b

105�


GAME

Matho (page 3 of 4)

Chapter 5 Review ExercisesFor Exercises 14–19, answer each question and find thebest answer on your Matho Board. You may want tosketch a picture first.

14. The midsegment of a trapezoid measures 3 cm. Find the length ofone base if the other base measures 4.2 cm.

15. A student cuts a polygon along two lines and forms three congruentequilateral triangles. What shape was the original polygon?

16. A student reflects an obtuse triangle across the side opposite itsobtuse angle. What quadrilateral consists of the original triangle plusthe image triangle?

17. A kite has angles of measures 80° and 140°. Find the measure ofanother of its angles.

18. The measure of an exterior angle of a regular polygon is 30°. Findthe number of sides of the polygon.

19. Describe polygon ABCD if A has coordinates (2, 1),B has coordinates (1, 2), C has coordinates (4, 5),and D has coordinates (4, 3).

In Exercises 20–24, find the measure ofeach angle in the diagram.

20. a

21. b

22. c

23. d

24. eM

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Discovering and Proving PolygonProperties

C H A P T E R

5Chapter 5 extends the explorations of triangle properties from the previous chapterto examine properties shared by all polygons. Students begin by investigating thesums of interior and exterior angles of any polygon. The chapter then focuses onquadrilaterals, which are polygons with four sides. Students explore relationshipsamong the sides, angles, and diagonals of different special quadrilaterals, includingthe parallelogram family.

PolygonsThe chapter begins with conjectures about polygons in general. Students experimentto form conjectures about the sum of the angles of any polygon, and the sum of theexterior angles of any polygon. They write a paragraph proof of the first conjecture,relying on the Triangle Sum Conjecture from Chapter 4.

QuadrilateralsThe book considers properties of three categories of quadrilaterals,as in the diagram: kites, trapezoids, and parallelograms.

Students explore two kinds of parallelograms, rhombuses and rectangles, as well as squares, which are both rhombuses and rectangles. Students discover properties of all types of quadrilaterals,including how their diagonals are related. In the case of trapezoids,students investigate midsegments, which they relate to midsegments of triangles.

Properties of various quadrilaterals can be seen from their symmetry.A kite has reflectional symmetry across the diagonal through its vertex angles; an isosceles trapezoid has reflectional symmetry across the line through the midpoints of the parallel sides; and a parallelogram has 2-fold rotational symmetry about the point at which its diagonals intersect.These symmetries can help explain why certain pairs of segments or angles arecongruent or perpendicular.

Summary ProblemMake a copy of the quadrilaterals diagram above, but with large boxes. Write in eachbox the properties of that kind of figure as you encounter them in the book.

Questions you might ask in your role as student to your student:

● If you add a box above the picture for polygons in general, what properties canyou put into that box?

● What other kinds of polygons might go into an expanded diagram?

● Where might isosceles trapezoids be added to your diagram?

● Where might darts be added to your diagram?

● What properties can you think of that aren’t already in your diagram?

● Do you see any patterns in what properties are shared by various kinds ofpolygons?

Quadrilateral

TrapezoidKite

Parallelogram

Square

RectangleRhombus

Discovering Geometry A Guide for Parents CHAPTER 5 23©2008 Key Curriculum Press

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Chapter 5 • Discovering and Proving Polygon Properties (continued)

Sample Answers

24 CHAPTER 5 Discovering Geometry A Guide for Parents


A rhombus has all the characteristics of aparallelogram. Four sides are congruent.

Diagonals are perpendicular.It has two lines of symmetry and

2-fold rotational symmetry.

A kite has exactly two distinct pairs of congruent sides. The nonvertex angles are congruent.

The diagonals are perpendicular. The diagonalbetween the vertex angles bisects the other

diagonal. It has exactly one line of symmetry.

A rectangle has all the characteristics of aparallelogram. Angles are all right angles.

It has two lines of symmetry.

A quadrilateral is a polygon with four sides.The sum of the interior angles is 360�.

Quadrilateral

A trapezoid has exactly one pair ofparallel sides. Two pairs of adjacent

angles are supplementary.

Trapezoid

An isosceles trapezoid has two pairs of congruentangles and at least two congruent sides.Its diagonals are congruent. It has one

line of symmetry.

Isosceles trapezoid

Rectangle

A square has all the characteristics of a rhombus and of a rectangle.

It has four lines of symmetry.It has 4-fold rotational symmetry.

Square

Rhombus

Kite

The opposite sides of a parallelogramare congruent and parallel.

The opposite angles are congruent.Adjacent angles are supplementary.

The diagonals bisect each other.

Parallelogram

The sum of the exterior angles is 360�.A polygon of n sides has diagonals.

Polygons

n(n � 3)________2

Making each box in the shape of the polygon whose properties it contains mightmake the chart more interesting.

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Chapter 5 • Review Exercises

Name Period Date

1. (Lessons 5.1, 5.2) Find the sum of the measures of the interior angles of aregular 14-gon. Then find the sum of the exterior angles.

(Lessons 5.1, 5.2, 5.4) For Exercises 2 and 3, find the lettered measures in each figure.

2.

3. Given that CD � AF,

BE � ?—.

m�ABE � ?—.

m�CDF � ?—.

4. (Lesson 5.3) Given kite ABCD, find the missing measures.

5. (Lesson 5.5) The perimeter of parallelogram ABCD is 46 in. Find the lengths of the sides.

6. (Lesson 5.6, 5.7) Draw a diagram and write a paragraph proof to show that thediagonals of a rectangle are congruent.

A B

CD

x � 4

3x – 1

B

CA

c

a

D

b

60�

10 cm

D C

F A

BE 75� 70�

32 cm

15 cm

a

b

c

Discovering Geometry A Guide for Parents CHAPTER 5 25©2008 Key Curriculum Press

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5. 2(3x � 1) � 2(x � 4) � 46 Opposite sides of aparallelogram arecongruent.

6x � 2 � 2x � 8 � 46 Distributive property.

8x � 6 � 46 Combine like terms.

8x � 40 Subtraction.

x � 5 Division.

AB � 3(5) � 1 � 14 in. Substitution.

AD � 5 � 4 � 9 in. Substitution.

6. Sample answer:

By definition, all angles of a rectangle are congruent,so �ABC � �DCB. A rectangle, like any parallelo-gram, has opposite sides congruent, so AB�� DC��.Because it is the same segment, BC�� BC�. Thus�ABC � �DCB by SAS, and AC�� DB� by CPCTC.Therefore, the diagonals of a rectangle are congruent.

A B

CD

1. Interior Angles:

Interior angle sum � 180 (n � 2) � 180 (14 � 2)� 2160°

Exterior Angles � 360° for all polygons

2. For the hexagon:

Interior angle sum � 180 (n � 2) � 180 (6 � 2) � 720°

Each angle � �72

60°� � 120°

a � 120°

b � 60° Linear pair.

c � 60° Triangle sum.

3. BE � �15 �

232

� � 23.5 cm Midsegment.

m�ABE � 110° Linear pair.

m�CDF � 105° Supplementary angles.

4. a � 90° Diagonals of a kite are perpendicular.

b � 10 cm Definition of a kite.

c � 30° Triangle sum.

26 CHAPTER 5 Discovering Geometry A Guide for Parents


S O L U T I O N S T O C H A P T E R 5 R E V I E W E X E R C I S E S

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Quiz 6 (For use anytime after Discovering Geometry, Using Your Algebra Skills 2.)

Name Period Date

Discovering Mathematics STANDARDIZED TEST PREPARATION 21©2004 Key Curriculum Press

1. The cylinder shown at right is intersected by a plane perpendicular tothe cylinder’s bases. What shape is the cross-section formed by theintersection?

circle

oval

rectangle

trapezoid

triangle

2. A diagram of a steering wheel is shown at right. Which line in the diagram represents a line of symmetry?

1

2

3

4

None of these

3. An employee at a music store is making a display of CDs, as shown inthe picture at right. If the pattern continues, how many CDs will be in the display after step 4 is completed?

36

44

52

56

64E

D

C

B

A

K

J

H

G

F

E

D

C

B

A

(continued)

1

2

3

4

Step 1

Step 2

Step 3

CD Display

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22 STANDARDIZED TEST PREPARATION Discovering Mathematics


4. In the figure at right, CE___

bisects �BAD. Show your work,explain in words, or provide a proof showing that �BAE is congruent to �DAE.

5. A scuba diver measures the visibility at various depths in a lake. Thegraph shows the results.

a. Determine the slope of a line connecting the points on the graph.Give mathematical evidence to justify your answer.

Slope __________________________________________________________________

b. Explain what the slope of the graph represents.

________________________________________________________

________________________________________________________

________________________________________________________

30100 20

Depth (feet)

Vis

ibil

ity

(fee

t)

Lake Visibility

100

80

60

40

20

040 50 60

B

C

E

DA

50˚50˚

Quiz 6 (continued)

Name Period Date

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