Extra Practice ........................................................................................................

_'Describe a pattern in the sequence of numbers. Predict the next number. (Lesson 1.1)

1. 16,8,4,2,1, .... 11 2.1,2,4,7,11,... 3.1,5,25,125, ... powers of 5; 625 multiply by -r 2' add 1. add 2. add 3. add 4....; 16

4.7,2,2,8,2,2,9,2,2, . . . 5.32,48,72, 108, . . . 6.2, -6, 18, -54, ... multiply by -3; 162 add 1.2.2. add 1; 10 multiply by 1.5; 162

7. Complete the conjecture based on the pattern you observe in the specific cases. (Lesson 1.1)

Conjecture: Any negative number cubed is _7_. negative

-13 =-1 -73 = -343 -33 = -27 -93 = -729 -53 =-125 -1l3=-1331

8. Show that n" + I > (n + 1)" for the values n = 3,4, and 5. Then show that the values n = 1 and n = 2 are counterexamples to the conjecture that

34n" + I > (n + 1)". (Lesson 1.1) = 81 > 43 =64; 45 =1024> 54 =625; 56 =15.625 > 65 =7776; 12 = 1 >- 2' = 2; 23 =8 ? 32 = 9

S,ketch the points. lines. segments, planes, and rays. (Lesson 1.2) 9-13. Check sketches.

9. Draw four collinear points A, B, C, and D.

Draw two opposite rays MN and MP. 1. Draw a plane that contains two intersecting lines.

Draw three points E, F, and G that are coplanar, but are not collinear.

Draw two points, Rand S. Then sketch !?S. Add a point T on the ray so that S is between Rand T.

diagram of the collinear points, AE =24, C is the midpoint of AE, 8, and DE = 5. Find each length. (Lesson 1.3) '2 G! 7

4 15. AD 19 • !·I

A B c D E 11 17. AC 12

7 19. BE 16

the Distance Formula to decide whether HM =ML. (Lesson 1.3)

21. H(3, -1) 22. H(-5, 2) M(8,2) M(-4,6) L(3,5) yes L(-6, 2) no

the vertex and sides of the angle. then write two names for the angle. (Lesson 1.4)

25.24. A

B

Ca -->- -->- -->- -->­ -->- -->­

a; ap ,aR ; L paR. L Rap F; FG. FE; LGFE. LEFG B; BA • BC; LABC. L CBA Extra Practice

Use the Angle Addition Postulate to find the measure of the unknown angle. (Lesson 1.4)

26. mL STR = --'1 70° 27. mLHJK = _?_ 65°

H

28. mLDEF = _?_ 90°

.­---~';::"'~J

L

s

State whether the angle appears to be acute, right, obtuse, or straight. Then estimate its measure: (Lesson1.4)

29.~. 3·· L 31 .

obtuse; =150° acute; ""25° right; =90°

F

D

Find the coordinates of the midpoint of a segment with the given endpoints. (Lesson 1.5)

32. P(-4, 2) 33. P(-l, 3.5) 34. P(-12, 4) Q(8, -4) (2, -1) Q(7, -5.5) (3, -1) Q(-3, -6) (-7.5, -1)

xv is the angle bisector of L UXB. Find mL UXY. (Lesson 1.5)

35. 42° UL

84° X B

Find the measure of each angle.

36. 25°

(Lesson 1.6)

X

37. 74°

38. Two vertical angles are complementary. Find the measu!"e of each angle. 45°

39. The measure of one angle of a linear pa~r is 3 times the measure of the other angle. Find the measures of the two angles. 45°,135°

40. The supplement of an angle is l30°. Find the complement of the angle. 40°

Find the perimeter (or circumference) and area of the figure. (Where necessary, use 1T "" 3.14.) (Lesson 1.7)

41.

0' 42.

,~ 43.~ 12 10

4 13

28; 49 12; 6 36; 60 45.

G 46. 0, 47.

o'~ 9.25

31 .4; 78.5 12; 9 33; 67.0625

Student Resources

44.

D·18

52; 144

~8 12

37.68; 113.04

IAIPTER 2

{rite the conditional statement in if-then form. (lesson 2.1)

It must be true if you read it in a newspaper. If you read it in a newspaper, then it must be true.

An apple a day keeps the doctor away. If you have an apple a day, then it will keep the doctor away.

The square of an odd number is odd. If a number is odd, then its square is odd.

te the inverse, converse, and contra positive of the conditional ement. (lesson 2.1)

2If x = 12, then x = 144. If x *" 12, then x2 *" 144; If,'(2 = H4, then x = 12; If x2 *" H4, then)( *" 12.

If you are indoors, then you are not caught in a rainstorm. See margin.

If four points are collinear, then they are coplanar. If four points are not collinear, then they are not coplanar; If 4 points are coplanar, then they are collinear; If four points are not coplanar, then they are not collinear. If two angles are vertieal angles, then they are congruent.

See margin. te the converse of the true statement. Decide whether the converse is lor false. If false, provide a counterexample. (lesson 2.1)

If two angles form a linear pair, then they are supplementary. See margin.

If 2x - 5 = 7, then x = 6. 1It }( = 6, then 2]( - 5 = 7; true.

~rite the biconditional statement as a conditional statement and its verse. (lesson 2.2)

Two segments have the same length if and only if they are congruent. If tvvo segments have the same length, then they are congruent; If two segments are congruent, then they have the same length."

Two angles are right angles if ana only if they are supplem(wtary. If two angles are right angles, then they . . 2 are supplementary; If two angles are supplementary then they are right angles.

x = 10 If and only If x = 100. 2If x = 10, then x2 =100; If x = 100, then x = 10.

ermine whether the statement can be combined with its converse to n a true biconditional statement. (lesson 2.2)

If LABC is a right angle, then AB .1 Be. yes

If L 1 and L2 are adjacent, supplementary angles, then L I and L2 form a linear pair. yes

If two angles are vertieal angles, then they are congruent. no

ng p, q, r, and s below, write the symbolic statement in words. (lesson 2.3) 16-21. See margin.

p: We go shopping. r: We stop at the bank. 17: We need a shopping list. s: We see our friends.

17. ~r~ ~s 18. r~ s

20. ~p~ ~s 21. P ~ r

en that the statement is of the form p --t q, write p and q. Then write the !rse and the contrapositive of p --t q both symbolically and in words. (lesson 2.3) 22-24. See margin.

If it is hot, May will go to the beach.

If the hockey team wins the game tonight, they will play in the championship.

If John misses the bus, then he will be late for school.

Extra Practice

5. If you are not indoors, then you are caught in a rainstorm; If you are not caught in a rainstorm, then you are indoors; If you are caught in a rainstorm, then you are not indoors.

7. If two angles are not lIertical angles, then they are not congru ­ent; If two angles are congruent. then they are vertical angles; If two angles are not congruent, then they are not vertical angles.

8. If two angles are supplementary, then they f01m a Ii near pai r; false. For example, two oonsecutive angles of a parallelogram are supplementary, but they do not form a linear pair.

16. If we go shopping, then We need a shopping list.

17. If we don't stop at the bank, then we won't see our friends.

18. If we stop at the bank, then we see our friends.

19. We go shopping if and only if we need a shopping list.

20. If we don't go shoppilJg, then we won't see our fri ends.

21. We go shopping if and only if we stop althe bank.

22. p: It is hot. q: May will go to the beach. -p --7 -q; If it is not hOI, May Wi ll not go to the beach. .. -q --7 "'p; If May does not goto the beach, thell it is not hot.

23. p: The .hockey t8am wins the game tonighJ.. q: They WUI play in the Championship RoulJd. -p --7 ~q; Ifthe hockey team doesn't win the game t6IJight, they won't play in the Championship Round. -q -4 "'P; If the bockey team doesn't play in the Championship Round, then they didn't win the game tonight.

24. p: John misses the bus. q: He will be late for school.

. -p --7 -q; If John doesn't miss the bus, then he won't be late for school. -q --7 -p; If John isn't late for sphoo!, then he did not miss t~e bus.

805

~tements (Reasons) , L1 e L3 (Vertical angles are congruent)

. L4 ill ..&.2 (Vertical angles are congruent)

i. L1 and L411,e complementary (Given)

I. IJIL1+mL4 =90° (Definition of complementary)

i. mL1 = mL3 (Definition of congruence)

ii. mL 4 =mL2 (Definition of conumence)

1. mL3 + mL2 =90° (SiJbstitution property of equality)

8. L 3 and L2 are complementary (Definition of complementary)

Given

(Lesson 2.6)

GIVEN ... L I and L 4 are complementary, LDBE is a right angle.

o PROVE L 2 and L 3 are complementary.

See margin.

Student Resources

Use the p~operty to complete the stat~ment. (Lesson 2.4)

25. Reflexive property of equality: AB = _?_. AB

26. Symmetric property of equality: If ED = DF, then ~. OF = EO . . 27. Transitive property of equality: If AB = AC and AC = DF, then _?_.AB:: OF

28. Division property of equality: If 2x = 3y, then ~ =~. 3:

29. Subtraction property of equality: If x = 6, then x - 4 = _?_ . 6 _ 4 (or 2)

Copy and complete the proof using the diagram and the given information. (Lesson 2.5)

30. GIVEN '" PD == PC, P is the midpoint of AC and BD

PROVE ... AP == BP

Statements Reasons

1. P is the midpoint of AC and BD. 1. _?_

2. AP = PC 2. _?_ Definition of a midpoint

3. BP = PD 3. _?_ Definition of a midpoint

4. ~? _ PO == PC 4. Given

5. PD = PC 5. _?_ Definition of congruent segments

6. _?_ AP= BP 6. Transitive property of equality

7. AP == BP 7. Definition of congruent segments

In Exercises 31-32, use the diagram to complete the statement. (Lesson 2.6)

31. L2 and _?_ are vertical angles. L6

32. L QWR is supplementary to _?_. L vwa or L RWU

33. In the diagram. suppose that L3 and L4 ·are complementary and that L4 and V L 5 are complementary. Prove that L!3 == L 5. L3.., L5 by the Congruent Complements Theorem

Solve for each variable. (Lesson 2.6)

34. 35.

z:: 13; x :: 10 b =8; c :: 27

36.

37. Write a two-column proof. (Lesson 2.6) r =25; s =.10

c

806

HAPTER 3 ...­___ _ ..... E

:Think of each segment in the diagram as part of a line. ill in the blank with parallel, skew, or perpendicular. Ufsson 3.1)

1. HA and EC are _ ?_. para'lIel

---­ <----> ?2. FD and AD are _._. perpendicular

skew

Think of each segment in the diagram as part of a line. here may be more than one right answer. (Lesson 3.1)

t---? of--+ +---+ +---+ 4. Name a line parallel to AD. Sample answers: HF, Be, GE

N I· d' I G<->B +--+ +--+ +--+ +->5, arne a me perpen ICU ar to . Sample answers: AB, Be, GE, HG ~ +--+ +--+ +--+ +--+

6. Name a line skew to EC. Sample answers: GH, AB, HF, AD

c

7. arne a plane para lei to GBC. Samp e answers: HAD. ADF. DFH. FHJ~-"""--""""--"""-~..:o._.........:........

Complete the statement with corresponding, alternate interior, alternate exterior, or consecutive interior. (Lesson 3.1)

8. L 3 and L 7 are _?_ angles. corresponding

9. L 4 and L 6 are _ ?_ angles. alternate interior

~O . L 8 and L 2 are _?_ angles. alternate exterior

11. L4 and L5 are _?_ angles. consecutive interior

12. L5 and L 1 are ~?_ angles. corresponding \

~3. Fiij'ln the blanks to complete the proof. (Lesson 3.2)

GIVEN' . AB 1. BC, jjjj bisects.LABC

mLABD = 45°

Statements Reaso ns

B

1. AB .lBC 1. _?_ Given

c

2. _ ?_ LABe is a right L. 2. Detinition of perpendicular lines

3. mLABC = 90° 3. _?_ Definition of right L

4. jjjj bisects LABC 4. _?_ Given

5. mLABD = mLDBC 5 . _?_ Definition of L bisector

6. mLABD + mLDBC = 90°

7. mLABD + ? = 90° 8. 2(mLABD) = 900 mLABD

6. _?_ If 2 sides of 2 adj. acute .1! are .t, then the .1! are complementary.

7. Substitution property of equality

8. _?_ Distributive property

9. mLABD = 45° 9. _?_ Division property of equality

Extra Practice

807

:3. Corresponding angl!ts a're congruent.

:4. Consecutive angles are supplementary.

:5. Alteroate interior angles are congruent.

:6. All slo.pes are 1; AB II CO II EF '1. AB: 0; CO: -t; £I:- ~!; none

:8. All lines are vertical; )

AB ll COllff

B08

Find the values of x and y. Explain your reasoning . (Lesson 3.3)

14. 15.

x =30; Y =150

16.

x=95;y=85

17. 19.

x =110; y =70 x = 118; Y = Which lines, if any, are parallel? Explain. (Lesson 3.4)

21.20. 22........ 9=".----i=,..­

none

Explain how you would show that a II b. State any theorems or postulates that you would use. (Lesson 3.5) 23-25. See margin.

25, 23.

130"24'M 70" 70·51r.

+-+ +-+ +-+Find the slopes of AB, CD, and EF. Wl;1ich lines are parallel, if any? (Lesson 3.6) 26-28. See margin.

26. A(3, 7), 8(1,5) 27. A(-4, 1),8(3, 1) 28. A(-3 , 2), 8(-3, 5) C(4, 1), D(9, 6) C(-2, -1), D(4, -3) C(7, -1), D(7, 7) E(2, 5), F(-8, -5) E(-1O, 3), F(4, -8) E(4, -11 ), F(4, -6)

Write an equation of the line that passes through point P and is parallel to the line with the given equation. (Lesson 3.6) y = _.!x _ 2

1 2 29. P(-4, -5),), = 6x - 7 30. P(2, -3),), = -Z-x + 4 31. P(-9,8), x = -12

y = 6x + 19 x =-9 Decide whether lines p, and P2 are perpendicular. (Lesson 3.7)

32.linepl : -7), + 3x = 6 33. line PI: 3y + 12x = 15 34. 1inepl : 16)' - 2x = 11 line P2: -9y - 21x = 3 yes line P2: 8)' - 2x = 9 yes linep2: -12x - 2y = 6 no

Line i is perpendicular to the line with the given equation and line i passes through P. Write an equation of line j. (Lesson 3.7)

135. Y = -2x + 1, P(4, -1) 36. 2x + 5)' = 20, P(4 , 10) 37. Y = Z-x + 6, P(-2, -7)

1 5 y = IX - 3 Y=IX y=-2x-11

Student Resources

118

1

10 Exercises 1-4, the variable expressions represent the angle measures of a triangle. Find the measure of each angle. Then classify the triangle by its <8ngles. (Lesson 4.1)

2. mLH = 60° 3. mLP = XO 4. mLS = (2x)0 mLF = 3xo mLK = XO mLQ = (2x + 10)° mLT= (2x - 4)° 'mLG = 5xo mLL = XO mLR = (x + 10)° mL U = (2x - 2)° 20,60,100; obtuse 60,60; equiangular 40,90,50; right 62, 58, 60; acute

5. The measure of an exterior angle of a right triangle is 135°. Find the me:lsures of the interior angles of the triangle. (Lesson 4.1) 90°,45°,45°

8. none; corresponding sides ·Utltl••ntifv any figures that can be proved congruent. For those that can be are congruent, but no

proved congruent, write a congruence statement. (Lesson 4.2) information is given about the corresponding angles. t:,ABC == t:, FED ABGH ==

8. B H

~D F 0 M K

E L A~ 7AtS !~~': AOe GOJ

A F H G F

9. Use the tri:lngles in Exercise 6 above. Identify all pairs of congruent conesponding angles and corresponding sides. (Lesson 4.2) LA, L F; L B, L E; L C, L D

Use the given information to find the value of x. (Lesson 4.2) AB, FE; BC, ED; AC, FD

10.LQ==LT. LR==LH 11. LE == LK. LF== L M 12. LA == L C. LBDA == L BDC 13 M J '6 B

E~ l~~A"V'D ~ ~K 0 eA s R T G-~ F

_1III[)ecide whether enough information is given to prove that the triangles are If there is enough information, state the congru"!nce postulate_

would use. (Lesson 4.3) .

.6XVY.6ZVW 14. 6MPN. 6QPR 15. 6BCE == 6DCE

Xt2sJ MyN A Y WaR 0 B SSS no SAS

16. Use the diagram in Exercise 13. Prove that 6XYW == 6 ZWY. See margin.

possible to prove that the triangles are congruent? If so, state the ,unl"""'" postulate or theorem you would use. (Lesson 4.4) .

18.

Extra Practice : , •

16. Statements (Reasons)

1. XV :i lV, VY e VW!Given) 2. XV=ZIl, VY~ VW(Definition of

cOJlgruent se.gments ) 3. XV + VW = Xw. ZV + VY= lY

(Segment Addition Postulate) 4. ZV+ VY~ XW(SUbstitution

property of equality) 5. XW= lY(Substitution ptoperty

of equality)

6. XW e zy (Definition of congru­ent segments)

7. XYe lW (Given)

8. YW =: YW (Refl exive property of equality)

9. 6XYW =: ,e,zwr (SSS Congruence Postu late)

SO!

\ 20. Statements (Reasons)

1.~O II BC (Gil'en) 2. LADE ;;! L CBE (If 211 lines are

cut by a transv., then alt. int. Ll are =.)

3. LAEO =LeEB(Vertical'Ll are D.)

4. AC bisects BO (Given) 5. BE=EO (Def. of segment

bisector) , 6. LAEO~ LCEB

(ASA Congruence Postulate) 24. Paragraph proof: Given that

LBAF. .. AFBO,LAFB = e.BOF by corresp. parts of s It,: are "'.

25. Paragraph proof: Given that

LCBO s ABAF. BC =AB by cor­resp. parts of g '& are =.

26. Paragraph proof: Given that LFBOs LOFE,FO= OEby corre­sponding parts of =.&.are s.

30. Sample answer: ii!' rrTi Vl~' -t- --+-, - .:.....tll I!- A(-5.~Ti ~( ~ 1.' 2) , '-- .-_. ·";1-L -- --1]I

! ! !

Write a two-colum", proof or a paragraph proof. (Lesson 4.4)

20. GIVEN II- AD " BC, AC bisects BD A~

PROVE ... LAED: LCEB See margin, o - e

State which postulate or theorem you can use to prove that the triangles are congruent. Then explain how proving that the triangles are congruent proves the given statement. (Lesson 4.5)

21. PROVE II- AB: CD 22. PROVE . L GEF : L GHJ 23. PROVE II- LRQT== LRST

/5;7'............... '\/4DeE J

ASA; Corresp, parts of : .& are :, SAS; corresp, parts of := .& are:=, SSS; corresp. parts of := .& are :=.

Use the diagram and the information given below. (Lesson 4.5) 24--26. See margin. e

GIVEN II- LCBD == LBAF LBAF: LFBD LFBD == LDFE

24. PROVE .. LAFB =' LBDF

25. PROVE BC : AB A E

26. PROVE .. FD : DE

Find the values of x and y. (Lesson 4.6)

27. r::-::::-------:~ 28. yO29'~

XO

x = 60; Y = 60 x = 65; y = 77.5 x = 45; y = 45

Place the figure in a coordinate plane. Label the vertices and give the coordinates of each vertex. (Lesson 4.7) 30.31 , See margin.

30. A 4 unit by 3 unit rectangle with one vertex at (-5 , 2)

31. A square with side length 6 and one vertex at (3, -4)

In the diagram, !iEFG is a right triangle. Its base is 80 units and its height is )'

G60 units. (Lesson 4.7)

32. Give the coordinates of points F and G. F( -30. -20); G( -30.40)

33. Find the length of the hypotenuse of LEFG. 100

Place the figure in a coordinate plane and find the given information. (Lesson 4.7)

34. A rectangle with length 6 units and width 3 units; find the length of a diagonal of the rectangle. 3\/5

F

35. An isosceles right triangle with legs of 7 units; find the length of the hypotenuse, Nz

Student Resources

810

CHAPTER 5

Use the diagram shown. (lesson 5.1)

1. In the diagram, DB 1. AC and BA == BC. Find BC. 12 EI~C

2. In the diagram, Dl3 1. AC and BA == BC. Find DC. 20 '8 0

13 12 . 20

.3. In the diagram, i5iJ is the perpendicular bisector of AC. _A Because EA = EC = 13, what can you conclude about the point E? Eis on DB.

Use the diagram shown. (lesson 5.1)

4. In the diagram, m L FEH = mLGEH = 30°, mLHGE = mLHFE = 90°, and HF = 5. Find HG. 5

5. In the diagram, Eil bisects L JEM, mLEJK = m LEMK = 90° and JK = MK = 10. What can you conclude about point K? K is on EH

In each case, find the indicated measure. (Lesson 5.2)

6. The perpendicular bisectors of 7. The perpendicular bisectors of 6.ABC meet at point D. Find AC. 16 6.EFG meet at point H. Find HJ. 9

&EI J 16 ~ ~24--jA 6 C

8. The angle bisectors of 6.RST 9. The angle bisectors of 6.AEC meet at point Q. Find WS. 24 meet at point G. Find GF. 10

T

., R

Use the figure below and the given information . (Lesson 5.3) T is the centroid of 6.ABC, BT = 14, XC = 24, and TZ = 8.5.

21

11. Find the length of TX.. 8

17 A x 8

and label a large triangle of the given type and construct altitudes. Theorem 5.8 by showing that the lines containing the altituijes are

Concurrent and label the orthocenter. (lesson 5.3) 13-15. Check sketches.

14. an equilateral 6.DEF 15. a right isosceles 6.STR

The orthocenter should be The orthocenter should be at the equidistant from the three vertex of the right angle of the vertices of the triangle. triangle.

Extra Practice

811

812

Use t::.ABC, where X, Y, ·and Z are midpoints of the sides. (Less,?n 5.41

16. CB II -.L . ZX 17. XY II-?_. AC

18. IfAB = 8, then yz = _?_. 4

. 19. IfAC = 10, thenXY = _ 7 _ . 5

20. IfXZ = 6, then BC = _?_ . 12

21. If yz = 4x - 11 andAB = 3x + 3, then YZ = _7_ . 9

22. IfAZ = 4x - 5 and XY = 2x + 1, then AC = _?_ . 14

Name the shortest and longest sides of the triangle. (Lesson 5.51

23.

A BC.AC

C

24. 0 ~:::::----==-7 E

OF. DE

A -

~ C Y B

25.

J

GJ. GH Name the smallest and largest angles of the triangle. (Lesson 5.51

28.26.

K~L 27.

~: a 10 P

DT 7 S

LL.LM L D.LP LT,LR

Complete with >, <, or =. (Lesson 5.61

29. AC .1.. DF < 30. QS.1.. TU . < 31. mL 1.1.. mL2

C &8 S U

32. MN.1.. PR > 33. mL 1.1.. mL 2 = 34. mL 1.1.. m L 2 >

. P

M ~R ~N o

35. JK.1.. ST 36. XY .1.. WV < 37. mL 1 .1.. mL 2 <

Student Resources