state whether each sentence is true or false. if false, replace

State whether each sentence is true or false. Iffalse, replace the underlined term to make atrue sentence.

1. The first part of an if-then statement is theconjecture.

SOLUTION:

The first part of an if-then statement is thehypothesis. So, the sentence is false.

ANSWER:

false; hypothesis

2. The contrapositive is formed by negating thehypothesis and conclusion of a conditional.

SOLUTION:

The contrapositive is formed by negating both thehypothesis and the conclusion of the converse of theconditional. So, the sentence is false. The inverse isformed by negating the hypothesis and conclusion ofa conditional.

ANSWER:

false; inverse

3. A conjunction is formed by joining two or morestatements with the word and.

SOLUTION:

True

ANSWER:

true

4. To show that a conjecture is false, you would providea disjunction.

SOLUTION:

To show that a conjecture is false, you would providea counter example. So, the sentence is false.

ANSWER:

false; counterexample

5. The inverse of a statement p would be written in theform not p.

SOLUTION:

The inverse is formed by negating both the hypothesisand conclusion of the conditional. So, the sentence isfalse. The negation of a statement p would be writtenin the form not p.

ANSWER:

false; negation

6. In a two-column proof, the properties that justify eachstep are called reasons.

SOLUTION:

True

ANSWER:

true

7. The slope-intercept form of a linear equation is .

SOLUTION:

The slope-intercept form is y = mx + b. So, thesentence is false. The point-slope form of a line is

.

ANSWER:

false; point-slope

8. The distance from point X to the line q iss the lengthof the segment perpendicular to line q from X.

SOLUTION:

True

ANSWER:

true

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Chapter 2 Study Guide and Review

9. Explain the difference between inductive anddeductive reasoning.

SOLUTION: Inductive reasoning is a conjecture reached based onobservations of previous patterns, while deductivereasoning uses the law of mathematics to reachlogical conclusions from given statements.

ANSWER: Inductive reasoning is a conjecture reached based onobservations of previous patterns, while deductivereasoning uses the law of mathematics to reachlogical conclusions from given statements.

10. Explain how to prove two lines cut by a transversalare parallel.

SOLUTION: Show that a pair of corresponding angles iscongruent, show that a pair of alternate interiorangles is congruent, show that a pair of alternateexterior angles is congruent, or show that a pair ofconsecutive interior angles is supplementary.

ANSWER: Show that a pair of corresponding angles iscongruent, show that a pair of alternate interiorangles is congruent, show that a pair of alternateexterior angles is congruent, or show that a pair ofconsecutive interior angles is supplementary.

Determine whether each conjecture is true orfalse. If false, give a counterexample.

11. If are supplementary angles, then form a linear pair.

SOLUTION:

Two supplementary angles form a linear pair only ifthey share a common side. So, the sentence is false.Counter example: Two nonadjacent supplementaryangles.

ANSWER:

false; two nonadjacent supplementary angles

12. If W(–3, 2), X(–3, 7), Y(6, 7), Z(6, 2), thenquadrilateral WXYZ is a rectangle.

SOLUTION:

The sides are horizontal lines and

are vertical lines. So, the quadrilateralhas four right angles. Therefore, it is a rectangle, bydefinition.

ANSWER:

true

13. PARKS Jacinto enjoys hiking with his dog in theforest at his local park. While on vacation in SmokyMountain National Park in Tennessee, he wasdisappointed that dogs were not allowed on mosthiking trails. Make a conjecture about why his localpark and the national park have differing rules withregard to pets.

SOLUTION:

Sample answer: The national park may be home towildlife species not found in the local park. Dogs orother pets may threaten or chase these unfamiliaranimals or insects.

ANSWER:

Sample answer: Dogs or other pets may threaten orchase wildlife that might not be present in his localpark.



Use the following statements to write acompound statement for each conjunction ordisjunction. Then find its truth value. Explain.p: A plane contains at least three noncollinearpoints.q: A square yard is equivalent to three squarefeet.r: The sum of the measures of twocomplementary angles is 180.

14.

SOLUTION:

Negate q finding the opposite truth values. Then findthe disjunction . A disjunction is true if atleast one of the statements is true. A square yard is not equivalent to three square feetor the sum of the measures of two complementaryangles is 180º. Here, ~q is a true statement. Sinceone of the statements is true, the disjunction is alsotrue.

ANSWER:

A square yard is not equivalent to three square feetor the sum of the measures of two complementaryangles is 180º; true.

15.

SOLUTION:

Negate r finding the opposite truth values. Then findthe conjunction . A conjunction is true onlywhen both statements that form it are true. A plane contains at least three noncollinear points isTrue. The sum of the measures of twocomplementary angles is not 180 is true. Since boththe statements are true, the conjunction is also true.

ANSWER:

A plane contains at least three noncollinear points andthe sum of the measures of two complementaryangles is not 180; true.

16.

SOLUTION:

Negate p finding the opposite truth values. Then findthe disjunction . A disjunction is true if atleast one of the statements is true. The statement "A plane does not contain at leastthree noncollinear points" is false. The statement "asquare yard is equivalent to three square feet" isfalse. Since both the statements are false, thedisjunction is also false.

ANSWER:

A plane does not contain at least three noncollinearpoints or a square yard is equivalent to three squarefeet; false.

Determine the truth value of each conditionalstatement. If true, explain your reasoning. Iffalse, give a counterexample.

17. If you square an integer, then the result is a positiveinteger.

SOLUTION:

The conditional statement "If you square an integer,then the result is a positive integer." is false. Whenthis hypothesis is true "you square an integer" , theconclusion "the result is a positive integer" is also true,except when the integer being squared is 0, since thesquare of 0 is 0 which is neither positive nornegative.

ANSWER:

false; If the integer is 0, its square is also 0 which isnot a positive integer



18. If a hexagon has eight sides, then all of its angles willbe obtuse.

SOLUTION:

A conditional with a false hypothesis is always true.Here, the hypothesis a hexagon has eight sides isfalse as a hexagon has six sides. Therefore, theconditional statement is true.

ANSWER:

true

19. Write the converse, inverse, and contrapositive of thefollowing true conditional. Then, determine whethereach related conditional is true or false. If astatement is false, find a counterexample.If two angles are congruent, then they have thesame degree measure.

SOLUTION:

Converse: The converse is formed by exchangingthe hypothesis and conclusion of the conditional.If two angles have the same degree measure, thenthey are congruent. The statement is true by thedefinition of congruence. Inverse: The inverse is formed by negating both thehypothesis and conclusion of the conditional.If two angles are not congruent, then they do nothave the same degree measure. The statement is trueby the definition of congruence. Contrapositive: The contrapositive is formed bynegating both the hypothesis and the conclusion of theconverse of the conditional.If two angles do not have the same degree measure,then they are not congruent. The statement is true bythe definition of congruence.

ANSWER:

Converse: If two angles have the same degreemeasure, then they are congruent. True.Inverse: If two angles are not congruent, then they donot have the same degree measure. True.Contrapositive: If two angles do not have the samedegree measure, then they are not congruent. True.

Draw a valid conclusion from the givenstatements, if possible. Then state whether yourconclusion was drawn using the Law ofDetachment or the Law of Syllogism. If no validconclusion can be drawn, write no validconclusion and explain your reasoning.

20. Given: If a quadrilateral has diagonals that bisecteach other, then it is a parallelogram.The diagonals of quadrilateral PQRS bisect eachother.

SOLUTION:

If p → q is a true statement then if p is true, then q.Here, the statement “if a quadrilateral has diagonalsthat bisect each other, then it is a parallelogram” is atrue statement and the diagonals of quadrilateralPQRS bisect each other. So, PQRS is a parallelogramby the Law of Detachment.

ANSWER:

PQRS is a parallelogram; Law of Detachment.

21. Given: If Liana struggles in science class, then shewill receive tutoring.If Liana stays after school on Thursday, then she willreceive tutoring.

SOLUTION:

By the Law of Syllogism, if p → q and q → r aretrue statements, then p → r is a true statement. TheLaw of Syllogism does not apply since the conclusionof the first statement is not the hypothesis of thesecond statement. The conclusion is invalid.

ANSWER:

Invalid; the Law of Syllogism does not apply since theconclusion of the first statement is not the hypothesisof the second statement.



22. EARTHQUAKES Determine whether the statedconclusion is valid based on the given information. Ifnot, write invalid. Explain.Given: If an earthquake measures a 7.0 or higher onthe Richter scale, then it is considered a majorearthquake that could cause serious damage. The1906 San Francisco earthquake measured 8.0 on theRichter scale.Conclusion: The 1906 San Francisco earthquake wasa major earthquake that caused serious damage.

SOLUTION:

If p → q is a true statement then if p is true, then q.Here, the statement “if an earthquake measures a 7.0or higher on the Richter scale, then it is considered amajor earthquake that could cause serious damage”is a true statement and the 1906 San Franciscoearthquake measured 8.0 on the Richter scale. So,the 1906 San Francisco earthquake was a majorearthquake that caused serious damage is a validconclusion by the Law of Detachment.

ANSWER:

Valid; Law of Detachment

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

23. Two planes intersect at a point.

SOLUTION:

If two planes intersect, they form a line. So, thestatement "Two planes intersect at a point." is nevertrue.

ANSWER:

Never; if two planes intersect, they form a line.

24. Three points are contained in more than one plane.

SOLUTION:

If the three points are non-collinear, then there existsa plane containing all the three points. But if thepoints are collinear, we can find three different planeswith each plane containing one of the three points.So, the statement is sometimes true.

ANSWER:

Sometimes; if the three points are collinear, they willbe contained in multiple planes, but if they arenoncollinear, they will be contained in only one plane.

25. If line m lies in plane X and line m contains a point Q,then point Q lies in plane X.

SOLUTION:

If a plane contains a line, then every point of that linelies in the plane. Therefore, the statement " If line mlies in plane X and line m contains a point Q, thenpoint Q lies in plane X." is always true.

ANSWER:

Always; if a plane contains a line, then every point ofthat line lies in the plane.



26. If two angles are complementary, then they form aright angle.

∠1 and ∠2 are complementary.

SOLUTION:

Two complementary angles form a right angle only ifthey are adjacent. Otherwise they do not form a rightangle. So, the statement is sometimes true.

ANSWER:

Sometimes; if the angles are adjacent, they will forma right angle, but if they are not adjacent, they willnot.

27. NETWORKING Six people are introduced at abusiness convention. If each person shakes handswith each of the others, how many handshakes willbe exchanged? Include a model to support yourreasoning.

SOLUTION:

The first person will shake hands with the other 5people and the second person with the other 4 peopleas the hand shake between the first and the secondpersons has already been counted. Similarly, the thirdperson will shake hands with 3 other people and soon. So, the total number of hand shakes will be 5 + 4+ 3 + 2 + 1 = 15. So, a total of 15 handshakes will beexchanged at the convention.

ANSWER:

15 handshakes;



Write a two-column proof.

28. Given: X is the midpoint of Prove: VW = ZY

SOLUTION:

You need to walk through the proof step by step.Look over what you are given and what you need toprove. Here, you are given the midpoint of twosegments Use the properties that you have learnedabout congruent segments, midpoints, and equivalentexpressions in algebra to walk through the proof.

Given: X is the midpoint of Prove: VW = ZY

Proof:Statements (Reasons)

1. X is the midpoint of (Given)

2. (Definition. of midpoint)3. WX = YX, VX = ZX (Definition of congruence)4. VX = VW + WX, ZX = ZY + YX (Segment. AdditionPostulate.)5. VW + WX = ZY + YX (Substitution)6. VW = ZY (Subtraction Prop.)

ANSWER:

Statements (Reasons)

1. X is the midpoint of (Given)

2. (Def. of midpoint)3. WX = YX, VX = ZX (Def. of )4. VX = VW + WX, ZX = ZY + YX (Seg. Add. Post.)5. VW + WX = ZY + YX(Subs.)6. VW = ZY (Subt. Prop.)

29. Given: AB = DCProve: AC = DB

SOLUTION:

You need to walk through the proof step by step.Look over what you are given and what you need toprove. Here, you are given two segments of equallength. Use the properties that you have learnedabout congruent segments and equivalent expressionsin algebra to walk through the proof. Given: AB = DCProve: AC = DBProof:Statements (Reasons)1. AB = DC (Given)2. BC = BC (Reflexive Property)3. AB + BC = DC + BC (Addition Property)4. AB + BC = AC, DC + BC = DB (SegmentAddition Property)5. AC = DB (Substitution)

ANSWER:

Statements (Reasons)1. AB = DC (Given)2. BC = BC (Refl. Prop.)3. AB + BC = DC + BC (Add. Prop.)4. AB + BC = AC, DC + BC = DB (Seg. Add. Post.)5. AC = DB (Subs.)



30. GEOGRAPHY Leandro is planning to drive fromKansas City to Minneapolis along Interstate 35. Themap he is using gives the distance from Kansas Cityto Des Moines as 194 miles and from Des Moines toMinneapolis as 243 miles. What allows him toconclude that the distance he will be driving is 437miles from Kansas City to Minneapolis? Assume thatInterstate 35 forms a straight line.

SOLUTION:

Interstate 35 forms a straight line and Des Moines isbetween Kansas City and Minneapolis. So, bySegment Addition Postulate the total distance fromKansas City to Minneapolis is the sum of thedistances from Kansas City to Des Moines and DesMoines to Minneapolis, that is 437 miles.

ANSWER:

Seg. Add. Post.

Find the measure of each angle.

31.

SOLUTION:

Since the measure of the linear pair of ∠5 is 90,

ANSWER:

90

32.

SOLUTION:

Angle 6 and the angle with measure 53 form a linearpair. So,

ANSWER:

127

33.

SOLUTION:

The ∠7 and the angle with measure 53 are verticalangles. So, they are congruent.

ANSWER:

53



34. PROOF Write a two-column proof.Given: Prove:

SOLUTION:

You need to walk through the proof step by step.Look over what you are given and what you need toprove. Here, you are given two sets of congruentangles. Use the properties that you have learnedabout congruent angles and equivalent expressions inalgebra to walk through the proof.Given: Prove: Proof:Statements (Reasons)1. (Given)2. (Definition ofcongruence)3. (Addition Property)4. (Angle Addition Postulate)5. (Substitution)6. (Definition. of congruence)

ANSWER:

Statements (Reasons)1. (Given)2. (Def. of )3. (Add. Prop.)4. (∠Add. Post.)5. (Subs.)6. (Def. of )

Classify the relationship between each pair ofangles as alternate interior, alternate exterior,corresponding, or consecutive interior angles.

35. 1 and 5

SOLUTION:

and lie on the same side of the transversaland on the same side of the other two lines. They arecorresponding angles.

ANSWER:

corresponding

36. 4 and 6

SOLUTION:

and are nonadjacent interior angles that lie onopposite sides of the transversal. They are alternateinterior angles.

ANSWER:

alternate interior

37. 2 and 8

SOLUTION:

and are nonadjacent exterior angles that lieon opposite sides of the transversal. They arealternate exterior angles.

ANSWER:

alternate exterior

38. 4 and 5

SOLUTION:

and are interior angles that lie on the sameside of the transversal. They are consecutive interiorangles.

ANSWER:

consecutive interior



39. BRIDGES The Roebling Suspension Bridgeextends over the Ohio River connecting Cincinnati,Ohio to Covington, Kentucky. Describe the type oflines formed by the bridge and the river.

SOLUTION:

The bridge and the river are represented by lines thatdo not intersect and are not coplanar. They are skewlines.

ANSWER:

skew lines

In the figure, m 1 = 123. Find the measure ofeach angle. Tell which postulate(s) ortheorem(s) you used.

40. 5

SOLUTION:

In the figure, angles 1 and 5 are alternate exteriorangles.Alternate Exterior Angles Theorem:If two parallel lines are cut by a transversal, theneach pair of alternate exterior angles is congruent.By the Alternate Exterior Angles Theorem,

.

ANSWER:

123; Alt. Ext. s Thm

41. 14

SOLUTION:

In the figure, angles 1 and 5 are alternate exteriorangles, angles 5 and 13 are corresponding angles, andangles 13 and 14 form a linear pair. By the Alternate Exterior Angles Theorem,

. By the Corresponding Angles Postulate,

. By the definition of a linear pair, . Substitute.

ANSWER:

57; 5 13 by Corr. s Post. and 13 and 14 form a linear pair.

42. 16

SOLUTION:

In the figure, angles 1 and 9 are corresponding anglesand angles 9 and 16 form a linear pair.By the Corresponding Angles Postulate, m∠9 =m∠1 = 123. Since angles 9 and 16 form a linear pair, they aresupplementary.

ANSWER:

57; 1 9 by Corr. s Post. and 9 and 16form a linear pair.



43. 11

SOLUTION:

In the figure, angles 1 and 5 are alternate exteriorangles, and angles 5 and 11 are alternate interiorangles. By the Alternate Exterior Angles Theorem, m∠5 =m∠1 = 123. By the Alternate Interior Angles Theorem, m∠11 =m∠5 = 123 .So, m∠5 = 123.

ANSWER:

123; 11 5 by Alt. Int. s Thm and 5 1 by Alt. Ext ∠s Thm.

44. 4

SOLUTION:

In the figure, angles 1 and 5 are alternate exteriorangles and angles 4 and 5 form a linear pair. Since alternate interior angles of parallel lines arecongruent m∠1 = m∠5. Since m∠1 = 123, bysubstitution, m∠5 = 123. Since angles 4 and 5 form a linear pair, they aresupplementary. Substitute.

ANSWER:

57; 1 5 by Alt. Ext. s Thm and 4 and 5 form a linear pair

45. 6

SOLUTION:

In the figure, angles 1 and 3 are corresponding anglesand angles 3 and 6 form a linear pair.By the Corresponding Angles Postulate, m∠3 =m∠1 = 123.Since a linear pair of angles is supplementary, m∠3+ m∠6 = 180.Substitute.

ANSWER:

57; 1 3 by Corr. s Post. and 3 and 6form a linear pair

46. MAPS The diagram shows the layout of Elm, Plum,and Oak streets. Find the value of x.

SOLUTION:

By the Consecutive Interior Angles Theorem, .

Solve for x.

ANSWER:

125

Determine whether and are parallel,perpendicular, or neither. Graph each line toverify your answer.



47. A(5, 3), B(8, 0), X(–7, 2), Y(1, 10)

SOLUTION:

Substitute the coordinates of the points in slopeformula to find the slopes of the lines.

The product of the slopes of the lines is –1.Therefore, the lines are perpendicular.Graph the lines on a coordinate plane to verify theanswer.

ANSWER:

perpendicular

48. A(–3, 9), B(0, 7), X(4, 13), Y(–5, 7)

SOLUTION:


The two lines neither have equal slopes nor theirproduct is –1. Therefore, the lines are neither parallelnor perpendicular.Graph the lines on a coordinate plane to verify theanswer.

ANSWER:

neither



49. A(8, 1), B(–2, 7), X(–6, 2), Y(–1, –1)

SOLUTION:


The two lines have equal slopes, . Therefore, the

lines are parallel.Graph the lines on a coordinate plane to verify theanswer.

ANSWER:

parallel

Graph the line that satisfies each condition.

50. contains (–3, 4) and is parallel to with A(2, 5)and B(9, 2)

SOLUTION:

Find the slope of the line

The required line is parallel to So, the slope of

the required line is .

Start at (–3, 4). Move three units down and sevenunits right to reach the point (4, 1). Join the two pointsand extend.

ANSWER:



51. contains (1, 3) and is perpendicular to with P(4,–6) and Q(6, –1)

SOLUTION:

Find the slope of the line .

The required line is perpendicular to . So, the

slope of the required line is

Start at (1, 3). Move two units down and five unitsright to reach the point (6, 1). Join the two points andextend.

ANSWER:

52. AIRPLANES Two Oceanic Airlines planes areflying at the same altitude. Using satellite imagery,each plane’s position can be mapped onto acoordinate plane. Flight 815 was mapped at (23, 17)and (5, 11) while Flight 44 was mapped at (3, 15) and(9, 17). Determine whether their paths are parallel,perpendicular, or neither.

SOLUTION:

Substitute the coordinates of the points in slopeformula to find the slopes of the lines. Let and

be the slopes of the paths of flight 815 and flight44 respectively.

The paths are parallel, since they have the sameslope.

ANSWER:

parallel



Write an equation in point-slope form of the linehaving the given slope that contains the givenpoint.

53. m = 2, (4, –9)

SOLUTION:

The point-slope form of a line is

where m is the slope and is a point on the

line.Here, m = 2 and (x1, y1) = (4, –9).

ANSWER:

y + 9 = 2(x – 4)

54.

SOLUTION:

The point-slope form of a line is y – y1 = m(x – x1)

where m is the slope and (x1, y1) is a point on the

line.

Here, and (x1, y1) = (8, –1).

So, the equation of the line is

ANSWER:

Write an equation in slope-intercept form of theline having the given slope and y-intercept.

55. m: 5, y-intercept: –3

SOLUTION:

The slope-intercept form of a line of slope m and y-intercept b is given by y = mx + b.Here, and y-intercept = –3.So, the equation of the line is

ANSWER:

y = 5x – 3

56. y-intercept: 4

SOLUTION:

The slope-intercept form of a line of slope m and y-intercept b is given by y = mx + b.

Here, and y-intercept = 4.


ANSWER:



Write an equation in slope-intercept form foreach line.

57. (–3, 12) and (15, 0)

SOLUTION:

Use the slope formula to find the slope of the line.

Use the slope and one of the points to write theequation of the line in point-slope form.The point-slope form of a line is


line.

Here, and (x1, y1) = (15, 0).


ANSWER:

58. (–7, 2) and (5, 8)

SOLUTION:

Use the slope formula to find the slope of the line.

Use the slope and one of the points to write theequation of the line in point-slope form.The point-slope form of a line is


line.

Here, and (x1, y1) = (5, 8).

ANSWER:



59. WINDOW CLEANING Ace Window CleaningService charges $50 for the service call and $20 foreach hour spent on the job. Write an equation inslope-intercept form that represents the total cost Cin terms of the number of hours h.

SOLUTION:

The slope-intercept form of a line with slope m and y-intercept b is given by y = mx + b.Here, and y-intercept = 50.The total cost C in terms of the number of hours h is

.

ANSWER:

C = 20h + 50

Given the following information, determinewhich lines, if any, are parallel. State thepostulate or theorem that justifies your answer.

60. m∠7 + m∠10 = 180

SOLUTION:

If angles 7 and 10 are supplementary, than ; bythe Converse of the Consecutive Interior AnglesTheorem.

ANSWER:

; Consecutive Interior Angles Converse Thm.

61. 2 10

SOLUTION:

If angles 2 and 10 are congruent this does not implyany of the lines are parallel.

ANSWER:

none

62. 1 3

SOLUTION:

and are corresponding angles of lines w and x.Since ,w || x by the Converse of Corresponding AnglesPostulate.

ANSWER:

w || x; Corresponding Angles Converse Postulate

63. 3 11

SOLUTION:

and are alternate exterior angles of lines vand z. Since ,v || z by the Converse of Alternate Exterior AnglesTheorem.

ANSWER:

v || z; Alternate Exterior Angles Converse Thm.

64. Find x so that . Identify the postulate or theoremyou used.

SOLUTION:

Since ,l || m by the Converse of Consecutive Interior AnglesTheorem.

ANSWER:

9; converse of Cons. Int. s Thm.



65. LANDSCAPING Find the measure needed for mADC that will make if m BAD = 45.

SOLUTION:

By the Consecutive Interior Angles Theorem, .

Substitute.

ANSWER:

135

Copy each figure. Draw the segment thatrepresents the distance indicated.

66. X to

SOLUTION:

A perpendicular line from X to VW represents theshortest distance.

ANSWER:



67. L to

SOLUTION:

A perpendicular line from L to JK represents theshortest distance.

ANSWER:

68. HOME DÉCOR Scott wants to hang two rows offramed pictures in parallel lines on his living roomwall. He first spaces the nails on the wall in a line forthe top row. Next, he hangs a weighted plumb linefrom each nail and measures an equal distance beloweach nail for the second row. Why does this ensurethat the two rows of pictures will be parallel?

SOLUTION:

Hanging a weighted plumb and measuring an equaldistance ensures that the second row is is the samedistance at every point from the first row. The secondrow of nails is equidistant at all points from the firstrow.

ANSWER:

The second row of nails is equidistant at all pointsfrom the first row.



state whether each sentence is true or false. if false, replace

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