Chapter 2 Study Guide and Review State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The first part of an if-then statement is the conjecture. SOLUTION: The first part of an if-then statement is the hypothesis. So, the sentence is false. ANSWER: false; hypothesis 2. The contrapositive is formed by negating the hypothesis and conclusion of a conditional. SOLUTION: The contrapositive is formed by negating both the hypothesis and the conclusion of the converse of the conditional. So, the sentence is false. The inverse is formed by negating the hypothesis and conclusion of a conditional. ANSWER: false; inverse 3. A conjunction is formed by joining two or more statements with the word and. SOLUTION: True ANSWER: true 4. To show that a conjecture is false, you would provide a disjunction. SOLUTION: To show that a conjecture is false, you would provide a counter example. So, the sentence is false.
5. The inverse of a statement p would be written in the form not p. SOLUTION: The inverse is formed by negating both the hypothesis and conclusion of the conditional. So, the sentence is false. The negation of a statement p would be written in the form not p. ANSWER: false; negation 6. In a two-column proof, the properties that justify each step 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, the sentence 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 length of the segment perpendicular to line q from X. SOLUTION: True ANSWER: true
ANSWER: false; counterexample
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Chapter 2 Study Guide and Review 9. Explain the difference between inductive and deductive reasoning. SOLUTION: Inductive reasoning is a conjecture reached based on observations of previous patterns, while deductive reasoning uses the law of mathematics to reach logical conclusions from given statements.
12. If W(–3, 2), X(–3, 7), Y(6, 7), Z(6, 2), then quadrilateral WXYZ is a rectangle. SOLUTION:
ANSWER: Inductive reasoning is a conjecture reached based on observations of previous patterns, while deductive reasoning uses the law of mathematics to reach logical conclusions from given statements. 10. Explain how to prove two lines cut by a transversal are parallel. SOLUTION: Show that a pair of corresponding angles is congruent, show that a pair of alternate interior angles is congruent, show that a pair of alternate exterior angles is congruent, or show that a pair of consecutive interior angles is supplementary. ANSWER: Show that a pair of corresponding angles is congruent, show that a pair of alternate interior angles is congruent, show that a pair of alternate exterior angles is congruent, or show that a pair of consecutive interior angles is supplementary. Determine whether each conjecture is true or false. If false, give a counterexample. 11. If are supplementary angles, then form a linear pair. SOLUTION: Two supplementary angles form a linear pair only if they share a common side. So, the sentence is false. Counter example: Two nonadjacent supplementary angles.
The sides
are horizontal lines and
are vertical lines. So, the quadrilateral has four right angles. Therefore, it is a rectangle, by definition. ANSWER: true 13. PARKS Jacinto enjoys hiking with his dog in the forest at his local park. While on vacation in Smoky Mountain National Park in Tennessee, he was disappointed that dogs were not allowed on most hiking trails. Make a conjecture about why his local park and the national park have differing rules with regard to pets. SOLUTION: Sample answer: The national park may be home to wildlife species not found in the local park. Dogs or other pets may threaten or chase these unfamiliar animals or insects. ANSWER: Sample answer: Dogs or other pets may threaten or chase wildlife that might not be present in his local park.
ANSWER: false; two nonadjacent supplementary angles
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Chapter 2 Study Guide and Review Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. Explain. p: A plane contains at least three noncollinear points. q: A square yard is equivalent to three square feet. r: The sum of the measures of two complementary angles is 180. 14. SOLUTION: Negate q finding the opposite truth values. Then find the disjunction . A disjunction is true if at least one of the statements is true. A square yard is not equivalent to three square feet or the sum of the measures of two complementary angles is 180º. Here, ~q is a true statement. Since one of the statements is true, the disjunction is also true. ANSWER: A square yard is not equivalent to three square feet or the sum of the measures of two complementary angles is 180º; true. 15. SOLUTION: Negate r finding the opposite truth values. Then find the conjunction . A conjunction is true only when both statements that form it are true.
A plane contains at least three noncollinear points is True. The sum of the measures of two complementary angles is not 180 is true. Since both the statements are true, the conjunction is also true.
16. SOLUTION: Negate p finding the opposite truth values. Then find the disjunction . A disjunction is true if at least one of the statements is true. The statement "A plane does not contain at least three noncollinear points" is false. The statement "a square yard is equivalent to three square feet" is false. Since both the statements are false, the disjunction is also false. ANSWER: A plane does not contain at least three noncollinear points or a square yard is equivalent to three square feet; false. Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counterexample. 17. If you square an integer, then the result is a positive integer. SOLUTION: The conditional statement "If you square an integer, then the result is a positive integer." is false. When this hypothesis is true "you square an integer" , the conclusion "the result is a positive integer" is also true, except when the integer being squared is 0, since the square of 0 is 0 which is neither positive nor negative. ANSWER: false; If the integer is 0, its square is also 0 which is not a positive integer
ANSWER: A plane contains at least three noncollinear points and the sum of the measures of two complementary angles is not 180; true.
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Chapter 2 Study Guide and Review 18. If a hexagon has eight sides, then all of its angles will be obtuse. SOLUTION: A conditional with a false hypothesis is always true. Here, the hypothesis a hexagon has eight sides is false as a hexagon has six sides. Therefore, the conditional statement is true. ANSWER: true 19. Write the converse, inverse, and contrapositive of the following true conditional. Then, determine whether each related conditional is true or false. If a statement is false, find a counterexample. If two angles are congruent, then they have the same degree measure. SOLUTION: Converse: The converse is formed by exchanging the hypothesis and conclusion of the conditional. If two angles have the same degree measure, then they are congruent. The statement is true by the definition of congruence. Inverse: The inverse is formed by negating both the hypothesis and conclusion of the conditional. If two angles are not congruent, then they do not have the same degree measure. The statement is true by the definition of congruence.
Contrapositive: The contrapositive is formed by negating both the hypothesis and the conclusion of the converse of the conditional. If two angles do not have the same degree measure, then they are not congruent. The statement is true by the definition of congruence.
Draw a valid conclusion from the given statements, if possible. Then state whether your conclusion was drawn using the Law of Detachment or the Law of Syllogism. If no valid conclusion can be drawn, write no valid conclusion and explain your reasoning. 20. Given: If a quadrilateral has diagonals that bisect each other, then it is a parallelogram. The diagonals of quadrilateral PQRS bisect each other. SOLUTION: If p → q is a true statement then if p is true, then q. Here, the statement “if a quadrilateral has diagonals that bisect each other, then it is a parallelogram” is a true statement and the diagonals of quadrilateral PQRS bisect each other. So, PQRS is a parallelogram by the Law of Detachment. ANSWER: PQRS is a parallelogram; Law of Detachment. 21. Given: If Liana struggles in science class, then she will receive tutoring. If Liana stays after school on Thursday, then she will receive tutoring. SOLUTION: By the Law of Syllogism, if p → q and q → r are true statements, then p → r is a true statement. The Law of Syllogism does not apply since the conclusion of the first statement is not the hypothesis of the second statement. The conclusion is invalid. ANSWER: Invalid; the Law of Syllogism does not apply since the conclusion of the first statement is not the hypothesis of the second statement.
ANSWER: Converse: If two angles have the same degree measure, then they are congruent. True. Inverse: If two angles are not congruent, then they do not have the same degree measure. True. Contrapositive: If two angles do not have the same degree measure, then they are not congruent. True. eSolutions Manual - Powered by Cognero
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Chapter 2 Study Guide and Review 22. EARTHQUAKES Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain. Given: If an earthquake measures a 7.0 or higher on the Richter scale, then it is considered a major earthquake that could cause serious damage. The 1906 San Francisco earthquake measured 8.0 on the Richter scale. Conclusion: The 1906 San Francisco earthquake was a 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 7.0 or higher on the Richter scale, then it is considered a major earthquake that could cause serious damage” is a true statement and the 1906 San Francisco earthquake measured 8.0 on the Richter scale. So, the 1906 San Francisco earthquake was a major earthquake that caused serious damage is a valid conclusion 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, the statement "Two planes intersect at a point." is never true.
24. Three points are contained in more than one plane. SOLUTION: If the three points are non-collinear, then there exists a plane containing all the three points. But if the points are collinear, we can find three different planes with each plane containing one of the three points. So, the statement is sometimes true.
ANSWER: Sometimes; if the three points are collinear, they will be contained in multiple planes, but if they are noncollinear, 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 line lies in the plane. Therefore, the statement " If line m lies in plane X and line m contains a point Q, then point Q lies in plane X." is always true.
ANSWER: Never; if two planes intersect, they form a line.
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ANSWER: Always; if a plane contains a line, then every point of that line lies in the plane.
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Chapter 2 Study Guide and Review 26. If two angles are complementary, then they form a right angle.
∠1 and ∠2 are complementary. SOLUTION: Two complementary angles form a right angle only if they are adjacent. Otherwise they do not form a right angle. So, the statement is sometimes true. ANSWER: Sometimes; if the angles are adjacent, they will form a right angle, but if they are not adjacent, they will not.
27. NETWORKING Six people are introduced at a business convention. If each person shakes hands with each of the others, how many handshakes will be exchanged? Include a model to support your reasoning. SOLUTION: The first person will shake hands with the other 5 people and the second person with the other 4 people as the hand shake between the first and the second persons has already been counted. Similarly, the third person will shake hands with 3 other people and so on. So, the total number of hand shakes will be 5 + 4 + 3 + 2 + 1 = 15. So, a total of 15 handshakes will be exchanged at the convention.
ANSWER: 15 handshakes;
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Chapter 2 Study Guide and Review 29. Given: AB = DC Prove: AC = DB
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 to prove. Here, you are given the midpoint of two segments Use the properties that you have learned about congruent segments, midpoints, and equivalent expressions 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. Addition Postulate.) 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.)
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SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given two segments of equal length. Use the properties that you have learned about congruent segments and equivalent expressions in algebra to walk through the proof. Given: AB = DC Prove: AC = DB Proof: 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 (Segment Addition 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.)
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Chapter 2 Study Guide and Review 30. GEOGRAPHY Leandro is planning to drive from Kansas City to Minneapolis along Interstate 35. The map he is using gives the distance from Kansas City to Des Moines as 194 miles and from Des Moines to Minneapolis as 243 miles. What allows him to conclude that the distance he will be driving is 437 miles from Kansas City to Minneapolis? Assume that Interstate 35 forms a straight line. SOLUTION: Interstate 35 forms a straight line and Des Moines is between Kansas City and Minneapolis. So, by Segment Addition Postulate the total distance from Kansas City to Minneapolis is the sum of the distances from Kansas City to Des Moines and Des Moines to Minneapolis, that is 437 miles.
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 linear pair. So, ANSWER: 127 33. SOLUTION: The ∠7 and the angle with measure 53 are vertical angles. So, they are congruent.
ANSWER: Seg. Add. Post.
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ANSWER: 53
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Chapter 2 Study Guide and Review 34. PROOF Write a two-column proof. Given: Prove:
Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.
35. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given two sets of congruent angles. Use the properties that you have learned about congruent angles and equivalent expressions in algebra to walk through the proof. Given: Prove: Proof: Statements (Reasons) 1. (Given) 2. (Definition of congruence) 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 )
1 and
5
SOLUTION: and lie on the same side of the transversal and on the same side of the other two lines. They are corresponding angles. ANSWER: corresponding 36.
4 and
6
SOLUTION: and are nonadjacent interior angles that lie on opposite sides of the transversal. They are alternate interior angles. ANSWER: alternate interior 37.
2 and
8
SOLUTION: and are nonadjacent exterior angles that lie on opposite sides of the transversal. They are alternate exterior angles. ANSWER: alternate exterior
(∠ 38.
4 and
5
SOLUTION: and are interior angles that lie on the same side of the transversal. They are consecutive interior angles. ANSWER:
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Chapter 2 Study Guide and Review 39. BRIDGES The Roebling Suspension Bridge
41.
extends over the Ohio River connecting Cincinnati, Ohio to Covington, Kentucky. Describe the type of lines formed by the bridge and the river.
SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles, angles 5 and 13 are corresponding angles, and angles 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.
SOLUTION: The bridge and the river are represented by lines that do not intersect and are not coplanar. They are skew lines. ANSWER: skew lines In the figure, m 1 = 123. Find the measure of each angle. Tell which postulate(s) or theorem(s) you used.
40.
ANSWER: 57; 5 13 by Corr. s Post. and 13 and 14 form a linear pair.
5 SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles. Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. By the Alternate Exterior Angles Theorem, . ANSWER: 123; Alt. Ext.
14
42.
16 SOLUTION: In the figure, angles 1 and 9 are corresponding angles and 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 are supplementary.
s Thm
ANSWER: 57; 1 9 by Corr. form a linear pair.
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s Post. and 9 and
16
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Chapter 2 Study Guide and Review 43.
11
45.
SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles, and angles 5 and 11 are alternate interior angles.
By the Alternate Exterior Angles Theorem, m∠5 = m∠1 = 123. By the Alternate Interior Angles Theorem, m∠11 = m∠5 = 123 .
6 SOLUTION: In the figure, angles 1 and 3 are corresponding angles and 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.
So, m∠5 = 123.
ANSWER: 123; 11 5 by Alt. Int. 1 by Alt. Ext ∠s Thm.
ANSWER: 57; 1 3 by Corr. form a linear pair
44.
s Thm and
5
4 SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles and angles 4 and 5 form a linear pair.
s Post. and 3 and
6
46. MAPS The diagram shows the layout of Elm, Plum, and Oak streets. Find the value of x.
Since alternate interior angles of parallel lines are congruent m∠1 = m∠5. Since m∠1 = 123, by substitution, m∠5 = 123.
Since angles 4 and 5 form a linear pair, they are supplementary. Substitute.
ANSWER: 57; 1 5 by Alt. Ext. 5 form a linear pair
s Thm and
4 and
SOLUTION: By the Consecutive Interior Angles Theorem, . Solve for x.
ANSWER: 125 Determine whether and are parallel, perpendicular, or neither. Graph each line to verify your answer.
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Chapter 2 Study Guide and Review 47. A(5, 3), B(8, 0), X(–7, 2), Y(1, 10) SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines.
48. A(–3, 9), B(0, 7), X(4, 13), Y(–5, 7) SOLUTION: Substitute the coordinates of the points in slope formula 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 the answer.
ANSWER: perpendicular
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The two lines neither have equal slopes nor their product is –1. Therefore, the lines are neither parallel nor perpendicular. Graph the lines on a coordinate plane to verify the answer.
ANSWER: neither
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Chapter 2 Study Guide and Review
49. A(8, 1), B(–2, 7), X(–6, 2), Y(–1, –1)
Graph the line that satisfies each condition.
SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines.
50. contains (–3, 4) and is parallel to and B(9, 2)
with A(2, 5)
SOLUTION: Find the slope of the line
The required line is parallel to the required line is
So, the slope of
.
Start at (–3, 4). Move three units down and seven units right to reach the point (4, 1). Join the two points and extend. The two lines have equal slopes,
. Therefore, the
lines are parallel. Graph the lines on a coordinate plane to verify the answer.
ANSWER:
ANSWER: parallel
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Chapter 2 Study Guide and Review
51. contains (1, 3) and is perpendicular to –6) and Q(6, –1)
with P(4,
SOLUTION: Find the slope of the line
.
The required line is perpendicular to
. So, the
52. AIRPLANES Two Oceanic Airlines planes are flying at the same altitude. Using satellite imagery, each plane’s position can be mapped onto a coordinate 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 slope formula to find the slopes of the lines. Let and be the slopes of the paths of flight 815 and flight 44 respectively.
slope of the required line is Start at (1, 3). Move two units down and five units right to reach the point (6, 1). Join the two points and extend.
ANSWER: The paths are parallel, since they have the same slope. ANSWER: parallel
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Chapter 2 Study Guide and Review Write an equation in point-slope form of the line having the given slope that contains the given point. 53. m = 2, (4, –9)
Write an equation in slope-intercept form of the line 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 yintercept b is given by y = mx + b. Here, and y-intercept = –3. So, the equation of the line is
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 = 5x – 3
y-intercept: 4
56. ANSWER: y + 9 = 2(x – 4)
54.
SOLUTION: The slope-intercept form of a line of slope m and yintercept b is given by y = mx + b. Here,
and y-intercept = 4.
SOLUTION: The point-slope form of a line is y – y1 = m(x – x1)
So, the equation of the line is
where m is the slope and (x1, y1) is a point on the
ANSWER:
line. Here,
and (x1, y1) = (8, –1).
So, the equation of the line is
ANSWER:
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Chapter 2 Study Guide and Review Write an equation in slope-intercept form for each line. 57. (–3, 12) and (15, 0)
58. (–7, 2) and (5, 8) SOLUTION: Use the slope formula to find the slope of the line.
SOLUTION: Use the slope formula to find the slope of the line.
Use the slope and one of the points to write the equation of the line in point-slope form. The point-slope form of a line is where m is the slope and
is a point on the
line. Here,
Use the slope and one of the points to write the equation of the line in point-slope form. The point-slope form of a line is where m is the slope and
is a point on the
line. Here,
and (x1, y1) = (5, 8).
and (x1, y1) = (15, 0).
So, the equation of the line is
ANSWER: ANSWER:
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Chapter 2 Study Guide and Review 59. WINDOW CLEANING Ace Window Cleaning Service charges $50 for the service call and $20 for each hour spent on the job. Write an equation in slope-intercept form that represents the total cost C in terms of the number of hours h. SOLUTION: The slope-intercept form of a line with slope m and yintercept 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, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer.
62.
1
3
SOLUTION: and are corresponding angles of lines w and x. Since , w || x by the Converse of Corresponding Angles Postulate. ANSWER: w || x; Corresponding Angles Converse Postulate 63.
3
11
SOLUTION: and are alternate exterior angles of lines v and z. Since , v || z by the Converse of Alternate Exterior Angles Theorem. ANSWER: v || z; Alternate Exterior Angles Converse Thm. 64. Find x so that you used.
. Identify the postulate or theorem
60. m∠7 + m∠10 = 180 SOLUTION: If angles 7 and 10 are supplementary, than ; by the Converse of the Consecutive Interior Angles Theorem. ANSWER: ; Consecutive Interior Angles Converse Thm. 61.
2
10
SOLUTION: If angles 2 and 10 are congruent this does not imply any of the lines are parallel.
SOLUTION:
Since , l || m by the Converse of Consecutive Interior Angles Theorem. ANSWER: 9; converse of Cons. Int.
s Thm.
ANSWER: none
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Chapter 2 Study Guide and Review 65. LANDSCAPING Find the measure needed for m ADC that will make
if m BAD = 45.
Copy each figure. Draw the segment that represents the distance indicated. 66. X to
SOLUTION: By the Consecutive Interior Angles Theorem, . Substitute.
SOLUTION:
A perpendicular line from X to VW represents the shortest distance.
ANSWER: 135 ANSWER:
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Chapter 2 Study Guide and Review
67. L to
SOLUTION:
A perpendicular line from L to JK represents the shortest distance.
ANSWER:
68. HOME DÉCOR Scott wants to hang two rows of framed pictures in parallel lines on his living room wall. He first spaces the nails on the wall in a line for the top row. Next, he hangs a weighted plumb line from each nail and measures an equal distance below each nail for the second row. Why does this ensure that the two rows of pictures will be parallel? SOLUTION: Hanging a weighted plumb and measuring an equal distance ensures that the second row is is the same distance at every point from the first row. The second row of nails is equidistant at all points from the first row. ANSWER: The second row of nails is equidistant at all points from the first row.
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