Chapter 2 Study Guide and Review

Transcription

1 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 The first part of an if-then statement is the hypothesis So, the sentence is false 2 The contrapositive is formed by negating the hypothesis and conclusion of a conditional 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 3 A conjunction is formed by joining two or more statements with the word and True 4 To show that a conjecture is false, you would provide a disjunction 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 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 6 In a two-column proof, the properties that justify each step are called reasons True 7 The slope-intercept form of a linear equation is The slope-intercept form is y = mx + b So, the sentence is false The point-slope form of a line is 8 The distance from point X to the line q iss the length of the segment perpendicular to line q from X True 9 Explain the difference between inductive and deductive reasoning 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 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 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 esolutions Manual - Powered by Cognero Page 1

2 12 If W( 3, 2), X( 3, 7), Y(6, 7), Z(6, 2), then quadrilateral WXYZ is a rectangle The sides are horizontal lines and are vertical lines So, the quadrilateral has four right angles Therefore, it is a rectangle, by definition 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 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 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 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 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 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 esolutions Manual - Powered by Cognero Page 2

3 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 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 18 If a hexagon has eight sides, then all of its angles will be obtuse 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 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 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 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 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 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 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 70 or higher on the Richter scale, then it is considered a major earthquake that could cause serious damage The 1906 San Francisco earthquake measured 80 on the Richter scale Conclusion: The 1906 San Francisco earthquake was a major earthquake that caused serious damage If p q is a true statement then if p is true, then q Here, the statement if an earthquake measures a 70 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 80 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 esolutions Manual - Powered by Cognero Page 3

4 Determine whether each statement is always, sometimes, or never true Explain 23 Two planes intersect at a point If two planes intersect, they form a line So, the statement "Two planes intersect at a point" is never true 25 If line m lies in plane X and line m contains a point Q, then point Q lies in plane X 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 26 If two angles are complementary, then they form a right angle 24 Three points are contained in more than one plane 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 1 and 2 are complementary 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 esolutions Manual - Powered by Cognero Page 4

5 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 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 = 15 So, a total of 15 handshakes will be exchanged at the convention Write a two-column proof 28 Given: X is the midpoint of Prove: VW = ZY 29 Given: AB = DC Prove: AC = DB 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) 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) esolutions Manual - Powered by Cognero Page 5

6 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 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 33 The 7 and the angle with measure 53 are vertical angles So, they are congruent 34 PROOF Write a two-column proof Given: Prove: 31 Find the measure of each angle 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) Since the measure of the linear pair of 5 is 90, 32 Angle 6 and the angle with measure 53 form a linear pair So, esolutions Manual - Powered by Cognero Page 6

7 Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles In the figure, m 1 = 123 Find the measure of each angle Tell which postulate(s) or theorem (s) you used 35 1 and 5 and lie on the same side of the transversal and on the same side of the other two lines They are corresponding angles 36 4 and 6 and are nonadjacent interior angles that lie on opposite sides of the transversal They are alternate interior angles 37 2 and 8 and are nonadjacent exterior angles that lie on opposite sides of the transversal They are alternate exterior angles 38 4 and 5 and are interior angles that lie on the same side of the transversal They are consecutive interior angles 39 BRIDGES The Roebling Suspension Bridge extends over the Ohio River connecting Cincinnati, Ohio to Covington, Kentucky Describe the type of lines formed by the bridge and the river The bridge and the river are represented by lines that do not intersect and are not coplanar They are skew lines 40 5 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, 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 esolutions Manual - Powered by Cognero Page 7

8 42 16 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 45 6 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 In the figure, angles 1 and 5 are alternate exterior angles, and angles 5 and 11 are alternate interior angles 46 MAPS The diagram shows the layout of Elm, Plum, and Oak streets Find the value of x 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 = In the figure, angles 1 and 5 are alternate exterior angles and angles 4 and 5 form a linear pair By the Consecutive Interior Angles Theorem, Solve for 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 esolutions Manual - Powered by Cognero Page 8

9 Determine whether and are parallel, perpendicular, or neither Graph each line to verify your answer 47 A(5, 3), B(8, 0), X( 7, 2), Y(1, 10) 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) 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 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 esolutions Manual - Powered by Cognero Page 9

10 49 A(8, 1), B( 2, 7), X( 6, 2), Y( 1, 1) Substitute the coordinates of the points in slope formula to find the slopes of the lines Graph the line that satisfies each condition 50 contains ( 3, 4) and is parallel to with A(2, 5) and B(9, 2) 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 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 esolutions Manual - Powered by Cognero Page 10

11 51 contains (1, 3) and is perpendicular to with P(4, 6) and Q(6, 1) 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 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 The paths are parallel, since they have the same slope Write an equation in point-slope form of the line having the given slope that contains the given point 53 m = 2, (4, 9) The point-slope form of a line is where m is the slope and is a point on the line Here, m = 2 and (x 1, y 1 ) = (4, 9) esolutions Manual - Powered by Cognero Page 11

12 54 The point-slope form of a line is y y 1 = m(x x 1 ) where m is the slope and (x 1, y 1 ) is a point on the line Write an equation in slope-intercept form for each line 57 ( 3, 12) and (15, 0) Use the slope formula to find the slope of the line Here, and (x 1, y 1 ) = (8, 1) So, the equation of the line is Write an equation in slope-intercept form of the line having the given slope and y-intercept 55 m: 5, y-intercept: 3 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 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 (x 1, y 1 ) = (15, 0) So, the equation of the line is 56 y-intercept: 4 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 So, the equation of the line is esolutions Manual - Powered by Cognero Page 12

13 58 ( 7, 2) and (5, 8) Use the slope formula to find the slope of the line Given the following information, determine which lines, if any, are parallel State the postulate or theorem that justifies your answer 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 (x 1, y 1 ) = (5, 8) 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 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 60 m 7 + m 10 = 180 If angles 7 and 10 are supplementary, than ; by the Converse of the Consecutive Interior Angles Theorem If angles 2 and 10 are congruent this does not imply any of the lines are parallel and are corresponding angles of lines w and x Since, w x by the Converse of Corresponding Angles Postulate and are alternate exterior angles of lines v and z Since, v z by the Converse of Alternate Exterior Angles Theorem esolutions Manual - Powered by Cognero Page 13

14 64 Find x so that Identify the postulate or theorem you used Copy each figure Draw the segment that represents the distance indicated 66 X to A perpendicular line from X to VW represents the shortest distance Since, l m by the Converse of Consecutive Interior Angles Theorem 65 LANDSCAPING Find the measure needed for m ADC that will make if m BAD = L to By the Consecutive Interior Angles Theorem, Substitute A perpendicular line from L to JK represents the shortest distance esolutions Manual - Powered by Cognero Page 14

15 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? 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 esolutions Manual - Powered by Cognero Page 15