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All Textbook Solutions for Elementary Geometry for College Students
In Exercises 1 and 2, which sentences are statements? If a sentence is a statement, classify it as true or false. aWhere do you live? b 4+75 cWashington was the first U.S. president. d x+3=7 when x=5.In Exercises 1 and 2, which sentences are statements? If a sentence is a statement, classify it as true or false. a Chicago is located in the state of Illinois. b Get out of here c x6 read as x is less than 6 when x=10. d Babe Ruth is remembered as a great football player.In Exercises 3 and 4, give the negation of each statement. 3. a Christopher Columbus crossed the Atlantic Ocean. b All jokes are funny.In Exercises 3 and 4, give the negation of each statement. a No one likes me. b Angle 1 is a right angle.In Exercises 5 to 10, classify each statement as simple, conditional, a conjunction, or a disjunction. If Alice plays, the volleyball team will win.In Exercises 5 to 10, classify each statement as simple, conditional, a conjunction, or a disjunction. Alice played and the team won.In Exercises 5 to 10, classify each statement as simple, conditional, a conjunction, or a disjunction. The first-place trophy is beautiful.In Exercises 5 to 10, classify each statement as simple, conditional, a conjunction, or a disjunction. An integer is odd or it is even.In Exercises 5 to 10, classify each statement as simple, conditional, a conjunction, or a disjunction. Matthew is paying shortstop.In Exercises 5 to 10, classify each statement as simple, conditional, a conjunction, or a disjunction. You will be in trouble if you dont change your ways.In Exercises 11 to 18, state the hypothesis and the conclusion of each statement. If you go to the game, then you will have a great time.In Exercises 11 to 18, state the hypothesis and the conclusion of each statement. If two chords of a circle have equal lengths, then the arcs of the chords are congruent.In Exercises 11 to 18, state the hypothesis and the conclusion of each statement. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.14EIn Exercises 11 to 18, state the hypothesis and the conclusion of each statement. Corresponding angles are congruent if two parallel lines are cut by a transversal.In Exercises 11 to 18, state the hypothesis and the conclusion of each statement. Vertical angles are congruent when two lines intersect.In Exercises 11 to 18, state the hypothesis and the conclusion of each statement. All squares are rectangles.In Exercises 11 to 18, state the hypothesis and the conclusion of each statement. Base angles of an isosceles triangle are congruent.In Exercises 19 to 24, classify each statement as true or false. If a number is divisible by 6, then it is divisible by 3.In Exercises 19 to 24, classify each statement as true or false. Rain is wet and snow is cold.In Exercises 19 to 24, classify each statement as true or false. Rain is wet or snow is cold.In Exercises 19 to 24, classify each statement as true or false. If Jim lives in Idaho, then he lives in Boise.In Exercises 19 to 24, classify each statement as true or false. Triangles are round or circles are square.In Exercises 19 to 24, classify each statement as true or false. Triangles are square or circles are round.In Exercises 25 to 32, name the type of reasoning if any used. While participating in an Easter egg hunt, Sarah notices that each of the seven eggs she has found is numbered. Sarah concludes that all eggs used for the hunt are numbered.In Exercises 25 to 32, name the type of reasoning if any used. You walk into your geometry class, look at the teacher, and conclude that you will have quiz today.In Exercises 25 to 32, name the type of reasoning if any used. Lucy knows the rule If a number is added to each side of an equation, then the new equation has the same solution set as the given equation. Given the equation x-5=7, Lucy concludes that x=12.In Exercises 25 to 32, name the type of reasoning if any used. You believe that Anyone who plays major league baseball is a talented athlete. Knowing that Duane Gibson has just been called up to the major leagues, you conclude that Duane Gibson is a talented athlete.In Exercises 25 to 32, name the type of reasoning if any used. As a handcuffed man is brought into the police station, you glance at him and say to your friend, That fellow looks guilty to me.In Exercises 25 to 32, name the type of reasoning if any used. While judging a science fair project, Mr. Cange finds that each of the first 5 projects is outstanding and concludes that all 10 will be outstanding.In Exercises 25 to 32, name the type of reasoning if any used. You know the rule If a person lives in the Santa Rosa Junior College district, then he or she will receive a tuition break at Santa Rosa. Emma tells you that she has received a tuition break. You conclude that she resides in the Santa Junior College district.32EIn Exercises 33 to 36, use intuition to state a conclusion. You are told that the opposite angles formed when two lines cross are vertical angles. In the figure, angles 1 and 2 are vertical angles. Conclusion?34EIn Exercises 33 to 36, use intuition to state a conclusion. The two triangles shown are similar to each other. Conclusion?In Exercises 33 to 36, use intuition to state a conclusion. Observe but do not measure the following angles. Conclusion?37EIn Exercises 37 to 40, use induction to state a conclusion. On Monday, Matt says to you, Andy hit his little sister at school today. On Tuesday, Matt informs you, Andy threw his math book into the wastebasket during class. On Wednesday, Matt tells you, Because Andy was throwing peas in the school cafeteria, he was sent to the principals office. Conclusion?39EIn Exercises 37 to 40, use induction to state a conclusion. At a friends house, you see several food items, including apples, pears, grapes, oranges, and bananas. Conclusion?In Exercises 41 to 50, use deduction to state a conclusion, if possible. If the sum of the measures of two angles is 90, then these angles are called complementary. Angle 1 measures 27 and angle 2 measures 63. Conclusion?In Exercises 41 to 50, use deduction to state a conclusion, if possible. If a person attends college, then he or she will be a success in life. Kathy Jones attends Dade County Community College. Conclusion?In Exercises 41 to 50, use deduction to state a conclusion, if possible. All mathematics teachers have a strange sense of humor. Alex is a mathematics teacher. Conclusion?In Exercises 41 to 50, use deduction to state a conclusion, if possible. All mathematics teachers have a strange sense of humor. Alex has a strange sense of humor. Conclusion?In Exercises 41 to 50, use deduction to state a conclusion, if possible. If Stewart Powers is elected president, then every family will have an automobile. Every family has an automobile. Conclusion?In Exercises 41 to 50, use deduction to state a conclusion, if possible. If Tabby is meowing, then she is hungry. Tabby is hungry. Conclusion?47EIn Exercises 41 to 50, use deduction to state a conclusion, if possible. If a student is enrolled in a literature course, then he or she will work very hard. Bram Spiegel digs ditches by hand six days a week. Conclusion?In Exercises 41 to 50, use deduction to state a conclusion, if possible. If a person is rich and famous, then he or she is happy. Marilyn is wealthy and well known. Conclusion?In Exercises 41 to 50, use deduction to state a conclusion, if possible. If you study hard and hire a tutor, then you will make an A in this course. You make an A in this course. Conclusion?In Exercises 51 to 54, use Venn diagrams to determine whether the argument is valid or not valid. 1. If an animal is a cat, then it makes a meow sound. 2. Tipper is a cat. C Then Tipper makes a meow sound.In Exercises 51 to 54, use Venn diagrams to determine whether the argument is valid or not valid. 1 If an animal is a cat, then it makes a meow sound. 2 Tipper makes a meow sound. C Then Tipper is a cat.In Exercises 51 to 54, use Venn diagrams to determine whether the argument is valid or not valid. 1 All Boy Scouts serve the United States of America. 2 Sean serves the United States of America. C Sean is a Boy Scout.In Exercises 51 to 54, use Venn diagrams to determine whether the argument is valid or not valid. 1 All Boy Scouts serve the United States of America. 2 Sean is a Boy Scout. C Sean serves the United States of America.55EIn Exercises 55 and 56, P is a true statement, while Q and R are false statements. Classify each of the following statements as true or false. a PandQorR b PorQandRIn Exercises 55 and 56, P is a true statement, while Q and R are false statements. Classify each of the following statements as true or false. a PandQor~R b PorQand~RIf line segment AB and line segment CD are drawn to scale, what does intuition tell you about the length of these segments?If angles ABC and DEF were measured with a protractor, what does intuition tell you about the degree measures of these angles?How many endpoints does a line segment have? How many midpoints does a line segment have?Do the points A, B, and C appear to be collinear?How many lines can be drawn that contain both points A and B? How many lines can be drawn that contain points AB and C?Consider noncollinear points A, B, and C. If each line must contain two of the points, what is the total number of lines that are determined by these points?Name all the angles in the figure.Which of the following measures can an angle have? 23, 90, 200, 110.5, -15Must two different points be collinear? Must three or more points be collinear? Can three or more points be collinear?Which symbols correctly expresses the order in which the points A, B and X lie on the given line, A-X-B or A-B-X?Which symbols correctly name the angle shown?ABC, ACB, CBAA triangle is named ABC. Can it also be named ACB? Can it be named BAC?Consider rectangle MNQ. Can it also be named rectangle PQMN? Can it be named rectangle MNQP?Suppose ABC and DEF have the same measure. Which statements are expressed correctly? a mABC=mDEF b ABC=mDEF c mABCmDEF d mABCmDEFSuppose AB and CD have the same length. Which statements are expressed correctly? a AB=CD b AB=CD c ABCD d ABCDWhen two lines cross intersect, they have exactly one point in common. In the drawing, what is the point of intersection? How do the measure of 1 and 2 compare?Judging from the ruler shown not to scale, estimate the measure of each line segment. a AB b CDJudging from the ruler, estimate the measure of each line segment. a EF b GH19E20EConsider the square at the right, RSTV. It has four right angles and four sides of the same length. How are sides RS and ST related? How are sides RS and VT related?Square RSTV has diagonals RT and SV not shown. If the diagonals are drawn, how will their lengths compare? Do the diagonals of a square appear to be perpendicular?Use a compass to draw a circle. Draw a radius, a line segment that connects the center to a point on the circle. Measure the length of the radius. Draw other radii and find their lengths. How do the lengths of the radii compare?Use a compass to draw a circle of radius 1 inch. Draw a chord, a line segment that joins two points on the circle. Draw other chords and measure their lengths. What is the largest possible length of a chord in this circle?The sides of the pair of angles are parallel. Are 1 and 2 congruent?The sides of the pair of angles are parallel. Are 3 and 4 congruent?The sides of the pair of angles are perpendicular. Are 5 and 6 congruent?The sides of the pair of angles are perpendicular. Are 7 and 8 congruent?On a piece of paper, use your compass to construct a triangle that has two sides of the same length. Cut the triangle out of the paper and fold the triangle in half so that the congruent sides coincide one lies over the other. What seems to be true of two angles of that triangle?On a piece of paper, use your protractor to draw a triangle that has two angles of the same measure. Cut the triangle out of the paper and fold the triangle in half so that the angles of equal measure coincide one lies over the other. What seems to be true of two of the sides of that triangle?A trapezoid is a four-sided figure that contains one pair of parallel sides. Which sides of the trapezoid MNPQ appear to be parallel?In the rectangle shown, what is true of the lengths of each pair of opposite sides?33EAn angle is bisected if its two parts have the same measure. Use three letters to name the angle that is bisected.35E36E37E38EABC is straight angle. Using your protractor, you can show that m1+m2=180.Find m1 if m2=56.40E41EIn Exercises 41 to 44,m1+m2=mABC. Find m1 if mABC=68 and m1=m2.43E44EA compass was used to mark off three congruent segment, AB, BC, and CD. Thus, AD has been trisected at points B and C. If AD=32.7, how long is AB?Use your compass and straightedge to bisect EF.In the figure, m1=x and m2=y. If x-y=24, find x and y. HINT: m1+m2=180.In the drawing, m1=x and m2=y. If mRSV=67 and x-y=17, find x and y. (HINT: m1+m2=mRSV).For Exercises 49 to 50, use the following information. Relative to its point departure or some other point of reference, the angle that is used to locate the position of a ship or airplane is called its bearing. The bearing may also be used to describe the direction in which the airplane or ship is moving. By using an angle between 0 and 90, a bearing is measured from the North-South line toward the East or West. In the diagram, airplane A which is 250 miles from Chicagos O Hare airports control tower has a bearing of S 53 W. Find the bearing of airplane B relative to the control tower.For Exercises 49 to 50, use the following information. Relative to its point departure or some other point of reference, the angle that is used to locate the position of a ship or airplane is called its bearing. The bearing may also be used to describe the direction in which the airplane or ship is moving. By using an angle between 0 and 90, a bearing is measured from the North-South line toward the East or West. In the diagram, airplane A which is 250 miles from Chicagos O Hare airports control tower has a bearing of S 53 W. Find bearing of airplane C relative to the control tower.In Exercises 1 and 2, complete the statement. Exercises 1, 2 AB+BC=?_In Exercises 1 and 2, complete the statement. Exercises 1, 2 If AB=BC, then B is the ?_ of AC.In Exercises 3 and 4, use the fact that 1foot=12inches. Convert 6.25 feet to a measure in inches.4E5EIn Exercises 5 and 6, use the fact that 1meter3.28feet measure is approximate. Convert 16.4 feet to meters.7EA cross-country runner jogs at a rate of 15 feet per second. If she runs 300 feet from A to B, 450 feet from B to C, and then 600 feet from C back to A. how long will it take her to return to point A? See the figure for Exercise 7. Exercises 7, 8In Exercises 9 to 28, use the drawings as needed to answer the following questions. Name three points that appear to be a collinear b noncollinear. Exercises 9, 10In Exercises 9 to 28, use the drawings as needed to answer the following questions. How many lines can be drawn through a point A? b points A and B? c points A, B, and C? d points A, B, and D? Exercises 9, 10In Exercises 9 to 28, use the drawings as needed to answer the following questions. Give the meanings of CD,CD,CD,andCD.In Exercises 9 to 28, use the drawings as needed to answer the following questions. Explain the difference, if any, between a CDandDC. c CDandDC. b CDandDC. d CDandDC.In Exercises 9 to 28, use the drawings as needed to answer the following questions. Name two lines that appear to be a parallel. b nonparallel.In Exercises 9 to 28, use the drawings as needed to answer the following questions. Classify as true or false: a AB+BC=AD d AB+BC+CD=AD b ADCD=AB e AB=BC c ADCD=AC Exercises 1417In Exercises 9 to 28, use the drawings as needed to answer the following questions. Exercises 1417 Given: M is the midpoint of AB AM=2x+1 and MB=3x2 Find: x and AMIn Exercises 9 to 28, use the drawings as needed to answer the following questions. Exercises 1417 Given: M is the midpoint of AB AM=2(x+1) and MB=3(x2) Find: x and ABIn Exercises 9 to 28, use the drawings as needed to answer the following questions. Exercises 1417 Given: AM=2x+1, MB=3x+2, and AB=6x4 Find: x and ABIn Exercises 9 to 28, use the drawings as needed to answer the following questions. Can a segment bisect a line? a segment? Can a line bisect a segment? a line?In Exercises 9 to 28, use the drawings as needed to answer the following questions. In the figure. name a two opposite rays. b two rays that are not opposite.20EIn Exercises 9 to 28, use the drawings as needed to answer the following questions. Make a sketch of a two intersecting lines that are perpendicular. b two intersecting lines that are not perpendicular. c two parallel lines.22E23E24EIn Exercises 9 to 28, use the drawings as needed to answer the following questions. Suppose that points A, R, and V are collinear. If AR=7 and RV=5, then which point cannot possibly lie between the other two?In Exercises 9 to 28, use the drawings as needed to answer the following questions. Points A, B, C, and D are coplanar; B, C, and D are collinear; point E is not in plane M. How many planes contain a points A, B, and C? b points B, C, and D? c points A, B, C, and D? d points A, B, C, and E?27EIn Exercises 9 to 28, use the drawings as needed to answer the following questions. Consider the figure for Exercise 27. Given that B is the midpoint of AC and C is the midpoint of BD, what can you conclude about the lengths of a ABandCD? c ACandCD? b ACandBD? Exercises 27, 2829EIn Exercises 29 to 32, use only a compass and a straightedge to complete each construction. Exercises 29, 30 Given: ABandCD(ABCD) Construct: EF on line l so that EF=ABCDIn Exercises 29 to 32, use only a compass and a straightedge to complete each construction. Given: AB as shown in the figure Construct: PQ on line n so that PQ=3(AB) Exercises 31, 3232ECan you use the construction for the midpoint of a segment to divide a line segment into a three congruent parts? c six congruent parts? b four congruent parts? d eight congruent parts?Generalize your findings in Exercise 33. 33. Can you use the construction for the midpoint of a segment to divide a line segment into a three congruent parts? c six congruent parts? b four congruent parts? d eight congruent parts?Consider points A, B, C, and D, no three of which are collinear. Using two points at a time such as A and B, how many lines are determined by these points?Consider noncoplanar points A, B, C, and D. Using three points at a time such as A, B, and C, how many planes are determined by these points?Line l is parallel to plane P that is, it will not intersect P even if extended. Line m intersects line l.What can you conclude about m and P?AB and EF are said to be skew lines because they neither intersect nor are parallel. How many planes are determined by a parallel lines AB and DC? b intersecting lines AB and BC? c skew lines AB and EF? d lines AB, BC, and DC? e points A, B, and F? f points A, C, and H? g points A, C, F, and H? Exercises 3840Exercises 3840 In the box shown for Exercise 38, use intuition to answer each question. a Are AB and DC parallel? b Are AB and FE skew line segments? c Are AB and FE perpendicular?Exercises 3940 In the box shown for Exercise 38, use intuition to answer each question. a Are AG and BC skew line segments? b Are AG and BC congruent line segments? c Are GF and DC parallel?Let AB=a and BC=b. Point M is the midpoint of BC. If AN=23(AB), find the length of NM in terms of a and b.What type of angle has the given measure? a 47 b 90 c 137.3What type of angle has the given measure? a 115 b 180 c 30What relationship, if any, exists between two angles a with measures of 37 and 53? b with measures of 37 and 143?What relationship, if any, exists between two angles a with equal measures? b that have the same vertex and a common side between them?In Exercises 5 to 8, describe in one word the relationship between the angles. ABD and DBCIn Exercises 5 to 8, describe in one word the relationship between the angles. 7 and 8In Exercises 5 to 8, describe in one word the relationship between the angles. 1 and 2In Exercises 5 to 8, describe in one word the relationship between the angles. 3 and 4Use drawings as needed to answer each of the following questions. Must two rays with a common endpoint be coplanar? Must three rays with a common endpoint be coplanar?Suppose that AB,AC,AD,AE, and AF are coplanar, Exercises 10-13 Classify the following as true or false: a mBAC+mCAD=mBAD b BACCAD c mBAEmDAE=mBAC d BACand DAE are adjacent e mBAC+mCAD+mDAE=mBAEExercises 10-13 Without using a protractor, name the type of angle represented by: a BAE b FAD c BAC d FAEExercises 10-13 What, if anything, is wrong with the claim mFAB+mBAE=mFAEExercises 10-13 FAC and CAD are adjacent and AF and AD are opposite rays. What can you conclude regarding FAC and CAD.For Exercises 14 and 15, let m1=x and m2=y. Using variables x and y, write an equation that expresses the fact that 1 and 2 are: a supplementary b congruentFor Exercises 14 and 15, let m1=x and m2=y. Using variables x and y, write an equation that expresses the fact that 1 and 2 are: a complementary b verticalGiven: mRST=39 mTSV=23 Find: mRSV Exercises 1624Exercises 1624 Given: mRSV=59 mTSV=17 Find: mRSTExercises 1624 Given: mRST=2x+9 mTSV=3x2 mRSV=67 Find: xExercises 1624 Given: mRST=2x10 mTSV=x+6 mRSV=4(x6) Find: xandmRSVExercises 1624 Given: mRST=5(x+1)3 mTSV=4(x2)+3 mRSV=4(2x+3)7 Find: xandmRSVExercises 1624 Given: mRST=x2 mTSV=x4 mRSV=45 Find: xandmRSTExercises 1624 Given: mRST=2x3 mTSV=x2 mRSV=49 Find: xandmTSVExercises 1624 Given: STbisectsRSV mRST=x+y mTSV=2x2y mRSV=64 Find: xandyExercises 1624 Given: STbisectsRSV mRST=2x+3y mTSV=3xy+2 mRSV=80 Find: xandyGiven: AB and AC in plane P as shown AD intersects P at points A CABDAC DACDAB What can you conclude?Two angles are complementary. One angle is 12 larger than the other. Using two variable x and y, find the size of each angle by solving a system of equations.Two angles are supplementary. One angle is 24 more than twice the other. Using two variable x and y, find the measure of each angle.For two complementary angles, find an expression for the measure of the second angle if the measure of first is: a x b (3x12) c (2x+5y)Suppose that the two angles are supplementary. Find expressions for the supplements, using the expressions provided in Exercise 28, parts a to c. For two complementary angles, find an expression for the measure of the second angle if the measure of first is: a x b (3x12) c (2x+5v)On the protractor shown, NP bisects MNQ. Find x. Exercises 30,31Exercises 30,31 On the protractor shown for Exercise 30, MNP and PNQ are complementary. Find x.Classify as true or false: a If points P and Q lie in the interior of ABC, then PQ lies in the interior of ABC. b If points P and Q lie in the interior of ABC, then PQ lies in the interior of ABC. c If points P and Q lie in the interior of ABC, then PQ lies in the interior of ABC.33E34EIn Exercises 33 to 40, use only a compass and a straightedge to perform the indicated constructions. Exercises 33-35 Given: Obtuse MRP Construct: Rays RS,RT, and RU so that MRP is Divided into four angles.36EDraw a triangle with three acute angles. Construct angle bisectors for each of the three angles. On the basis of the appearance of your construction, what seems to be true?38E39E40E41EIf mTSV=38,mUSW=40, and mTSW=61, find mUSV. Exercises 44, 45Exercises 44, 45 If mTSU=x+2z,mUSV=xz, and mVSW=2xz, find x if mTSW=60. Also, find z if mUSW=3x6.44E45EWith 0x90, an acute angle has a measure x. Find the difference between the measure of its supplement and the measure of its complement.In Exercises 1 to 6, which property justifies the conclusion of the statement? If 2x=12, then x=6.In Exercises 1 to 6, which property justifies the conclusion of the statement? If x+x=12, then 2x=12.In Exercises 1 to 6, which property justifies the conclusion of the statement? If x+5=12, then x=7.4E5EIn Exercises 1 to 6, which property justifies the conclusion of the statement? If 3x2=13, then 3x=15.In Exercises 7 10, state the property or definition that justifies the conclusion the then clause. Given that s1 and s2 are supplementary, then m1+m2=180.In Exercises 7 10, state the property or definition that justifies the conclusion the then clause. Given that m3+m4=180, then s3 and 4 are supplementary.9EIn Exercises 7 10, state the property or definition that justifies the conclusion the then clause. Given that mRST=mTSV, then ST bisects RSV.11E12E13E14E15E16EIn Exercises 11 to 22, use the Given information to draw a conclusion based on the stated property or definition. Given: 2x3=7; Addition Property of Equality.18E19E20EIn Exercises 11 to 22, use the Given information to draw a conclusion based on the stated property or definition. Given: 3(2x1)=27; Distributive Property22EIn Exercises 23 to 24, fill in the missing reasons for the algebraic proof. Given: 3(x5)=21 Prove: x=12 PROOF Statements Reasons 1. 3(x5)=21 1. ? 2. 3x15=21 2. ? 3. 3x=36 3. ? 4. x=12 4. ?In Exercises 23 to 24, fill in the missing reasons for the algebraic proof. Given: 2x+9=3 Prove: x=3 PROOF Statements Reasons 1. 2x+9=3 1. ? 2. 2x=6 2. ? 3. x=3 3. ?25E26E27EIn Exercises 27 to 30, fill in the missing reasons for each geometric proof. Given: E is the midpoint of DF Prove: DE=12(DF) Exercises 27, 28 PROOF Statements Reasons 1. E is the midpoint of DF 1. ? 2. DE=EF 2. ? 3. DE+EF=DF 3. ? 4. DE+DE=DF 4. ? 5. 2(DE)=DF 5. ? 6. DE=12(DF) 6. ?In Exercises 27 to 30, fill in the missing reasons for each geometric proof. Given: BD bisects ABC Prove: mABD=12(mABC) Exercises 29, 30 PROOF Statements Reasons 1. BD bisects ABC 1. ? 2. mABD=mDBC 2. ? 3. mABD+mDBC=mABC 3. ? 4. mABD+mABD=mABC 4. ? 5. 2(mABD)=mABC 5. ? 6. mABD=12(mABC) 6. ?In Exercises 27 to 30, fill in the missing reasons for each geometric proof. Given: ABC and BD Prove: mABD=mABCmDBC PROOF Statements Reasons 1. ABC and BD 1. ? 2. mABD+mDBC=mABC 2. ? 3. mABD=mABCmDBC 3. ? Exercises 29, 30In Exercises 31 and 32, fill in the missing statements and reasons. Given: M-N-P-Q on MQ Prove: MN+NP+PQ=MQ PROOF Statements Reasons 1. ? 1. ? 2. MN+NQ=MQ 2. ? 3. NP+PQ=NQ 3. ? 4. 4. Substitution Property of EqualityIn Exercises 31 and 32, fill in the missing statements and reasons. Given: TSW with SU and SV Prove: mTSW=mTSU+mUSV+mVSW PROOF Statements Reasons 1. ? 1. ? 2. mTSW=mTSU+mUSW 2. ? 3. mUSW=mUSV+mVSW 3. ? 4. ? 4. Substitution Property of Equality33E34E35EThe Division Property of Inequality requires that we reverse the inequality symbol when dividing by a negative number. Given that 124, form the inequality that results when we divide each side by 4.37EWrite a proof for: If a=b and c=d, then ac=bd. HINT: Use Exercise 39 as a guide 39. Provide reasons for this proof. If a=b and c=d, then a+c=b+d. PROOF Statements Reasons 1. a=b 1. ? 2. a+c=b+c 2. ? 3. c=d 3. ? 4. a+c=b+d 4. ?In Exercise 1 and 2, supply reasons. Given: 13 Prove: MOPNOQ PROOF Statements Reasons 1. 13 1. ? 2. m1m3 2. ? 3. m1+m2=mMOP and m2+m3=mNOQ 3. ? 4. m1+m2=m2+m3 4. ? 5. mMOP=mNOQ 5. ? 6. MOPNOQ 6. ?In Exercise 1 and 2, supply reasons. Given: AB intersects CD at O so that 1 is a right Use the figure following Exercise 1. Prove: 2 and 3 are complementary. PROOF Statements Reasons 1. AB intersects CD at O. 1. ? 2. AOB is a straight , so mAOB=180 2. ? 3. m1+mCOB=mAOB 3. ? 4. m1+mCOB=180 4. ? 5. 1 is a right angle 5. ? 6. m1=90 6. ? 7. 90+mCOB=180 7. ? 8. mCOB=90 8. ? 9. m2+m3=mCOB 9. ? 10. m2+m3=90 10. ? 11. 2 and 3 are complementary 11. ?In Exercise 3 and 4, supply statements. Given: 12 and 23 Prove: 13 PROOF Statements Reasons 1. ? 1. Given 2. ? 2. Transitive Property of CongruenceIn Exercise 3 and 4, supply statements. Given: mAOBm1mBOCm1 Prove: OB bisects AOC PROOF Statements Reasons 1. ? 1. Given 2. ? 2. Substitution 3. ? 3. Angles with equal measures are congruent. 4. ? 4. If a ray divides an angle into two congruent angles, then the ray bisects the angle.In Exercise 5 to 9, use a compass and a straightedge to complete the constructions. Given: Point N on line s Construct: Line m through N so that msIn Exercises 5 to 9, use a compass and a straightedge to complete the constructions. Given: OA Construct: Right angle BOA HINT: Use a straightedge to extend OA to the left.In Exercise 5 to 9, use a compass and a straightedge to complete the constructions. Given: Line l containing point A Construct: A 45 angle with vertex A8E9E10EIn Exercise 11 and 12, provide the missing statements and reasons. Given: s 1 and 3 are complementary s 2 and 3 are complementary Prove: 12 PROOF Statements Reasons 1. s 1 and 3 are complementary s 2 and 3 are complementary 1. ? 2. m1+m3=90;m2+m3=90 2. The sum of the measures of complementary s is 90 3. m1+m3=m2+m3 3. ? 4. ? 4. Subtraction Property of Equality 5. ? 5. If two s are = in measure, they areIn Exercise 11 and 12, provide the missing statements and reasons. Given: 12;34 s 2 and 3 are complementary Prove: s 1 and 4 are complementary PROOF Statements Reasons 1. 13 and 34 1. ? 2. ? and ? 2. If two s are , then their measures are equal. 3. s 2 and 3 are complementary 3. ? 4. ? 4. The sum of the measures of complementary s is 90 5. m1+m4=90 5. ? 6. ? 6. If the sum of the measures of two angles is 90, then the angles are complementaryDoes the relation is perpendicular to have a reflexive property consider line l? A symmetric property consider lines l and m? A transitive property consider lines l, m and n?Does the relation is greater than have a reflexive property consider real number a? A symmetric property consider real numbers a and b? A transitive property consider real numbers a, b and cDoes the relation is complementary to for angles have a reflexive property consider one angle? A symmetric property consider two angles? A transitive property consider three angles?Does the relation is less than for a numbers have a reflexive property consider one number? A symmetric property consider two numbers? A transitive property consider three numbers?Does the relation is a brother of have a reflexive property consider one male? A symmetric property consider two males? A transitive property consider three males?Does the relation is in love with have a reflexive property consider one person? a symmetric property consider two people? a transitive property consider three people?19E20E21E22EProve the Extended Segment Addition Property by using the Drawing, the Given and the Prove that follow. Given: MNPQ on MQ Prove: MN+NP+PQ=MQThe Segment-Addition Postulate can be generalized as follows: The length of a line segment equals the sum of the length of its parts. State a general conclusion regarding AE based on the following figure.Prove the Extended Angle Addition Property by using the Drawing, the Given, and the Prove that follow. Given: TSW with SU and SV Prove: mTSW=mTSU+mUSV+mVSWThe Angle-Addition Postulate can be generalized as follows: The measure of an angle equals the sum of measures of its parts. State a general conclusion regarding mGHK based on the figure shown.27EIn the proof below, provide the missing reasons. Given: 1 and 2 are complementary 1 is acute Prove: 2 is also acute PROOF Statements Reasons 1. 1 and 2 are complementary 1. ? 1 2. m1+m2=90 2. ? 3. 1 is acute 3. ? 3 4. Where m1=x, 0x90 4. ? 2 5. x+m2=90 5. ? 5 6. m2=90x 6. ? 4 7. x090x 7. ? 7 8. 90x90180x 8. ? 7, 8 9. 090x90 9. ? 6, 9 10. 0m/290 10. ? 10 11. 2 is acute 11. ?29E30EIn Exercises 1 to 6, state the hypothesis H and the conclusion C for each statement. If a line segment is bisected, then each of the equal segments has half the length of the original segment.In Exercises 1 to 6, state the hypothesis H and the conclusion C for each statement. If two sides of a triangle are congruent, then the triangle is isosceles.In Exercises 1 to 6, state the hypothesis H and the conclusion C for each statement. All squares are quadrilaterals.In Exercises 1 to 6, state the hypothesis H and the conclusion C for each statement. Every regular polygon has congruent interior angles.5EIn Exercises 1 to 6, state the hypothesis H and the conclusion C for each statement. The lengths of corresponding sides of similar polygons are proportional.Name, in order, the five parts of the formal proof of a theorem.Which part hypothesis or conclusion of a theorem determines the a Drawing? b Given? c Prove?Which part Given or Prove of the proof depends upon the a hypothesis of theorem? b conclusion of theorem?Which of the following can be cited as a reason in a proof? a Given c Definition b Prove d PostulateWhen can a theorem be cited as a reason reason in a proof?Based upon the hypothesis of a theorem, do the drawings of different students have to be identical same names for vertices, etc.?13E14EFor each theorem stated in Exercises 13 to 18, make a Drawing. On the basis of your Drawing, write a Given and a Prove for the theorem. If two angles are complementary to the same angle, then these angles are congruent.16EFor each theorem stated in Exercises 13 to 18, make a Drawing. On the basis of your Drawing, write a Given and a Prove for the theorem. If two lines intersect, then the vertical angles formed are congruent.18E19E20E21E22E23E24EIn Exercises 19 to 26, use the drawing in which AC intersects DBat point O. If m2=x210 and m3=x3+40, find x and m2.In Exercises 19 to 26, use the drawing in which AC intersects DBat point O. If m1=x+20 and m4=x3, find x and m4.27E28EIn Exercises 27 to 35, complete the formal proof of each theorem. If two lines intersect, the vertical angles formed are congruent.30E31E32EIn Exercises 27 to 35, complete the formal proof of each theorem. If two angles are congruent, then their bisectors separate these angles into four congruent angles. Given: ABCEFG BD bisect ABC FH bisect EFG Prove: 1234In Exercises 27 to 35, complete the formal proof of each theorem. The bisectors of two adjacent supplementary angles form a right angle. Given: ABC is supplementary to CBD BE bisects ABC BF bisects CBD Prove: EBF is a right angleIn Exercises 27 to 35, complete the formal proof of each theorem. The supplement of an acute angle is an obtuse angle. HINT: Use Exercise 28 of Section 1.4 as a guide.Name the four components of a mathematical system.Name three types of reasoning.Name the four characteristics of a good definition.