1. a pair of parallel segments. 2. a pair of skew segments. 3. a pair of perpendicular segments. 4. a pair of parallel planes

Transcription

1 Practice Lines and Angles For Exercises 1 4, identify each of the following in the figure. 1. a pair of parallel segments 2. a pair of skew segments 3. a pair of perpendicular segments 4. a pair of parallel planes In Exercises 5 10, give one example of each from the figure. 5. a transversal 6. parallel lines 7. corresponding angles 8. alternate interior angles 9. alternate exterior angles 10. same-side interior angles Use the figure for Exercises The figure shows a utility pole with an electrical line and a telephone line. The angled wire is a tension wire. For each angle pair given, identify the transversal and classify the angle pair. (Hint: Think of the utility pole as a line for these problems.) and and and and 3

2 Practice Angles Formed by Parallel Lines and Transversals Find each angle measure. 1. m 1 2. m 2 3. m ABC 4. m DEF Complete the two-column proof to show that same-side exterior angles are supplementary. 5. Given: p q Prove: m 1 m Proof: Statements 1. p q 1. Given 2. a. 2. Lin. Pair Thm. Reasons b. 4. c. 4. Def. of s 5. d. 5. e. 6. Ocean waves move in parallel lines toward the shore. The figure shows Sandy Beaches windsurfing across several waves. For this exercise, think of Sandy s wake as a line. m 1 (2x 2y) and m 2 (2x + y). Find x and y. x = y =

3 Practice Proving Lines Parallel Use the figure for Exercises 1 8. Tell whether lines m and n must be parallel from the given information. If they are, state your reasoning. (Hint: The angle measures may change for each exercise, and the figure is for reference only.) m 3 (15x 22), m 1 (19x 10), x m 2 (5x 3), m 3 (8x 5), x m 8 (6x 1), m 4 (5x 3), x m 6 (x 10), m 2 (x 15) 9. Look at some of the printed letters in a textbook. The small horizontal and vertical segments attached to the ends of the letters are called serifs. Most of the letters in a textbook are in a serif typeface. The letters on this page do not have serifs, so these letters are in a sans-serif typeface. (Sans means without in French.) The figure shows a capital letter A with serifs. Use the given information to write a paragraph proof that the serif, segment HI, is parallel to segment JK. Given: 1 and 3 are supplementary. Prove: HI JK

4 Practice Perpendicular Lines For Exercises 1 4, name the shortest segment from the point to the line and write an inequality for x. (Hint: One answer is a double inequality.) Complete the two-column proof. 5. Given: m n Prove: 1 and 2 are a linear pair of congruent angles. Proof: Statements Reasons 1. a. 1. Given 2. b. 2. Def. of c. 4. m 1 m Add. Prop. of 5. d. 5. Def. of linear pair 6. The Four Corners National Monument is at the intersection of the borders of Arizona, Colorado, New Mexico, and Utah. It is called the four corners because the intersecting borders are perpendicular. If you were to lie down on the intersection, you could be in four states at the same time the only place in the United States where this is possible. The figure shows the Colorado-Utah border extending north in a straight line until it intersects the Wyoming border at a right angle. Explain why the Colorado-Wyoming border must be parallel to the Colorado New Mexico border.

5 Slopes of Lines Use the slope formula to determine the slope of each line. 1. AB 2. CD 3. EF 4. GH Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 5. IJ and KL for I(1, 0), J(5, 3), K(6, 1), 6. PQ and RS for P(5, 1), Q( 1, 1), R(2, 1), and L(0, 2) and S(3, 2) 7. At a ski resort, the different ski runs down the mountain are color-coded according to difficulty. Green is easy, blue is medium, and black is hard. Assume that the ski runs below are rated only according to their slope (steeper is harder) and that there is one green, one blue, and one black run. Assign a color to each ski run. 4 Emerald m 7 5 Diamond m 4 5 Ruby m 8

6 Practice Lines in the Coordinate Plane Write the equation of each line in the given form. 1. the horizontal line through (3, 7) in 2. the line with slope point-slope form point-slope form 8 through (1, 5) in 5 _ the line through, 2 2 and (2, 14) in 4. the line with x-intercept 2 and y-intercept slope-intercept form 1 in slope-intercept form _ Graph each line y 3 ( x 1) y x 2 3 Determine whether the lines are parallel, intersect, or coincide x 5y 0, y 1 ( x 5) 5 1 2y 2 x, x 1 y y 4( x 3), 4y x 4 4 An aquifer is an underground storehouse of water. The water is in tiny crevices and pockets in the rock or sand, but because aquifers underlay large areas of land, the amount of water in an aquifer can be vast. Wells and springs draw water from aquifers. 10. Two relatively small aquifers are the Rush Springs (RS) aquifer and the Arbuckle- Simpson (AS) aquifer, both in Oklahoma. Suppose that starting on a certain day in 1985, 52 million gallons of water per day were taken from the RS aquifer, and 8 million gallons of water per day were taken from the AS aquifer. If the RS aquifer began with 4500 million gallons of water and the AS aquifer began with 3000 million gallons of water and no rain fell, write a slope-intercept equation for each aquifer and find how many days passed until both aquifers held the same amount of water. (Round to the nearest day.)

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8 Problem Solving Lines and Angles Use the diagram of the rectangular box for Exercises 1 and 2. Refer to the diagram to help justify your answer. 1. Is the relationship is skew to transitive? 2. If a segment is skew to one of two parallel segments, must it be skew to the other? Use the flag of Puerto Rico for Exercises 3 and If DFC and ACF are same-side interior angles, identify the transversal. 4. Name a pair of alternate interior angles if the transversal is BE. Choose the best answer. 5. Describe the type of lines suggested by the two skis of a person water skiing. A intersecting lines B parallel lines C perpendicular lines D skew lines 6. Describe the type of lines suggested by the paths of two people at a fair when one person is riding the aerial ride from one end of the fair to the other, and the other person is walking in a different direction on the ground. F intersecting G parallel H perpendicular J skew 7. In the quilt pattern, which is a true statement about the angles formed by the transversal HK and HM and JL? A LSK and PHQ are corresponding angles. B JSQ and JQH are corresponding angles. C LSK and QSJ are same-side interior angles. D PHQ and RLS are same-side interior angles.

9 Problem Solving Angles Formed by Parallel Lines and Transversals Find each value. Name the postulate or theorem that you used to find the values. 1. In the diagram of movie theater seats, 2. In the diagram, roads a and b are parallel. the incline of the floor, f, is parallel to the seats, s. If m 1 68, what is x? What is the measure of PQR? 3. In the diagram of the gate, the horizontal bars are parallel and the vertical bars are parallel. Find x and y. Use the diagram of a staircase railing for Exercises 4 and 5. AG CJ and AD FJ. Choose the best answer. 4. Which is a true statement about the measure of DCJ? A It equals 30, by the Alternate Interior Angles Theorem. B It equals 30, by the Corresponding Angles Postulate. C It equals 50, by the Alternate Interior Angles Theorem. D It equals 50, by the Corresponding Angles Postulate. 5. Which is a true statement about the value of n? F It equals 25, by the Alternate Interior Angles Theorem. G It equals 25, by the Same-Side Interior Angles Theorem. H It equals 35, by the Alternate Interior Angles Theorem. J It equals 35, by the Same-Side Interior Angles Theorem.

10 Problem Solving Proving Lines Parallel 1. A bedroom has sloping ceilings as shown. Marcel is hanging a shelf below a rafter. If m 1 (8x 1), m 2 (6x 7), and x 4, show that the shelf is parallel to the rafter above it. 2. In the sign, m 3 (3y 7), m 4 (5y 5), and y 21. Show that the sign posts are parallel. Choose the best answer. 3. In the bench, m EFG (4n 16), m FJL (3n 40), m GKL (3n 22), and n 24. Which is a true statement? A FG HK by the Converse of the Corr. s Post. B FG HK by the Converse of the Alt. Int. s Thm. C EJ GK by the Converse of the Corr. s Post. D EJ GK by the Converse of the Alt. Int. s Thm. 4. In the windsurfing sail, m 5 (7c 1), m 6 (9c 1), m 7 17c, and c 6. Which is a true statement? F RV is parallel to SW. G SW is parallel to TX. H RT is parallel to VX. J Cannot conclude that two segments are parallel The figure shows Natalia s initials, which are monogrammed on her duffel bag. Use the figure for Exercises 5 and If m 1 (4x 24), m 2 (2x 8), 6. If m 3 (7x 13), m 4 (5x 35), and x 16, show that the sides of and x 11, show that the sides of the the letter N are parallel. letter H are parallel.

11 Problem Solving Perpendicular Lines A wall rack for holding CDs is shown. Use the figure for Exercises 1 and Explain why HK must be perpendicular to KL. 2. If JM HK, explain why JM GH. 3. The valve pistons on a trumpet are all perpendicular to the lead pipe. Explain why the valve pistons must be parallel to each other. Use the diagram of a bocce court for Exercises 4 and 5. Choose the best answer. 4. If m 1 m 2, what can you conclude? A BH GJ C BH CJ B AC BH D AC GJ 5. The pitch lines are parallel, and the first pitch line is perpendicular to the long sides of the court. Which is a correct conclusion? F BH CJ H EL AF G BH CJ J DK AF

12 Problem Solving Slopes of Lines Graph the line that represents each situation. Then find and interpret the slope of the line. 1. Mara is jogging at a constant speed. 2. A turtle swimming at a constant speed She jogs 2 miles in 14 minutes. After 35 travels 12 miles by 3:00 P.M. and 28 miles minutes, she has jogged 5 miles. Graph by 7:00 P.M. Graph the line that represents the line that represents Mara s distance the turtle s distance traveled. traveled. Choose the best answer. 3. A hang glider who started at 7:55 A.M. 4. The line represents the distance traveled has traveled at a constant speed as by an in-line skater traveling at a constant shown in the table. speed. What is the rate of change Time Distance Traveled 8:00 A.M. 2 mi 8:30 A.M. 14 mi If the line that represents the hang glider s distance traveled is graphed, which is a true interpretation of the slope? A The hang glider is traveling at an average speed of 24 miles per hour. B The hang glider is traveling at an average speed of 16 miles per hour. C The hang glider is traveling at an average speed of 12 miles per minute. D The hang glider is traveling at an average speed of 7 miles per minute. F 25 mi/h G 15 mi/h H 10 mi/h J 0.1 mi/h

13 Problem Solving Lines in the Coordinate Plane Use the following information for Exercises 1 and 2. Josh can order 1 color ink cartridge and 2 black ink cartridges for his printer for $78. He can also order 1 color ink cartridge and 1 black ink cartridge for $ Let x equal the cost of a color ink 2. What is the cost of each cartridge? cartridge and y equal the cost of a black ink cartridge. Write a system of equations to represent this situation. 3. Ms. Williams is planning to buy T-shirts for the cheerleading camp that she is running. Both companies total costs would be the same after buying how many T-shirts? Use a graph to find your solution. Art Creation Fee Cost per T-shirt Company A $70 $10 Company B $50 $12 Choose the best answer. 4. Two floats begin a parade at different 5. A piano teacher charges $20 for each half times, but travel at the same speeds. hour lesson, plus an initial fee of $50. Which is a true statement about the Another teacher charges $40 per hour, lines that represent the distance traveled plus a fee of $50. Which is a true statement by each float at a given time? A The lines intersect. about the lines that represent the total cost by each piano teacher? B The lines are parallel. F The lines intersect. C The lines are the same. G The lines are parallel. D The lines have a negative slope. H The lines are the same. 6. Serina is trying to decide between two J The lines have a negative slope. similar packages for starting her own Web site. Which is a true statement? A Both packages cost $ for 5 months. B Both packages cost $295 for 10 months. C Both packages cost $355 for 15 months. D The packages will never have the same cost. Design and Setup Monthly Fee to Host Package A $ $14.50 Package B $ $12.00

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