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Transcript
Chapter 4 Parallel and Perpendicular Lines Chapter Content Lessons Standards 4.1 G-CO.1; G-CO.9 Parallel Lines and Angles Parallel and Perpendicular Lines Corresponding Angles Alternate Interior and Exterior Angles Consecutive Interior Angles 4.2 More on Parallel Lines and Angles G-CO.9 Angle Relationships Parallel Lines and Angles Multi-Part Problem Practice 4.3 Perpendicular Lines G-CO.9; G-CO.12 Pairs of Perpendicular Lines Construction: Perpendicular and Parallel Lines Distance and Perpendicular Lines 4.4 Parallel Lines, Perpendicular Lines, and Slope G-GPE.5 Determining When Lines Are Parallel or Perpendicular Rotations and Perpendicular Lines Distance Between a Point and a Line Multi-Part Problem Practice 4.5 Parallel Lines and Triangles Interior and Exterior Angles Angles in a Right Triangle CHAPTER 4 REVIEW CUMULATIVE REVIEW FOR CHAPTERS 1–4 148 Chapter 4: : Parallel and Perpendicular Lines G-CO.10 Chapter Vocabulary alternate exterior angle coplanar parallel alternate interior angle corollary perpendicular bisector auxiliary line exterior angle skew line consecutive interior angle interior angle transversal LESSON 4.1 4.1 Parallel Lines and Angles Parallel and Perpendicular Lines Two lines that never intersect are either parallel lines or skew lines. If they are also coplanar—in the same plane— the lines are parallel. • Lines a and b are parallel, which is written as a i b. They are indicated by the pair of parallel lines in the diagram. c b M a a || b N If two lines are not in the same plane, and they never intersect, then they are called skew lines. • Lines b and c are skew lines. Two planes that do not intersect are parallel planes. • The planes M and N are parallel, which is written M i N. The symbol for parallel is a pair of vertical lines: i With parallel lines defined, we can state the parallel postulate. 4.1 Parallel Lines and Angles 149 Similar to the case for parallel lines, there is also a unique perpendicular line through a given point. Corresponding Angles A transversal is a line that intersects two other lines. Corresponding angles are formed by a transversal and located at the same position relative to the transversal. In other words, they are at the same location of each intersection, such as above a line and to the left of the transversal. In the diagram, the corresponding angles are the pairs A and E, B and F, C and G, and D and H. A B D C E F H G The corresponding angles postulate lets us define the relationship between four pairs of angles. MODEL PROBLEMS 1. A pair of corresponding angles created by a transversal and two lines have measures described by the expressions 2x 1 3 and 3x 2 11. If x 5 14, the lines must be A. B. C. D. Parallel. Perpendicular. Neither parallel nor perpendicular. Not planar. SOLUTION The answer is A. Evaluating the expressions with x 5 14, the result is 31 in both cases. The angles are congruent, and with congruent corresponding angles, the lines must be parallel. Model Problems continue . . . 150 Chapter 4: Parallel and Perpendicular Lines MODEL PROBLEMS continued a 2. What is mH? b SOLUTION A = 42° a and b are parallel lines. Angle H is a corresponding angle with D, so if we can calculate mD, we know mH , since the angles are congruent. A and D are supplementary D mA 1 mD 5 180° E C F H d G A and D are linear angles, which means they are supplementary and sum to 180°. Substitute for mA and solve. 42° 1 mD 5 180° mD 5 138° D and H are congruent B mH 5 mD 5 138° The two angles are corresponding angles and the lines are parallel, so mH 5 mD 5 138°. Alternate Interior and Exterior Angles A transversal crosses two lines. Alternate interior angles are inside the parallel lines on opposite sides of the transversal. C and E are alternate interior angles, as are D and F. They are interior to (inside) the parallel lines and on alternate (different) sides of the transversal. D E The pair of orange arrowheads indicate parallel lines. C F Alternate exterior angles are outside the parallel lines, on opposite sides of the transversal. A and G are alternate exterior angles, as are B and H. They are exterior to (outside) the parallel lines and on alternate (different) sides of the transversal. A H B G 4.1 Parallel Lines and Angles 151 Instead of stating that corresponding angles are congruent (when a transversal intersects parallel lines) as a postulate, we could state it as a theorem. To do this, assume that the alternate exterior angles theorem is a postulate. We use a paragraph proof to prove the theorem. Paragraph Proof With alternate exterior angles stated as a postulate, we have one set of angles shown to be congruent. Alternate interior angles would then be congruent too, since they are vertical angles to the alternate exterior angles, and vertical angles are congruent. The other pairs of corresponding angles would have to be congruent since they are supplementary to congruent angles. In this activity, prove the alternate exterior angles theorem converse by proving that lines m and n are parallel. 152 Chapter 4: Parallel and Perpendicular Lines MODEL PROBLEMS 1. MP 7 In the diagram, angles A and F have no direct relationship. What is mF? A = 130° B D C SOLUTION Look for angle that relates to F Start the problem by looking for an angle with a known relationship to F. For instance, B and F are corresponding angles, so if B is known, then F can be calculated. and B are A supplementary mA 1 mB 5 180° 130° 1 mB 5 180° m B 5 50° and F are B corresponding angles B F mB 5 mF 5 50° E F H G A and B are supplementary angles. Use an equation that says the angles sum to 180° and solve for mB. ince the two angles are congruent, they S have the same measure. 2. MP 1 mA is 50% greater than mD. What is mF? A B D C E F H G SOLUTION elationship of R A and D Calculate D Write equation Solve mA 5 1.5(mD) mA 1 mD 5 180° 1.5(mD) 1 mD 5 180° 2.5(mD) 5 180° mD 5 72° D and F are alternate interior angles F D mF 5 mD 5 72° mA is 50% greater than mD. This means mA equals 1.5 times mD. ngles A and D are supplementary, A so they sum to 180°. Replace mA with 1.5mD. olve for mD by combining like terms, S and then dividing both sides by 2.5. D and F are congruent since they are alternate interior angles. Consecutive Interior Angles Consecutive interior angles are angles on the same side of the transversal and between the lines. In the diagram, D and E are consecutive interior angles, and C and F are consecutive interior angles. D C E F 4.1 Parallel Lines and Angles153 MODEL PROBLEMS y x 1. What is mE? D = 35° A C B H E G F SOLUTION Look for a relationship between E and D. B is the connection between D and E. Vertical angles mB 5 mD 5 35° Angles B and D are vertical angles, and therefore congruent. Consecutive interior angles are supplementary mB 1 mE 5 180° Substitute D for B in the relationship of consecutive interior angles. MP 2, 4 3rd and 4th Avenues are parallel. The measure of the angle between 4th and 32nd Street is 65°. The measure of the angle between 3rd and Broadway is 85°. What is the measure of the angle between Broadway and 32nd Street? SOLUTION Identify consecutive interior angles (65° 1 x) 1 85° 5 180° Solve equation 65° 1 x 1 85° 5 180° 150° 1 x 5 180° x 5 30° 154 Broadway 32n x 85° 65° d Stre et The angles between Broadway and 4th Avenue and Broadway and 3rd Avenue are consecutive interior angles. Consecutive interior angles sum to 180° Chapter 4: Parallel and Perpendicular Lines 3rd Avenue Complete the calculation. 4th Avenue 35° 1 mE 5 180° mE 5 145° Solve 2. mD 1 mE 5 180° Consecutive interior angles sum to 180°. Solve for x. The measure of the angle between Broadway and 32nd Street is 30°. We take advantage of the consecutive interior angles theorem to solve this problem. PRACTICE 1. Non-coplanar lines that never intersect are called A. B. C. D. 7. /3 1 /5 5 148°. Determine the measure of /2. Parallel. Perpendicular. Transversal. Skew. 4 1 3 2 8 5 7 6 p 2. Lines j and k are skew. Which of these is true? Choose all that apply. A. B. C. D. E. j and k intersect. j and k do not intersect. j and k are in the same plane. j and k are in different planes. j and k are parallel. 3. On a plane, point A doesn’t lie on the line a. Three new lines are drawn through point A. How many of these lines could intersect a? A. B. C. D. Only one Only two All three Zero x A. B. C. D. y 32° 106° 74° 148° 8. Angles A and G are A B D C E F H G 4. On a plane, point A doesn’t lie on line b. Three new, distinct lines are drawn through point A. How many of these lines could be perpendicular to line b? A. One could be B. One or two could be C. All three could be A. B. C. D. Consecutive interior angles. Alternate interior angles. Alternate exterior angles. Corresponding angles. 9. Choose all true statements that apply to the diagram below. 5. Point A is not on line j. How many lines pass through A and are perpendicular to j? A. 0 B. 1 C. 2 D. An infinite number 6. Which of these are pairs of corresponding angles? Choose all that apply. B C F G A D E H A. A and B B. A and C C. A and E J K I L Lines l and m are skew lines. Lines l and m are perpendicular. Lines l and n are perpendicular. Lines l, m, and n are in the same plane, P. E. Lines m and n are skew lines. F. Lines m and n are perpendicular. G. Lines l and n are parallel. H. Lines m and n are in the same plane, P. A. B. C. D. D. A and H E. A and I Practice Problems continue . . . 4.1 Parallel Lines and Angles155 Practice Problems continued . . . 10. The lines m and n are intersected by a transversal d. List all pairs of corresponding angles formed in the diagram below. 14. m/FPM 5 34°. Find the measure of /NMP. m 1 4 2 3 5 6 8 M F N P n 7 d 11. In the diagram below, if m/6 5 100°, what is the measure of /2? m 2 1 n 5 6 8 15. Lines x and y are parallel. If m/C 5 120°, what is m/F? 3 B 4 A 7 12. The lines IJ and KL are parallel. Use the diagram below to answer the questions. A B C D E K D F E J H F G L aList all the pairs of alternate interior angles. bList all the pairs of alternate exterior angles. cIf m/D 5 40°, what is the measure of /E? dIf m/B 5 145°, what is the measure of /H? 13. MP 2 When two parallel lines are intersected by a transversal, one of the angles formed is equal to 150°. Make a sketch, labeling all the angles formed. Then find the measures of all the angles in your diagram. G H y x p I C 16. MP 2, 3 The lines a, b, c, and p are coplanar. a is perpendicular to p, b is perpendicular to p, and c intersects a. Do lines b and c intersect? Make a sketch and explain your answer. 17. MP 2, 4 5th and 10th Avenues are parallel; K Street runs parallel to J Street. Lincoln Way is perpendicular to 10th Avenue. K Street and Lincoln Way make an angle of 39°, and 5th Avenue and J Street make an angle of 129°. N W KS S reet J St 39° Lincoln Way 129° 5th Ave. t tree E 10th Ave. aAmid is going northeast on J Street and turning left on Lincoln Way. By what angle does he need to turn? bRachel goes north on 10th Avenue and needs to turn right onto K Street. By what angle does she need to turn? Practice Problems continue . . . 156 Chapter 4: Parallel and Perpendicular Lines Practice Problems continued . . . 18. MP 3 In the diagram below, line m intersects the sides of the angle M> at the >points A and B. Can both MA and MB be perpendicular to line m? Use the perpendicular postulate to explain your answer. 23. The parallel lines a and b are intersected by a transversal c. c a 1 4 b 2 3 5 A M B m 19. MP 3 m/2 1 m/8 5 46°. Find the measures of all angles in the diagram below. Show your work. 4 24. A B D C E F H G 2 b 3 6 5 7 8 Exercises 20–22: Lines x and y are parallel. x aWhat is m/C, if m/F = 110°? b Calculate the sum of m/D and m/E. 25. MP 3 The lines a and b intersect. Selena says she can draw a third line, c, that is parallel to both a and b. Is Selena right? Explain why or why not. c C B D A 7 aIf /2 = 108°, what is the measure of /5? bIf /3 = 45°, what is the measure of /8? cIf /1 = 82°, what is the measure of /6? dIf /7 = 29°, what is the measure of /1? a 1 8 6 F E G H y 20. m/B is 4 times m/A. What is m/F? 21. m/B is 3 times m/A. What is m/F? 22. List all pairs of consecutive interior angles. 26. In a triangle ABC, how many lines passing through the point C can be drawn so they are parallel to the side AB? Make a sketch and use the parallel postulate to explain your answer. 2 7. Lines a, b, and c are coplanar. The lines a and b are both perpendicular to line c. Show that a is parallel to b. Make a sketch to explain your answer. Hint: Suppose that line a is not parallel to line b, and use the perpendicular postulate to show that this leads to a contradiction. 28. MP 3, 4 The system of three weights is balanced as shown. All vertical strings are parallel. Show that m/A 1 m/B 5 m/C. A B C 4.1 Parallel Lines and Angles157 LESSON 4.2 4.2 More on Parallel Lines and Angles Angle Relationships We prove alternate exterior angles are congruent when a transversal intersects two parallel lines. Prove: /A > /G Given: m i n A B D C E F H G m n Statement Reason min Given We were told that m and n are parallel. /A > /E Corresponding angles postulate Since A and E are corresponding angles, the corresponding angles postulate says they are congruent. /E > /G Vertical angles Since E and G are vertical angles, congruence theorem they are congruent. /A > /G Transitive property of congruence The transitive property of congruence tells us that if /A > /E and /E > /G, then /A > /G. In this activity, you prove the alternate interior angles theorem. 158 Chapter 4: Parallel and Perpendicular Lines MODEL PROBLEM Find the measures of all the angles. Show two different ways to solve for the angles. The measure of angle G is 65°. a B A D C E H b F G SOLUTION Method 1 Calculate corresponding angles mC 5 mG 5 65° C and G are corresponding, so they are congruent. alculate C supplementary angles mC 1 mD 5 180° C and D are supplementary, so calculate mD. 65° 1 mD 5 180° mD 5 115° Calculate supplementary angles mG 1 mH 5 180° 65° 1 mH 5 180° alculate another angle using C a supplementary angle. mH 5 115° alculate vertical C angles mA 5 mC 5 mE 5 mG 5 65° Fill in the other angles. mB 5 mD 5 mF 5 mH 5 115° Method 2 Use alternate exterior angles mG 5 mA 5 65° G and A are alternate exterior angles, so they are congruent. Use vertical angles mA 5 mC 5 65° There are two pairs of vertical angles. mE 5 mG 5 65° Use supplementary angles mA 1 mB 5 180° mC 1 mD 5 180° Use four pairs of supplementary angles. mE 1 mF 5 180° mG 1 mH 5 180° Calculate supplementary angles 65° 1 mB 5 180° mB 5 115° mB 5 mD 5 mF 5 mH 5 115° We show how we calculate one angle. The remaining angles would be calculated the same way. 4.2 More on Parallel Lines and Angles159 Parallel Lines and Angles Earlier, we stated a postulate: If two parallel lines are intersected by a transversal, then the corresponding angles are congruent. The converse is true: If the corresponding angles are congruent, then the lines are parallel. Since converses are not always true, we write the converse as a postulate. This postulate lets us write a series of theorem converses: 160 Chapter 4: Parallel and Perpendicular Lines MODEL PROBLEMS 1. MP 3, 6 a If C F, which lines must be parallel? Explain. b If B D, which lines must be parallel? Explain. B C i D E o F a c If mB 1 mF 5 180°, which lines must be parallel? w SOLUTION a i i o because of the alternate interior angles theorem converse. This is a pair of alternate interior angles, with line w being the transversal. Since the angles are congruent, the lines the transversal intersects are parallel: i i o. b w i a because of the corresponding angles theorem converse. This is a pair of corresponding angles, with line i being the transversal. Since the angles are congruent, the lines the transversal intersects are parallel: w i a. c Since mB 1 mF 5 180° then lines i and o must be parallel, because angle B is a vertical angle with the unnamed angle that is a consecutive interior angle to F, making F and the unnamed angle supplementary. 2. MP 6 What do the angle measures have to equal for the lines to be parallel? 3(x – 10°) (x + 10°) SOLUTION If the lines are parallel, the angles must be supplementary 3(x 2 10°) 1 (x 1 10°) 5 180° The two angles whose measures are given by algebraic expressions are consectuve interior angles. Their measures must sum to 180° if the lines are parallel. Solve equation 3x 2 30° 1 x 1 10° 5 180° Distribute the 3. Combine like terms, and then solve for x. 4x 2 20° 5 180° x 5 50° Use value of x in expressions 3(x 2 10°) 5 3(50° 2 10°) 5 120° x 1 10° 5 50° 1 10° 5 60° Angle measures 5 120° and 60° Substitute the value of x into the expressions for the angles. Check the calculations: The angle measures sum to 180°. Model Problems continue . . . 4.2 More on Parallel Lines and Angles161 MODEL PROBLEMS continued 3. MP 2, 5, 7 A designer is trying to ensure that levels 1 and 2 are parallel in a game. If he is sketching out a diagram for the factory, how might he do so? SOLUTION Construct level 1 using a straightedge Construct an angle using a protractor The designer can use a postulate concerning corresponding angles and parallel lines, as well as a protractor and straightedge. Use the straightedge—a ruler, in this case—to draw a line segment that represents level 1. Place the straightedge to the left of level 1 and measure the angle with a protractor. This particular angle is 120°. Construct a corresponding angle using a protractor Move the protractor down along the straightedge, and measure a corresponding angle of 120°. Construct level 2 using the corresponding angles postulate converse Using a straightedge, construct level 2 at the 120° mark on the protractor. Since the corresponding angles are congruent, the levels are parallel. 162 Chapter 4: Parallel and Perpendicular Lines PRACTICE 1. If m1 5 146°, what must the sum of the angles 4 and 6 be so the lines a and b are parallel? 1 2 4 b A. 17° B. 34° 1 3 5 a 6 8 a A. 36° B. 54° A. 74° B. C. 75° c II. The sum of corresponding angles is 180°. III. Interior consecutive angles are congruent. IV. Interior consecutive angles are supplementary. I only I and III only I and II and IV only I and IV only 8 7 105° 5. In the diagram below, the lines a, b, and c are intersected by a transversal d. Which lines must be parallel? The diagram is not drawn to scale. 42° 140° 138° C. 126° D. 144° I. Alternate interior angles are congruent. 6 D. 106° d 3. Two lines are intersected by a transversal. Which of the following conditions must be true so the lines are parallel? A. B. C. D. 3 5 F p 4 b 2. What is the measure of angle F? 36° 2 a 7 C. 68° D. 146° b 4. If the lines a and b are parallel, and m3 is 40% greater than m4, what is m8? a b c d A. B. C. D. a and b a and c b and c None of the above 6. Two parallel lines are intersected by a transversal. One of the angles formed is 70°. Can any of the other angles be equal to 30°? A. Yes B. No C. Not enough information to decide Practice Problems continue . . . 4.2 More on Parallel Lines and Angles163 Practice Problems continued . . . 7. In the diagram below, m3 5 m4 5 128°, m5 5 52°. Which lines must be parallel? The diagram is not drawn to scale. Exercises 11–12: Solve for x. 11. x – 4° l 7(x – 8°) 3 m 2 5 1 12. n 4 2x + 61° p 3(x + 7°) A. B. C. D. m and n only n and p only m, n, and p are all parallel No parallel lines on the diagram 13. MP 3 In the diagram, the pairs of sides of the angles K and M are parallel: a is parallel to c, and b is parallel to d. Prove that mK 1 mM 5 180°. 8. According to the congruent angles marked in the diagram below, which lines must be parallel? a a c b m A. a and b B. m and n d C. All of the above D. None of the above 9. According to the congruent angles marked in the diagram below, which lines must be parallel? c d p q A. c and d B. p and q b K n L M Exercises 14–17: In the diagram, the lines x and y are parallel. x y t 1 4 2 3 5 8 6 7 C. All of the above D. None of the above 10. Suppose mG is known. Which of the following techniques could be used to find mA? Choose all that apply. A B D C E F H G a 14. If m4 5 124°, find all other angles. 15. If m6 1 m4 5 250°, find m7. 16. If m3 is 72° less than m8, find m2. 17. If m6 is 5 times m1, find m4. b A. Alternate exterior angles to find A. B. Vertical angles to find E, then corresponding angles to find A. C. Alternate interior angles to find F, then vertical angles to find A. D. Consecutive interior angles to find D, then corresponding angles to find A. 164 Chapter 4: Parallel and Perpendicular Lines Practice Problems continue . . . Practice Problems continued . . . Exercises 18–21: Use the diagram to answer the questions. 1 2 4 3 6 5 a 8 b 7 18. m2 5 45° and m7 is three times m4. Are the lines a and b parallel? Why or why not? 19. 2 measures 68°. What must the measure of angle 5 be so the lines a and b are parallel? 25. MP 2, 3 ABC is equal to 60°. BCD is equal< to> 120°.< Dilmah made a sketch and > said AB and CD must be parallel, since 120° 1 60° 5 180°. < Toshi > <also> sketched the angles and said AB and CD intersect. Try to create both the sketches Dilmah and Toshi could have made and explain who is right. 26. MP 2, 4 The sketch shows Jen’s favorite chair from the side. The black segments represent the backrest, seat, and one of the legs of the chair. The gray segments represent one of the chair’s arms. 20. 3 measures 100°. What must the m easure of angle 8 be so the lines a and b are parallel? 110˚ 110˚ 21. If m1 5 x 1 1 and m6 5 2x 1 2, what must m5 be so the lines a and b are parallel? 22. MP 3 Using the diagram, prove the alternate interior angles theorem converse. A B D C E m F n H G Prove: m n Given: D F 23. Explain how you would prove that if the alternate interior angles formed by two lines and a transversal are not congruent, then the two lines intersect. aAre the chair’s arms parallel to the chair’s seat? Why or why not? bJen likes to sit with her arms parallel to her thighs, so she decides to put a pillow under her hands. Finding the angle by which she needs to raise her arms will help Jen to figure out how thick the pillow should be. What must that angle be? 27. In the diagram, the corresponding sides of the angles K and M are parallel: a is parallel to c, and b is parallel to d. Prove that the angles K, L, and M are congruent. a 24. Are lines m and n parallel? Justify your answer. m A = 57° B D K n C F = 122° G E H c L M b d 28. Two parallel lines are intersected by a transversal. Prove that angle bisectors (lines dividing an angle into two adjacent congruent angles) of the alternate interior angles are parallel to each other. Make a sketch and label the angles, referring to them in your proof. Practice Problems continue . . . 4.2 More on Parallel Lines and Angles165 Practice Problems continued . . . 29. MP 5 To draw a line parallel to the line m, through the point M, Clive uses a straightedge and a drawing triangle, as shown in the picture. First, he sets the triangle so one side lies on the line m and draws a line along the longer side of the triangle so it passes through the point M. After that, he moves the triangle along this new line until its vertex is at the point M. He then draws a line along the original side of the triangle and continues this line using the straightedge. Explain why this method produces parallel lines. 30. Using the diagram provided and the knowledge that 3 10 and 1 6, prove that 15 13. Note that no information is given about whether lines are parallel or not. M c d 1 2 5 4 3 6 7 9 8 a 11 10 12 14 13 15 16 18 17 b m • Multi-Part PROBLEM Practice • MP 2, 4 Use the diagram to answer the questions. 136th Avenue is parallel to 130th Avenue. N W a Is Jefferson Boulevard parallel to 136th Avenue? Why or why not? S 110° y Wa set Jefferson Blvd. Sun 50° 63° 130th Ave. 136th Ave. b A lightrail track is planned to be built south from the intersection of Jefferson Boulevard and Sunset Way. What should the measure of the acute angle between the train route and Sunset Way be so the tracks run parallel to 136th Avenue? E La rk St re e t c What would be the measure of the acute angle between the lightrail route and Lark Street? LESSON 4.3 4.3 Perpendicular Lines Pairs of Perpendicular Lines 166 Chapter 4: Parallel and Perpendicular Lines This is a theorem about perpendicular lines and their relationship to right angles. The symbol for perpendicular lines is an upside-down T: These two theorems are about perpendicular transversals. m Prove: m y A B Statement We prove the perpendicular transversal theorem. x y Reason Diagram The diagram tells us that lines x and y are parallel, and that line m is perpendicular to line x. mA 5 90° Definition of perpendicular lines By the definition of perpendicular lines, we know that A is a right angle, and therefore its measure must be 90°. mA 5 mB 5 90° Angles A and B are corresponding angles A B mA 5 mB 5 90° Angles A and B are congruent, because they are corresponding angles of parallel lines. So they have the same measure, 90°. B is a right angle Definition of right angle Since angle B has a measure of 90°, angle B is a right angle by definition. my Definition of perpendicular lines Finally, by definition of perpendicular lines, line m is perpendicular to line y, and we have proved our theorem. xiy mx 4.3 Perpendicular Lines 167 MODEL PROBLEMS 1. a Is p parallel to q? b Is p perpendicular to r? q p r s SOLUTION a Yes, because of the lines perpendicular to a transversal theorem. The theorem states that if two lines (such as lines p and q) are perpendicular to the same line (such as line s), then they are parallel to each other. b Yes, because of the perpendicular transversal theorem. The theorem states that if two lines (such as r and s) are perpendicular to one of a pair of parallel lines (line q), then they are perpendicular to the other (line p). 2. Prove: o m m n o p SOLUTION Statement Reason pm Diagram This is shown in the diagram. oip Lines perpendicular to a transversal theorem Two lines, o and p, perpendicular to the same line, n, are parallel. om Perpendicular transversal theorem Since m is perpendicular to p, one of two parallel lines, m is also perpendicular to o, the other parallel line. 168 Chapter 4: Parallel and Perpendicular Lines Construction: Perpendicular and Parallel Lines We construct a perpendicular bisector, a perpendicular line that divides a line segment in half. To do so, we do two things at the same time. (1) We construct a line perpendicular to our original line using a compass and a straightedge. (2) We bisect (divide into two equal parts) the line segment defined by the two points on the line in the first step. Constructing a perpendicular line with a compass: 1. Start with two points on line. 2. Draw an arc. Keep the compass wider than distance halfway between the points. Put one end of the compass on a point and draw an arc with the other. 3. Draw another arc from second point. Keep compass open same amount. 4. Connect intersection points of arc. Use a straightedge. We can also construct a line that is perpendicular to our original line and that passes through a given point not on the original line: 1. Place compass at point. Draw arc through line. 4.3 Perpendicular Lines 169 2. Mark two intersection points. Create two arcs using points on line. Follow same steps as shown above. 3. Connect intersection points of arcs. Use a straightedge. Why does this construction create a perpendicular line? We add to our diagram to answer this question. An intersecting line that forms right angles is perpendicular to another line, so this proof explains the geometry behind the construction. Statement Reason AC DC and AB DB Each pair was drawn with same compass setting Both pairs of segments are congruent. BC BC Reflexive property The two triangles share this side. nABC nDBC SSS congruence The triangles have three pairs of congruent sides. (We discuss SSS in depth in the next chapter.) ACB DCB Definition of congruence They are two corresponding angles in congruent triangles. ACB, DCB are They are linear pair and right angles congruent. 170 Chapter 4: Parallel and Perpendicular Lines The measures of linear pair angles sum to 180° and the angles are congruent. Only 90° angles match this description. We construct a line parallel to our original line by constructing two perpendicular lines: 1. Start with a perpendicular line. Follow the same steps as shown above. 2. Construct another perpendicular line. Line is parallel to original line. Finally, we construct a parallel line through a point, C: 1. Connect point C and line using straightedge. Lines intersect at D. 2. Create two arcs. Set compass width greater than halfway between points C and D, but smaller than the distance between C and D. Draw arcs with compass placed at C and D. 4.3 Perpendicular Lines 171 3. Draw two small arcs. Set compass width to lower previously drawn arc. Draw arcs with compass placed at intersection points of the previously drawn arcs and the construction line. Label intersection point E. < > 4. Draw parallel line, CE . Use a straightedge. Why does the above construction create a parallel line? Statement Reason KC > MD and CE > DP and KE > MP Each pair was drawn with Each connects points drawn with same same compass setting compass setting. nKCD > nMDP SSS congruence Three pairs of corresponding sides are congruent. (We discuss SSS in depth in the next chapter.) /KCE > /MDP Definition of congruence They are two corresponding angles in congruent triangles. CD is a transversal Diagram Given in the diagram. /KCE and /MDP are corresponding angles Diagram They occupy corresponding positions relative to the lines. Lines are parallel Corresponding angles are congruent Since the corresponding angles are congruent, the lines must be parallel. 172 Chapter 4: Parallel and Perpendicular Lines Distance and Perpendicular Lines The distance between a point and a line is the shortest distance between the point and the line. That distance is the length of the perpendicular segment from the point to the line. MODEL PROBLEM What is the distance between the point and the line? y 5 4 3 2 1 –5 –4 –3 –2 –1 1 –1 2 3 4 x 5 –2 –3 –4 SOLUTION –5 Draw perpendicular segment y 5 4 3 2 1 –5 –4 –3 –2 –1 –1 1 2 3 4 5 x –2 –3 –4 –5 Use ruler postulate distance 5 uy2 2 y1u distance 5 u21 2 3u 5 4 PRACTICE 1. Lines a and b are parallel. Line c forms an angle of 91° with line a. Choose the true statement. A. b is parallel to c. B. c is perpendicular to b. C. None of the above. 2. If the angle between the lines a and b is 89°, and c is parallel to b, then A. B. C. D. a is perpendicular to c. c is parallel to a. b is perpendicular to a. None of the above. Practice Problems continue . . . 4.3 Perpendicular Lines 173 Practice Problems continued . . . 7. How do we know that lines g and h are parallel? 3. Which two lines must be parallel? a b g h c d j e k A. c and d B. a and e C. a and b 4. What is the distance between point M and line c? y 5 8. Lines m and n are perpendicular. Which of the following statements must be true? Choose all that apply. 4 3 2 m 1 –5 –4 –3 –2 –1 1 2 3 4 5 x –1 –2 –3 c A. B. C. D. –4 n F E G H M A. Angles E, F, G, and H are right angles. B. /E > /F C. Angles E and F are supplementary angles. D. Angles E and G are vertical angles. –5 3 4 5 24 5. If line c is perpendicular to one of a pair of perpendicular lines, and all lines are coplanar, then A. c is perpendicular to the other line of the pair. B. c is parallel to the other line of the pair. C. c is parallel to both lines. D. None of the above. 6. If point P is 8 units away from the y-axis, which of these coordinates may represent the point P? A. (2, 8) B. (26, 28) A. Given B. Perpendicular transversal theorem: g and h perpendicular to j C. Lines perpendicular to a transversal theorem: g and h perpendicular to k D. Perpendicular transversal theorem: g and h perpendicular to k C. (28, 3) D. (4, 4) 174 Chapter 4: Parallel and Perpendicular Lines 9. What is the distance between the point and the line? y 6 5 4 3 2 1 –6 –5 –4 –3 –2 –1 –1 1 2 3 4 5 6 x –2 –3 –4 –5 –6 Practice Problems continue . . . Practice Problems continued . . . 10. MP 3 Using the diagram, prove the lines perpendicular to a transversal theorem. m n A B D C E F H G 20. Find the distance between the point (27, 25) and the x-axis. 21. Find the distance between the point (20.57, 221) and the y-axis. p 22. Find the distance between the point (20.54, 27) and the y-axis. 23. Find the distance between the point A and the line a. y Prove: m n 10 Given: m p, and n p 11. MP 2, 3 Prove that the angle bisectors of adjacent supplementary angles are perpendicular to each other. Provide a sketch with your answer. 12. On a plane, a is perpendicular to b, b is parallel to c, and n is perpendicular to a. Is line n perpendicular to c? Explain your answer. Exercises 13–17: In the diagram, the measure of angle GDK is 35°, c ' a, d ' e, a i b, and e i f. c M d A a b e K f G F D B e C E H J 8 6 A 4 2 –10 –8 –6 –4 –2 2 –2 –4 4 6 8 10 x a –6 –8 –10 24. Cori says that the distance from a point (x, y) to the y-axis is the point’s x-coordinate. Using the point (24, 23), explain whether Cori’s statement is correct. 25. The coordinates of point M are (x, 23). If the distance from point M to the y-axis is 9 units, list all values of x. 26. The coordinates of point N are (5, y). If the distance from point N to the x-axis is 3 units, list all values of y. 15. Find the measure of /BCD. Exercises 27–30: The line going through point M and perpendicular to line a intersects line a at point C. The coordinates of the points are given. Find the distance from point M to line a. 16. Find the measure of /EHF. 27. M(8, 23), C(1, 211) 17. What is the measure of /CEJ? 28. M(7, 23), C(23, 210) 18. When two parallel lines are intersected by a transversal, the alternate exterior angles add up to 180°. What could be said about the angle between the lines and the transversal? Why? 29. M(7, 24), C(24, 210) 13. List all pairs of perpendicular lines. 14. What is the measure of /GDM? 30. M(10, 23), C(4, 210) Practice Problems continue . . . 19. Find the distance between the point (25, 23) and the x-axis. 4.3 Perpendicular Lines 175 Practice Problems continued . . . 31. Using information from the diagram, find the angle between the rays a and b. a b 116° 64° 154° m n 35. MP 4 Two mirrors are placed so they form a right angle. A light ray is directed into one mirror so it forms a 45° angle with it. Using the fact that the incoming and outgoing angles are congruent, prove that the ray reflected from the second mirror is parallel to the ray striking the first mirror. 32. MP 3, 5 Line m passes through point M. Explain how to construct a line p through point M and perpendicular to line m using a compass and a straightedge. Perform the constructions. 33. Describe the set of all points that are the same distance d from a given line a, and which are in a single plane passing through line a. 34. Describe the set of all points that are the same distance from parallel lines a and b and line in the plane containing the lines. incoming angle outgoing angle 45° 1 36. MP 4, 5 Using geometric construction tools, construct a map of Commercial Avenue, 2nd Street, and 3rd Street with the following constraints: 2nd and 3rd Streets are parallel to each other; Commercial Avenue intersects both 2nd and 3rd Streets, but not at right angles. LESSON 4.4 4.4 Parallel Lines, Perpendicular Lines, and Slope Determining When Lines Are Parallel or Perpendicular Parallel Lines y We discuss how to determine if two lines are parallel. (1) If both lines are vertical, then they are parallel. Any two horizontal lines are also parallel. 10 9 8 7 6 5 4 3 2 1 • Two vertical lines, such as x 5 4 and x 5 5, are parallel. • Two horizontal lines, such as y 5 2 and y 5 21, are parallel. (2) If both the lines are not conveniently vertical or horizontal, then you have to consider their slopes to determine if they are parallel. If their slopes are the same, the lines are parallel. • Lines with same slope, such as y 5 3x 1 2 and y 5 3x 2 1, are parallel. 176 Chapter 4: Parallel and Perpendicular Lines –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 y = 3x + 2 y = 3x – 1 1 2 3 4 5 6 7 8 9 10 x Perpendicular Lines As with parallel lines, the slopes of perpendicular lines have a specific mathematical relationship. (1) If one line is vertical and another horizontal, they are perpendicular. (2) If the lines are not vertical and horizontal, you have to consider their slopes to determine whether they are perpendicular. The slopes of perpendicular lines are negative reciprocals. • The two lines in the diagram are perpendicular because their 1 slopes, 22 and , are negative reciprocals. To put it another 2 way, the product of the slopes is 21. Rotations and Perpendicular Lines y 10 9 8 7 6 5 4 3 2 1 y= –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 Perpendicular lines have slopes that are negative reciprocals. We show how this is true with a 90° clockwise rotation. We call the perpendicular lines a and p. The equations for the lines will be y 5 max, and y 5 mpx. 1 2 x+2 1 2 3 4 5 6 7 8 9 10 x y = –2x – 1 y (xa, ya) (0, 0) a x p (xp , yp) = (ya, –xa) Slope of line a Rotate the line 90° clockwise Slopes of line a and p are negative reciprocals ma 5 Slopea 5 y2 2 y 1 x2 2 x 1 Slopea 5 y2 2 y 1 x2 2 x 1 Slopea 5 ya 2 0 ya 5 xa xa 2 0 (xa, ya) → (xp, yp) 5 (ya, 2xa) mp 5 yp 2xa 5 ya xp This is the equation for the slope of a line. We use (0, 0) for (x1, y1), (xa, ya) for (x2, y2), and calculate the slope. Now rotate the line using the coordinate relationship for a 90° clockwise rotation. The point remains on the line as it rotates. Use the equation for the slope again, substituting the coordinates of the point on the rotated line. Comparing it to our original slope, we see that the slopes are negative reciprocals. 4.4 Parallel Lines, Perpendicular Lines, and Slope 177 MODEL PROBLEMS 1. A line is rotated 90°. The slope of the original line and slope of the line created by the rotation, are A. Equal. B. Opposite. C. Negative reciprocals. D. None of the above. SOLUTION The answer is C. Rotating a line 90° will create a perpendicular line. Perpendicular lines have negative reciprocal slopes. 2. Write equations for b and c. 1 The equation for a is y 5 x 1 2. 2 y 10 a b –10 10 SOLUTION Equation for b –10 Equation for c 3. c Line b is parallel to a, because both lines are Same slope, passes through origin y 5 x 1 x 2 Slope is negative reciprocal y-intercept at y 5 2 y 5 22x 1 1 perpendicular to c. Thus, a and b have the same slope, 1 . The y-intercept of b is zero since it passes through 2 the origin. Line c is perpendicular to a. Thus, the slope of c is the 1 negative reciprocal of , the slope of a. The y-intercept 2 of c is 2, since c passes through the point (0, 2). MP 1, 2, 3 A teacher says: “Triangle ABC has vertex A at (2, 3), vertex B at (1, 10), and vertex C at (5, 7). ABC is a right triangle, with C its right angle.” Is he correct? SOLUTION Calculate slope AC Calculate slope CB Slopes are negative reciprocals We can use If the legs are at right angles, the slopes of two line they are perpendicular segments to and will have slopes that determine if are negative reciprocals. a triangle is a Calculate the slope of AC. right triangle. rise 7 2 10 3 Calculate the slope of the 5 5 2 Slope CB 5 run 521 4 other leg. 3 4 The slopes are negative ? a2 b 5 21 3 4 reciprocals: their product C is right angle is 21. This means the lines rise 723 4 Slope AC 5 5 5 run 522 3 are perpendicular and it is a right triangle. 178 Chapter 4: Parallel and Perpendicular Lines Distance Between a Point and a Line Earlier, we calculated the distance between a point and a horizontal or vertical line, or the shortest path between the two. Now we do the more difficult task of computing the distance between a point and a line that is neither horizontal nor vertical. To calculate the distance, we need to determine the perpendicular distance between the point and the line, since that will be the shortest path. This means we must first determine a line perpendicular to the line and passing through the point. MODEL PROBLEM Find the distance between the point and the line. y 10 5 –10 –5 5 10 x –5 SOLUTION Calculate slope of line –10 slope 5 y2 2 y1 21 2 3 24 5 5 5 22 Calculate the slope of the line. x2 2 x1 220 2 y 5 22x 1 3 Write equation y5 Write equation for perpendicular line Write in slope-intercept form. The line crosses the y-axis at 3, so b 5 3. 1 x1b 2 The distance will be along the perpendicular line. Its slope is the negative reciprocal of the slope of the given line. Calculate x of intersection point 1 (5) 1 b 2 1 b5 2 1 1 y5 x1 2 2 1 1 x 1 5 22x 1 3 2 2 x 5 1 Substitute y 5 22x 1 3 5 22(1) 1 3 5 1 Identify the intersection point (1, 1) Use distance formula distance 5 "(x2 2 x1)2 1 (y2 2 y1)2 Use the distance formula to calculate the distance between the point (5, 3), and the point of intersection (1, 1). Evaluate distance 5 "(2)2 1 (4)2 5 "20 distance 5 2"5 < 4.47 Square the terms and add. The d istance is about 4.47. Substitute point into line equation Calculate b by substituting the x- and y-values of the point we want to find the distance to into the equation. (3) 5 Set the two equations for the lines equal to each other. Where the lines intersect, their y- and x-values will be the same. distance 5 "(3 2 1)2 1 (5 2 1)2 Solve for y. 4.4 Parallel Lines, Perpendicular Lines, and Slope 179 PRACTICE 1. Which of the following equations represents a line parallel to the graph of the equation y 5 9x 1 0.4? A. y 5 0.4x 1 9 B. y 5 29x C. y 5 9x 2 3.6 2. Which of the following equations represents a line parallel to the graph of the equation y 5 3x 1 0.4? A. y 5 0.4x 1 3 B. y 5 23x C. y 5 3x 1 2.4 3. The graph of the equation y 5 1 is parallel to the graph of which of the following equations? Choose all that apply. A. B. C. D. E. F. G. y 5 1x y 5 21 y5x11 y55 x50 y50 x55 y 5 4x y 5 24 y5x14 y55 B. C. x 5 5.3 y 5 25.3 E. y 5 0 x 5 0.3 F. 7. A beetle sits at the origin on a coordinate plane. If he crawls along the shortest distance to the line y 5 2x 1 4, how far, in units, does he need to crawl? A. 2"2 B. 3 C. "2 D. 4 8. Which of the following equations r epresents the line that passes through the points (22, 24) and (3, 231)? C. y 5 211x 1 2 D. y 5 210x 2 6 E. x 5 0 F. y50 G. x 5 5 5. Which of the following equations represents a line perpendicular to the graph of the equation y 5 9x 1 4? A. B. C. A. y 5 212x 1 4 B. y 5 29x 2 3 4. The graph of the equation y 5 4 is parallel to the graph of which of the following equations? Select all that apply. A. B. C. D. 6. The graph of the equation y 5 5.3 is perpendicular to which of the following graphs? Select all that apply. 1 D. x 5 0 A. y 5 2 5.3 1 y 5 2 x 1 (22) 9 1 x14 9 y 5 29x y5 Exercises 9–11: Given three points, determine if they define a right triangle, a non-right triangle, or do not define a triangle. 9. (26, 25), (5, 22), and (22, 2) A. Not a right triangle B. Right triangle C. The points do not define a triangle 10. (24, 22), (6, 3), and (3, 9) A. Not a right triangle B. Right triangle C. The points do not define a triangle 11. (24, 22), (7, 22), and (23, 22) A. Not a right triangle B. Right triangle C. The points do not define a triangle Exercises 12–14: Write the equation of the line that passes through the points. 12. (24, 242) and (2, 18) 13. (23, 23) and (1, 21) 180 Chapter 4: Parallel and Perpendicular Lines Practice Problems continue . . . Practice Problems continued . . . 14. (21, 216) and (2, 20) Exercises 26–28: Find the equation of the line passing through a given point and having a given relationship with another line. 15. Show that the equation of a line that goes through the points (x1, y1) and (x2, y2) is: y 2 y2 x 2 x2 5 y2 2 y1 x2 2 x1 26. A line passes through the point (21, 9) and is perpendicular to the line y 5 –2x 1 1. 27. A line passes through the point (23, 8) and is perpendicular to the line y 5 27x 1 4. 16. Are the lines described by the equations y 5 5x 1 7 and y 5 2x 1 3 parallel? Justify your answer. 28. A line passes through the point (0, 1) and is parallel to y = –2x + 6. 17. What is the slope of a line perpendicular to y 5 2x? 29. Find the shortest distance between the point (5, 2) and the line y 5 3x 1 6. 18. What is the slope of a line perpendicular to y 5 0.5x 1 4? 30. The line m goes through the points (23, 5) and (1, 2). The line n goes through the points (0, 0) and (24, 23). Are these lines parallel? Why or why not? 19. What is the slope of a line perpendicular to y 5 24x 1 1? 20. Find the equation of the line that passes through (0, 25) and is perpendicular to 1 1 y52 x1 . 3 4 31. Use the diagram to answer the questions. y 10 9 8 7 6 5 4 3 2 1 21. Write the equation of the line shown. y 9 8 7 6 5 4 3 2 1 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 1 2 3 4 5 6 7 8 9 a b 1 2 3 4 5 6 7 8 9 10 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 x x aWhat is the slope of line segment a? bWhat is the slope of line segment b? 32. On the diagram, the triangles ABC and DBK are similar, that is, their corresponding angles are congruent. Prove that the lines have the same slope. 22. What is the slope of a line that passes through the origin and is parallel to 1 y 5 x 1 8? 5 23. What is the slope of a line that passes through the origin and is parallel to 1 y 5 x 1 8? 6 24. Give an equation of a line that is parallel to the line x 5 22.8 and passes through the point (27.2, 0). y A D B x C K 25. The line a goes through the points (6, 24) and (25, 0). The line b passes through the points (2, 27) and (29, 23). Are these lines parallel? Why or why not? Practice Problems continue . . . 4.4 Parallel Lines, Perpendicular Lines, and Slope 181 Practice Problems continued . . . 33. The sides of a triangle are given by the equations y 5 0.4x 2 34.5, y 5 24x 1 1.9, and y 5 22.5x 1 8. Determine if it is a right triangle. 37. State and graph the equation of a line that goes through the point (1, 2) and is perpendicular to the line that connects the points (4, 3) and (22, 1). 34. MP 3 Two vertices of a triangle are (0, 5) and (0, 25). Define the third vertex in such a way that the resulting triangle has a right angle. (Note that there are many possible correct answers.) 38. Two robots, Daisy and Chaise, are standing on the adjacent vertices of a playing field shaped as a quadrilateral. Coordinates of the four vertices are (2, 2), (5, 1), (3, 6), and (0, 3). On a signal, each robot sends a laser beam toward the opposite vertex. Find the coordinates of the point where these beams meet. 35. Graph the lines y 5 x 1 2 and y 5 x 2 4. On your graph, mark a segment showing the shortest distance between the lines. Find the distance between the lines to the nearest tenth of a unit. 36. State and graph the equation of a line that goes through the point (4, 3) and is parallel to the line that connects the points (1, 2) and (21,25). 39. MP 3 A teacher says: “Triangle ABC has vertex A at (25, 22), vertex B at (2, 210), and vertex C at (0, 0). ABC is a right triangle, with C its right angle.” Is he correct? Justify your answer. • Multi-Part PROBLEM Practice • MP 3, 7 Points A (24, 6), B (2, 10), C (11, 3), and D (2, 23) form a quadrilateral. a Graph the quadrilateral on a coordinate plane. b What type of quadrilateral is ABCD? Explain. c Move point C or D to make ABCD a parallelogram but not a rectangle. Justify your action. d Move point C or D to make ABCD a rectangle. Justify your action. e Is your rectangle from part d a square? Explain. LESSON 4.5 4.5 Parallel Lines and Triangles Interior and Exterior Angles Interior angles of a polygon are the angles inside a polygon. Exterior angles of a polygon are angles formed by a side of a polygon and the extension of its adjacent side. An interior angle Exterior and its exterior angles angle form a linear Exterior angle pair, so that the Interior Interior angles angles measures of the angles sum to 180°. 182 Chapter 4: Parallel and Perpendicular Lines You may already be familiar with the following theorem—that the measures of the angles in a triangle sum to 180°. We will prove that the measures of a triangle’s angles sum to 180°. To prove this theorem, we do a little planning first, adding a line not present in the diagram. An auxiliary line is a line added to help in a proof. Prove: m4 1 m2 1 m5 5 180° We can add a point or segment as an auxiliary in a proof. A 2 4 5 B C Draw a line parallel to segment BC: We want to prove the measures of the angles in a triangle sum to 180°. To do this, we start by drawing the line AD. We know that the measures of angles 1, 2, and 3 sum to 180°, since the angles form a line. Statement < > AD i BC A 1 2 4 B D 3 5 C If we can show that 1 and 3 are congruent to 4 and 5, the triangle’s interior angles, we will have proven that the sum of interior angles of a triangle is 180°. Reason Diagram The auxiliary line is parallel to BC. m1 1 m2 1 m3 5 180° Definition of a straight angle Since angles 1, 2, and 3 form a line, we know that their measures sum to 180°. 1 4 Alternate interior angles theorem AB is a transversal of the parallel line and line segment, so these alternate interior angles are congruent. 3 5 Alternate interior angles theorem AC is also a transversal of the parallel line and line segment, so these alternate interior angles are congruent too. Substitution Replace angles 1 and 3 from the previous steps with 4 and 5 to prove that the interior angle measures of a triangle sum to 180°. m4 1 m2 1 m5 5 180° 4.5 Parallel Lines and Triangles 183 We state the exterior angle theorem. MODEL PROBLEMS A 1. What is m/Z? 60° Z SOLUTION 40° B C Exterior angle theorem m/Z 5 m/A 1 m/C The exterior angle theorem states that the measure of the angle equals the sum of the measures of the two remote interior angles. Evaluate m/Z 5 60° 1 40° 5 100° Substitute and add. 2. What is m/N, given m/N 5 (2x 1 10°)? M 20° 2x + 10° N SOLUTION Exterior angle theorem K m/KPM 5 m/N 1 m/M The exterior angle theorem states that the 3x – 20° 5 (2x 1 10°) 1 20° measure of the exterior angle equals the sum of the measures of the two remote interior angles. Substitute the expressions and values shown in the diagram. Solve equation 3x 2 20° 5 2x 1 30° 3x 5 2x 1 50° x 5 50° Substitute 3x – 20° P Combine the constants, then isolate the variable. m/N 5 2x 1 10° Substitute the value of x into the expression for m/N 5 2(50°) 1 10° 5 110° m/N. 184 Chapter 4: Parallel and Perpendicular Lines Angles in a Right Triangle The corollary below follows from the theorem that a triangle’s angles sum to 180°. A corollary is a theorem easily proven from another theorem. To put it another way, the other two angles in a right triangle are complementary. MODEL PROBLEM What is z? SOLUTION J z + 15° K 2 (z – 12°) Use triangle sum corollary Distribute, combine L 2(z 2 12°) 1 z 1 15° 5 90° 2z 2 24° 1 z 1 15° 5 90° 3z 2 9 5 90° Solve equation Substitute 3z 5 99° z 5 33° We substitute our value for z into the expressions 2(z 2 12°) and z 1 15°. The angles equal 42° and 48°. They sum to 90°, as expected. PRACTICE 1. In a triangle ABC, A is a right angle. If the sum of angles A and B is less than 150°, which of the following best describes the measure of angle C? A. B. C. D. Greater than 150° Equal to 30° Greater than 30° Greater than 45° 2. In a right triangle XYZ, Y is a right angle and YW is perpendicular to XZ where W is a point on XZ. If X = 34°, then mZYW is: A. B. C. D. 34° 56° 90° 146° Practice Problems continue . . . 4.5 Parallel Lines and Triangles 185 Practice Problems continued . . . 3. Which angles are interior angles? Choose all that apply. e f d c h e f g h E. F. G. H. 4. Which angles are exterior angles? Choose all that apply. e f d c h g a b A. B. C. D. a b c d E. F. G. H. e f g h 5. In a triangle, the measure of the first angle is 12° less than the measure of the second one, and the measure of the third angle is twice as great as the measure of the first one. What are the measures of the angles of the triangle? A. B. C. D. 12. Angles’ ratio: 1 : 2 : 3 13. In a triangle, the measure of the second angle is twice as big as the measure of the first angle. The measure of the third angle is 6 times as big as the measure of the second one. Find the angles of the triangle. a b a b c d Exercises 11–12: Find the measures of all interior angles of a triangle, using the given ratio between them. 11. Angles’ ratio: 5 : 6 : 7 g A. B. C. D. 10. /A 5 60°, /B 5 60° 48°, 96°, 36° 54°, 108°, 42° 54°, 108°, 94° 54°, 84°, 42° Exercises 6–10: Given the measures of two interior angles A and B of a triangle ABC, find the measure of the third interior angle, C. 6. /A 5 50°, /B 5 30° 7. /A 5 75°, /B 5 40° 8. /A 5 55°, /B 5 80° 9. /A 5 35°, /B 5 120° 14. Dorothy says two angles in a triangle she drew have the measures 94° and 98°. Can Dorothy be right about her triangle? Why or why not? 15. MP 2 The measure of one of the exterior angles of a triangle is 40°, and the measure of one of the interior angles is 30°. Make a sketch, marking these angles, and find the measures of all interior angles of the triangle. 16. MP 3, 6 Can there be both an obtuse and a right angle in a triangle? If so, give an example. If not, explain why. 17. Using the diagram, find the measures of the angles of the triangle. (2x + 1)° (6x + 12)° (x – 4)° 18. Find the measure of exterior angle adjacent to interior angle C of the triangle ABC, if /A 5 40° and /B 5 84°. 19. One acute angle of a right triangle is 20° smaller than the other. Find the measure of the larger acute angle of the triangle. 20. One acute angle of a right triangle is 5 times as the other. Find the measures of the acute angles of the triangle. 21. One acute angle of a right triangle is 2 times as great as the other. Find the measures of the acute angles of the triangle. 22. Two exterior angles of a triangle are equal to 116° and 144°. Find the measures of all three interior angles of the triangle. Practice Problems continue . . . 186 Chapter 4: Parallel and Perpendicular Lines Practice Problems continued . . . 23. Two exterior angles of a triangle are equal to 101° and 141°. Find all three interior angles of the triangle. 30. In the diagram, ABC is a right triangle and CD ' AB. Prove that m/A 5 m/BCD. A D 24. The sum of the exterior angles adjacent to angles A and B of the triangle ABC is equal to 240°. Find the interior angle C of the triangle. 25. One of the angles of a triangle is 50°, and the difference of the other two angles is 20°. Find the angles of the triangle. 26. Two exterior angles of a triangle are equal to 120° and 135°. Find the third exterior angle of the triangle. 27. Two exterior angles of a triangle are equal to 100° and 145°. Find the third exterior angle of the triangle. 28. An exterior angle of a right triangle is equal to 135°. Find the acute angles of the triangle. 29. One acute angle of a right triangle is 10° smaller than the other. Find the larger acute angle of the triangle. C B 31. Find the angles of a right triangle with two congruent acute angles. 32. The sum of the exterior angles adjacent to angles A and B of the triangle ABC is equal to 230°. Find the interior angle C of the triangle. 33. MP 3 What is the sum of the three exterior angles of a triangle, where one angle is measured at each vertex? Justify your answer. 34. MP 3 The sum of any two interior angles of a particular triangle is greater than 90°. Determine the type of triangle: obtuse, right, or acute. Explain your answer. 35. MP 3, 6 Prove the sum of the measures of the exterior angles of a triangle is 360°. Chapter 4 Review 1. The two lines shown are parallel. Which expression must be true? K B F A. B. C. D. K5F K 1 B 5 180° K 1 F 5 180° K1F5B 2. Angle Z forms a vertical pair with angle B, and angle Z is also a corresponding angle with Y. Z and Y are angles formed by a transversal and two parallel lines. Angles B and Y must be A. B. C. D. Supplementary. Complementary. Congruent. Have no definite relationship. 3. Point A is not on line j. How many lines pass through A and are parallel to j? A. B. C. D. 0 1 2 An infinite number 4. The angles A and E are C B D A G F H E A. B. C. D. Consecutive interior. Alternate interior. Alternate exterior. Corresponding. Chapter Review continues . . . Chapter 4 Review 187 Chapter Review continued . . . 5. Point b is not on line z. You can correctly say: A. No line can pass through b and be parallel to z. B. Only one line passes through b and is perpendicular to z. C. b and z cannot be coplanar. D. None of the above. 10. When two parallel lines are intersected by a transversal, the difference of two consecutive interior angles formed is 50°. Find these angles. 11. In the diagram, assume the measure of angle A is twice the measure of angle B. What is the measure of angle G? A B D C E F H G 6. Two lines never intersect. This means: A. B. C. D. They must be coplanar. They must be parallel lines. They must be skew lines. None of the above. 7. If m i n, how do we know that p ' n? 12. Given m/1 5 (2y 2 1), m/2 5 (3x 2 52), m/5 5 (2x 1 34), and m/7 5 (3y 2 10), what is m/8? p m 2 1 3 4 5 q n m p 6 7 8 9 A. p is perpendicular to m, and m is parallel to n. B. p is perpendicular to m, and m is perpendicular to n. C. q is perpendicular to n. 8. Do the points (25, 4), (8, 3), and (22, 22) define a right triangle? n 13. Prove that the opposite angles of the quadrilateral ABCD are congruent. A 1 4 D 3 a B 2 C b A. No. B. Yes. C. The points do not define a triangle at all. D. It is not possible to tell. 9. Using the diagram, prove the consecutive interior angles theorem converse. A B D C E F H G Prove: m n m n Given: mD + mE = 180° Chapter Review continues . . . 188 Chapter 4: Parallel and Perpendicular Lines Chapter Review continued . . . 14. Two parallel lines are intersected by a transversal. Two consecutive interior angles have measures of 2(x 2 1) and 3x 2 3. What is the measure of the smaller angle? 15. When two parallel lines p and q are intersected by a transversal n, the interior consecutive angles A and B are congruent. Prove that the transversal is perpendicular to both parallel lines. 19. In the diagram, AB || DE and m/CDE 5 21°. Find the measure of /ABC. D E C B 16. Find the distance between the point M and the line t. y 5 4 3 2 t 1 –5 –4 –3 –2 –1 1 2 3 4 5 x –1 –2 –4 –5 17. Find the distance between the point P and the line l. 22. Using the diagram, prove the perpendicular lines and right angles theorem. Prove: Angles A, B, C, and D are right angles. Given: m ' n y 5 4 3 m 2 1 P –5 –4 –3 20. Lines b and d are perpendicular to each other. Line a is parallel to line b. Line c intersects line b, forming an angle of 142°. All lines are coplanar. Find the smaller of the angles that line c forms with line a. Then find the smaller of the angles that line c forms with line d. Provide a sketch with your answer. 21. MP 5 Point M does not lie on line m. Explain how to construct a line p through point M and perpendicular to line m using a compass and a straightedge. Perform the constructions. –3 M A –2 –1 1 2 3 4 5 x –1 A B D C –2 –3 n –4 –5 l 18. A snail moves along a straight line with a constant speed, changing direction by a right angle every 3 minutes. It follows this pattern for 2 hours and 45 minutes. During the last minute of its trip, will the snail be parallel to the line it started along or perpendicular to it? 23. What is the slope of a line perpendicular to 1 y 5 x 1 9? 4 24. What is the slope of a line perpendicular to 1 y 5 x 1 7? 6 25. Explain how to determine if three given points could be the vertices of a right triangle. Chapter Review continues . . . Chapter 4 Review 189 Chapter Review continued . . . 26. Kathy found the slopes of two lines that she thought might be perpendicular. Both slopes were negative. What should Kathy conclude about the angle between these lines? 27. Explain why a triangle cannot have two right angles. 28. What is the measure of angle B? 20° A B 40° 29. Find the measure of the exterior angle adjacent to interior angle C of the triangle ABC, if A 5 45° and B 5 110°. 190 Chapter 4: Parallel and Perpendicular Lines 30. The measure of the exterior angle adjacent to the interior angle M of the triangle KLM is 110°. The two non-adjacent interior angles of the triangle are congruent. Find all interior angles of the triangle. 31. Using the diagram, prove the triangle sum corollary. Prove: mA + mB 5 90° Given: mC 5 90° A C B Cumulative Review for Chapters 1–4 1. Two lines are described by the equations y 5 2zx 1 5 and y 5 zx 2 3. The lines are A. B. C. D. Parallel. Perpendicular. Overlapping. None of the above. E A F 2. How could the angle be written? Choose all that apply. B A C A. ABC B. CBA C. ACB D. A E. C < > < > 3. Choose all that apply. AB and AC are C A B F A. B. C. D. 5. How would the congruent triangles be described? Coplanar in plane F. Parallel. Perpendicular. None of the above. 4. Two of the lines shown are parallel. This means that angles V and W C B D A. nABC nDEF B. nABC nFDE C. nABC nFED 6. Consider the true statement: “If it is an even number, then it is divisible by 2.” Which of the following is also a true statement? Choose all that apply. A. Converse B. Inverse C. Contrapositive 7. Dr. Watson said: “If the patient has the boola-boola disease, he sneezes 4 times a minute. If a patient mumbles ‘Eli, Eli’ twice an hour, the patient has the boola-boola disease.” Dr. Watson should conclude A. If a patient sneezes 4 times a minute, he mumbles ‘Eli, Eli’ twice an hour. B. If a patient sneezes 4 times a minute, the patient has the boola-boola disease. C. A person who does not have the boolaboola disease never sneezes. D. If a patient mumbles ‘Eli, Eli’ twice an hour, he sneezes 4 times a minute. V W A. B. C. D. Are congruent. Are supplementary. Are complementary. Have no defined mathematical relationship. Chapters 1–4 191 8. Check the statement, or statements, that are true. A B A. B. C. D. 30° 120° C 60° 150° D Angles A and B are complementary. Angles A and B are supplementary. Angles B and C are complementary. Angles B and D are supplementary. 9. What is m/C? A = 127° D B C 10. Point P lies on YZ. YZ is 28 and YP is 21. What is PZ? HJ 8 5 , how far is 1. J is on HI, and HI 5 33. If 1 JI 3 J from H? 12. You want to translate a figure 8 units orizontally and 6 units vertically. What h constants do you add to the x- and y-coordinates for this translation? 13. MP 3 Prove the transitive property for parallel lines: If the lines a and b are p arallel, and the lines b and c are parallel, then a is also parallel to c. Hint: Suppose that line a is not parallel to line c, and use the parallel postulate to show that this leads to a contradiction. arallel to 14. In the diagram below, line a is p line b, /1 /3, and the measure of /9 5 120°. Prove the shaded triangle is equilateral. 1 2 3 6 5 4 a b 7 8 10 9 192 Cumulative Review 11 12 14 13 15. The measure of angle D is twice the measure of angle E. What is the measure of angle B? A B D C E F H G 16. Two parallel lines are intersected by a transversal. Prove that the angle bisectors of two consecutive interior angles are perpendicular. Make a sketch, label the angles, and refer to them in your proof. 17. A teacher says: “Triangle ABC has vertex A at (24, 21), vertex B at (2, 28), and vertex C at (0, 0). ABC is a right triangle, with C its right angle.” Is she correct? Justify your answer. 18. A graphic artist is rescaling a picture from 72 pixels to represent a square to 90 pixels. If the original picture had a width of 300 pixels, how many pixels wide is the new picture? 19. What is the measure of angle 2x? 3x 2x x 20. nABC is translated as shown. What is the translation? y 9 8 7 6 5 4 3 2 1 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 B A A’ B’ C 1 2 3 4 5 6 7 8 9 x C’ 21. Figure A is dilated by a scale factor of 5 to form figure B, which is then dilated by a 7 scale factor of to form figure C. What 4 is the scale factor that dilates figure A to figure C? 22. Two triangles are similar. Triangle ABC has 36 times the area of triangle DEF. How many times longer would a side of triangle ABC be than the corresponding side in triangle DEF? 23. Using the point (a, b), show algebraically that four 90° counterclockwise rotations about the origin return you to the original point (a, b). 24. What are the coordinates of the upper right vertex of a figure if it is dilated by a factor of 7? The original figure’s coordinates are (1, 3), (1, 5), (27, 3), and (27, 5), and the center of dilation is the origin. Exercises 25–28: Write the biconditional for each definition. 25. The real numbers are composed of the rational and irrational numbers. 26. An integer prime is a number with exactly 32. Kim’s office is 14 feet by 20 feet. She wishes to lengthen the walls of her office by a scale factor of 1.5. What will be the dimensions of the expanded office? Exercises 33–34: Determine whether the following statement is always true, sometimes true, or never true. Justify your answer. 33. Inductive reasoning leads to a correct conclusion. 34. Deductive reasoning based on true statements leads to a correct conclusion. 35. MP 2, 3 ,4 There were 437 homes in the county of Piranha Lake in 1995. Ten years later, the number of homes had increased to 761. Therefore, the number of homes in Piranha Lake in 2010 was exactly 923. Is this an example of inductive or deductive reasoning using a graph? Justify your answer. two positive distinct factors. 27. Two angles that sum to 180° are called supplementary angles. 28. Parallel lines are lines on a plane that do not intersect. 29. MP 5 Draw and label the sides and angles of two similar quadrilaterals. 30. Two rectangles are similar. Their widths are corresponding sides: the width of rectangle A is 5, and the width of rectangle B is 3. If the larger rectangle has a height of 15, what is the height of the smaller rectangle? 31. The corresponding sides of two similar octagons (8-sided figures) have a scale factor of 2.4. If the smaller octagon has a perimeter of 20, what is the perimeter of the larger octagon? Chapters 1–4 193