The line parallel to \(\overline{E F}\) is: \(\overline{D H}\), Question 2. The distance between the two parallel lines is: If two parallel lines are cut by a transversal, then the pairs of Corresponding angles are congruent. Perpendicular lines are intersecting lines that always meet at an angle of 90. 6-3 Write Equations of Parallel and Perpendicular Lines Worksheet. \(\begin{aligned} 6x+3y&=1 \\ 6x+3y\color{Cerulean}{-6x}&=1\color{Cerulean}{-6x} \\ 3y&=-6x+1 \\ \frac{3y}{\color{Cerulean}{3}}&=\frac{-6x+1}{\color{Cerulean}{3}} \\ y&=\frac{-6x}{3}+\frac{1}{3}\\y&=-2x+\frac{1}{3} \end{aligned}\). Hence, from the given figure, Answer: AP : PB = 3 : 2 The painted line segments that brain the path of a crosswalk are usually perpendicular to the crosswalk. (E) 1 and 8 are vertical angles y = \(\frac{1}{2}\)x + b (1) For example, if the equations of two lines are given as, y = -3x + 6 and y = -3x - 4, we can see that the slope of both the lines is the same (-3). Hence, from the above, So, This contradicts what was given,that angles 1 and 2 are congruent. Question 35. construction change if you were to construct a rectangle? The equation that is parallel to the given equation is: We can conclude that 4 5 and \(\overline{S E}\) bisects RSF. x + 2y = 10 So, We can conclude that the value of x is: 107, Question 10. The given figure is: The equation of a straight line is represented as y = ax + b which defines the slope and the y-intercept. (7x + 24) = 180 72 REASONING So, But, In spherical geometry, even though there is some resemblance between circles and lines, there is no possibility to form parallel lines as the lines will intersect at least at 1 point on the circle which is called a tangent y = 3x 5 Let A and B be two points on line m. Answer: What are the coordinates of the midpoint of the line segment joining the two houses? Which rays are not parallel? (5y 21) and 116 are the corresponding angles m = \(\frac{1}{2}\) (7x + 24) = 108 The distance between the meeting point and the subway is: Lines Perpendicular to a Transversal Theorem (Thm. a. Write the converse of the conditional statement. Find m1 and m2. Answer: Hence, from the above, y = \(\frac{3}{2}\)x + 2, b. Answer: Question 12. The representation of the perpendicular lines in the coordinate plane is: Question 19. The diagram shows lines formed on a tennis court. Now, Answer: The lines that are at 90 are Perpendicular lines PROBLEM-SOLVING We can observe that Hence, from the above, From the given figure, Compare the given equation with Similarly, in the letter E, the horizontal lines are parallel, while the single vertical line is perpendicular to all the three horizontal lines. Step 2: Substitute the slope you found and the given point into the point-slope form of an equation for a line. -2 = 0 + c b. m1 + m4 = 180 // Linear pair of angles are supplementary Answer: Answer: Question 40. 4 and 5 Compare the given points with (x1, y1), and (x2, y2) The slope of PQ = \(\frac{y2 y1}{x2 x1}\) Given Slope of a Line Find Slopes for Parallel and Perpendicular Lines Worksheets b. Alternate Exterior angles Theorem Begin your preparation right away and clear the exams with utmost confidence. So, c1 = 4 Use an example to support your conjecture. The diagram can be changed by the transformation of transversals into parallel lines and a parallel line into transversal Then use a compass and straightedge to construct the perpendicular bisector of \(\overline{A B}\), Question 10. Now, X (-3, 3), Y (3, 1) x + 2y = 10 Answer: Answer: So, Hence, from the above, So, The equation of a line is: Hence, from the above, Perpendicular to \(4x5y=1\) and passing through \((1, 1)\). We know that, The standard linear equation is: -2 = 3 (1) + c We can observe that the given angles are the consecutive exterior angles Example 2: State true or false using the properties of parallel and perpendicular lines. Answer: From the figure, Hence, from the above, Think of each segment in the diagram as part of a line. Answer: The given figure is: Each unit in the coordinate plane corresponds to 10 feet So, The given figure is: We know that, We know that, Yes, there is enough information to prove m || n Perpendicular Transversal Theorem A carpenter is building a frame. Slope (m) = \(\frac{y2 y1}{x2 x1}\) We know that, c = \(\frac{8}{3}\) For a horizontal line, Let the given points are: The missing information the student assuming from the diagram is: CRITICAL THINKING In Exercises 3 and 4. find the distance from point A to . y = \(\frac{2}{3}\)x + 1, c. 2. Now, Slope of AB = \(\frac{2}{3}\) 2x y = 18 Find a formula for the distance from the point (x0, Y0) to the line ax + by = 0. Write an equation for a line parallel to y = 1/3x - 3 through (4, 4) Q. (2x + 2) = (x + 56) So, The equation of the line that is perpendicular to the given line equation is: We can say that any intersecting line do intersect at 1 point The given figure is: 2y and 58 are the alternate interior angles Hence, Find the value of x that makes p || q. c.) Parallel lines intersect each other at 90. Answer: Answer: Question 4. Hence, Question 16. With Cuemath, you will learn visually and be surprised by the outcomes. Hence, The given point is: A (-1, 5) (x1, y1), (x2, y2) If the slope of two given lines are negative reciprocals of each other, they are identified as ______ lines. Hence, Answer: (x1, y1), (x2, y2) Slope of the line (m) = \(\frac{-1 2}{-3 + 2}\) We know that, m is the slope The angles formed at all the intersection points are: 90 0 = \(\frac{1}{2}\) (4) + c If the pairs of alternate interior angles are, Answer: y = \(\frac{10 12}{3}\) The coordinates of line 2 are: (2, -4), (11, -6) Write an equation of the line that is (a) parallel and (b) perpendicular to the line y = 3x + 2 and passes through the point (1, -2). a. We can conclude that the distance from the given point to the given line is: 32, Question 7. Expert-Verified Answer The required slope for the lines is given below. How are the Alternate Interior Angles Theorem (Theorem 3.2) and the Alternate Exterior b is the y-intercept y = -x + c y = -2x + 2, Question 6. The slope of first line (m1) = \(\frac{1}{2}\) The Converse of the Corresponding Angles Theorem says that if twolinesand a transversal formcongruentcorresponding angles, then thelinesare parallel. We can conclude that both converses are the same The given figure is: Now, The given point is: A (3, -1) b. We can conclude that the vertical angles are: FCA and __________ are alternate exterior angles. The slopes of the parallel lines are the same The given line equation is: But it might look better in y = mx + b form. Hence, from the above, We know that, No, we did not name all the lines on the cube in parts (a) (c) except \(\overline{N Q}\). y = -2x + b (1) 11. In Exercises 19 and 20, describe and correct the error in the reasoning. Compare the given equation with 12y = 138 + 18 The given table is: y = 2x To find the value of c, substitute (1, 5) in the above equation Answer: Name them. When we observe the Converse of the Corresponding Angles Theorem we obtained and the actual definition, both are the same PROVING A THEOREM Yes, there is enough information to prove m || n We know that, We know that, Given a b We know that, Now, The equation of a line is x + 2y = 10. Now, From the above figure, XY = \(\sqrt{(3 + 3) + (3 1)}\) We can conclude that the value of x is: 23. Hence, The construction of the walls in your home were created with some parallels. Each unit in the coordinate plane corresponds to 10 feet. c = -1 Explain your reasoning. Hence, from the above, So, line(s) PerPendicular to . Answer: Slope of RS = \(\frac{-3}{-1}\) The equation that is perpendicular to the given line equation is: Hence, Answer: x + 2y = 2 By comparing the given pair of lines with Substitute (-1, -9) in the given equation These worksheets will produce 6 problems per page. Answer: Parallel and perpendicular lines can be identified on the basis of the following properties: If the slope of two given lines is equal, they are considered to be parallel lines. Select the orange Get Form button to start editing. According to this Postulate, We can solve it by using the "point-slope" equation of a line: y y1 = 2 (x x1) And then put in the point (5,4): y 4 = 2 (x 5) That is an answer! We can conclude that the alternate exterior angles are: 1 and 8; 7 and 2. c is the y-intercept 1 = 32. 2 = 123 We know that, Answer: m || n is true only when x and 73 are the consecutive interior angles according to the Converse of Consecutive Interior angles Theorem x = 107 For parallel lines, Perpendicular Lines Homework 5: Linear Equations Slope VIDEO ANSWER: Gone to find out which line is parallel, so we have for 2 parallel lines right. Two lines are cut by a transversal. Proof: 4x + 2y = 180(2) d = \(\sqrt{(x2 x1) + (y2 y1)}\) Now, so they cannot be on the same plane. Answer: d. AB||CD // Converse of the Corresponding Angles Theorem. Compare the given points with So, Hence, We will use Converse of Consecutive Exterior angles Theorem to prove m || n 5x = 149 Follows 1 Expert Answers 1 Parallel And Perpendicular Lines Math Algebra Middle School Math 02/16/20 Slopes of Parallel and Perpendicular Lines y = \(\frac{1}{5}\)x + \(\frac{4}{5}\) Hence, from the above, \(\overline{D H}\) and \(\overline{F G}\) are Skew lines because they are not intersecting and are non coplanar, Question 1. Answer: Question 4. Answer: The completed table of the nature of the given pair of lines is: Work with a partner: In the figure, two parallel lines are intersected by a third line called a transversal. Answer Key Parallel and Perpendicular Lines : Shapes Write a relation between the line segments indicated by the arrows in each shape. If the corresponding angles are congruent, then the lines cut by a transversal are parallel Lines that are parallel to each other will never intersect. We can conclude that the plane parallel to plane LMQ is: Plane JKL, Question 5. This page titled 3.6: Parallel and Perpendicular Lines is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous. A(- \(\frac{1}{4}\), 5), x + 2y = 14 From the given figure, The coordinates of the school = (400, 300) The coordinates of x are the same. Compare the effectiveness of the argument in Exercise 24 on page 153 with the argument You can find the distance between any two parallel lines What flaw(s) exist in the argument(s)? -2 \(\frac{2}{3}\) = c Then, let's go back and fill in the theorems. Substitute (4, -3) in the above equation Answer: c = 2 Answer: Question 12. Which point should you jump to in order to jump the shortest distance? (x + 14)= 147 In Exploration 1, explain how you would prove any of the theorems that you found to be true. the equation that is perpendicular to the given line equation is: m1 and m3 Hence, The given equation is: The measure of 1 is 70. Simply click on the below available and learn the respective topics in no time. \(\frac{5}{2}\)x = \(\frac{5}{2}\) Hence, from the above, The parallel lines have the same slopes So, m2 = -2 The equation that is parallel to the given equation is: The given point is: A (3, -4) Possible answer: plane FJH plane BCD 2a. These Parallel and Perpendicular Lines Worksheets will show a graph of a series of parallel, perpendicular, and intersecting lines and ask a series of questions about the graph. Explain your reasoning. 132 = (5x 17) Perpendicular lines do not have the same slope. Proof: Question 17. The given figure is: Substitute A (3, -4) in the above equation to find the value of c So, AP : PB = 3 : 7 Question 39. d = \(\sqrt{(x2 x1) + (y2 y1)}\) The given perpendicular line equations are: Answer: Given \(\overrightarrow{B A}\) \(\vec{B}\)C Slope (m) = \(\frac{y2 y1}{x2 x1}\) (x1, y1), (x2, y2) They are not perpendicular because they are not intersecting at 90. So, The equation of the perpendicular line that passes through (1, 5) is: m is the slope 1 = 180 138 From the given figure, Hence, from the above, 42 + 6 (2y 3) = 180 an equation of the line that passes through the midpoint and is perpendicular to \(\overline{P Q}\). . It is given that (C) are perpendicular This line is called the perpendicular bisector. c. m5=m1 // (1), (2), transitive property of equality x = 97 A (x1, y1), B (x2, y2) \(\left\{\begin{aligned}y&=\frac{2}{3}x+3\\y&=\frac{2}{3}x3\end{aligned}\right.\), \(\left\{\begin{aligned}y&=\frac{3}{4}x1\\y&=\frac{4}{3}x+3\end{aligned}\right.\), \(\left\{\begin{aligned}y&=2x+1\\ y&=\frac{1}{2}x+8\end{aligned}\right.\), \(\left\{\begin{aligned}y&=3x\frac{1}{2}\\ y&=3x+2\end{aligned}\right.\), \(\left\{\begin{aligned}y&=5\\x&=2\end{aligned}\right.\), \(\left\{\begin{aligned}y&=7\\y&=\frac{1}{7}\end{aligned}\right.\), \(\left\{\begin{aligned}3x5y&=15\\ 5x+3y&=9\end{aligned}\right.\), \(\left\{\begin{aligned}xy&=7\\3x+3y&=2\end{aligned}\right.\), \(\left\{\begin{aligned}2x6y&=4\\x+3y&=2 \end{aligned}\right.\), \(\left\{\begin{aligned}4x+2y&=3\\6x3y&=3 \end{aligned}\right.\), \(\left\{\begin{aligned}x+3y&=9\\2x+3y&=6 \end{aligned}\right.\), \(\left\{\begin{aligned}y10&=0\\x10&=0 \end{aligned}\right.\), \(\left\{\begin{aligned}y+2&=0\\2y10&=0 \end{aligned}\right.\), \(\left\{\begin{aligned}3x+2y&=6\\2x+3y&=6 \end{aligned}\right.\), \(\left\{\begin{aligned}5x+4y&=20\\10x8y&=16 \end{aligned}\right.\), \(\left\{\begin{aligned}\frac{1}{2}x\frac{1}{3}y&=1\\\frac{1}{6}x+\frac{1}{4}y&=2\end{aligned}\right.\). = \(\frac{-4}{-2}\) The slope of first line (m1) = \(\frac{1}{2}\) The given figure is: The Perpendicular Postulate states that if there is a line and a point not on the line, then there is exactly one line through the point perpendicularto the given line. b is the y-intercept Prove: m || n The given coplanar lines are: 1 + 57 = 180 XZ = \(\sqrt{(7) + (1)}\) Hence, from the above, We know that, Answer: From the above figure, We can observe that the figure is in the form of a rectangle a. So, Hence, To find the value of c, Answer: 6 + 4 = 180, Question 9. Let the given points are: Question: ID Unit 3: Paraliel& Perpendicular Lines Homework 3: Proving Lines are Parolel Nome: Dnceuea pennon Per Date This is a 2-poge document Determine Im based on the intormation alven on the diogram yes, state the coverse that proves the ines are porollel 2 4. Now, We can conclude that Now, We can conclude that XY = \(\sqrt{(x2 x1) + (y2 y1)}\) We can conclude that \(\overline{K L}\), \(\overline{L M}\), and \(\overline{L S}\), d. Should you have named all the lines on the cube in parts (a)-(c) except \(\overline{N Q}\)? Draw the portion of the diagram that you used to answer Exercise 26 on page 130. WHICH ONE did DOESNT BELONG? The equation for another perpendicular line is: P = (4 + (4 / 5) 7, 1 + (4 / 5) 1) Is b c? 3.4). We can conclude that \(m_{}=\frac{4}{3}\) and \(m_{}=\frac{3}{4}\), 15. So, By the _______ . y = mx + c So, = \(\frac{3}{4}\) 1 = 2 The equation that is perpendicular to the given line equation is: Explain. Answer: The slopes of the parallel lines are the same The equation of the line that is parallel to the given line equation is: Question 5. We can conclude that the distance from point X to \(\overline{W Z}\) is: 6.32, Find XZ We know that, a. Hence, from the above, x = 54 Perpendicular lines are lines in the same plane that intersect at right angles (\(90\) degrees). We can conclude that 2 and 11 are the Vertical angles. We can observe that the product of the slopes are -1 and the y-intercepts are different (11y + 19) = 96 Answer: Question 26. We know that, c = -2 x = 2 Question 29. m is the slope Your friend claims that lines m and n are parallel. You can select different variables to customize these Parallel and Perpendicular Lines Worksheets for your needs. y = \(\frac{1}{3}\)x + 10 y = 180 35 (- 5, 2), y = 2x 3 y = \(\frac{1}{2}\)x + c Answer: Now, When we compare the actual converse and the converse according to the given statement, The two lines are Parallel when they do not intersect each other and are coplanar We can conclude that the distance of the gazebo from the nature trail is: 0.66 feet. x = -3 Compare the above equation with From the given figure, We can observe that, According to Contradiction, From the given figure, x = \(\frac{180}{2}\) So, So, = 1 It also shows that a and b are cut by a transversal and they have the same length The symbol || is used to represent parallel lines. = \(\frac{0 + 2}{-3 3}\) So, The consecutive interior angles are: 2 and 5; 3 and 8. Given: 1 2 Chapter 3 Parallel and Perpendicular Lines Key. if two lines are perpendicular to the same line. (5y 21) = (6x + 32) We can conclude that FCA and JCB are alternate exterior angles. From the given coordinate plane, The two pairs of supplementary angles when \(\overline{A B}\) and \(\overline{D C}\) are parallel is: ACD and BDC. Parallel and perpendicular lines worksheet answers key geometry - Note: This worksheet is supported by a flash presentation, under Mausmi's Math Q2: Determine. From the given figure, In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. y = \(\frac{1}{6}\)x 8 9 0 = b So, So, ERROR ANALYSIS b.) We can conclude that the top step is also parallel to the ground since they do not intersect each other at any point, Question 6. The points of intersection of intersecting lines: To find the coordinates of P, add slope to AP and PB plane(s) parallel to plane ADE Question 1. Substitute (3, 4) in the above equation Mark your diagram so that it cannot be proven that any lines are parallel. The slope of one line is the negative reciprocal of the other line. The point of intersection = (0, -2) For a pair of lines to be perpendicular, the product of the slopes i.e., the product of the slope of the first line and the slope of the second line will be equal to -1 The given figure is: The equation of the line that is parallel to the given line equation is: (B) Alternate Interior Angles Converse (Thm 3.6) y = -3 (0) 2 Substitute the given point in eq. c = -9 3 Transitive Property of Parallel Lines Theorem (Theorem 3.9),/+: If two lines are parallel to the same line, then they are parallel to each other. Question 42. Answer: We can conclude that X (-3, 3), Y (3, 1) 5 (28) 21 = (6x + 32) In Exercises 11 and 12, describe and correct the error in the statement about the diagram. 3y = x 50 + 525 The 2 pair of skew lines are: q and p; l and m, d. Prove that 1 2. According to the Perpendicular Transversal Theorem, Answer: Can you find the distance from a line to a plane? Explain your reasoning. The coordinates of line d are: (0, 6), and (-2, 0) The equation for another parallel line is: forming a straight line. then they are congruent. The coordinates of the subway are: (500, 300) From the figure, y = \(\frac{1}{2}\)x + 5 1 and 8 WHAT IF? -4 = -3 + c c = 2 + 2 One way to build stairs is to attach triangular blocks to angled support, as shown. (x1, y1), (x2, y2) Imagine that the left side of each bar extends infinitely as a line. XY = \(\sqrt{(4.5) + (1)}\) a. Hence, Answer: Hence, from the above, Hence, from the above, We know that, We can observe that the given lines are parallel lines Solve eq. m || n is true only when 147 and (x + 14) are the corresponding angles by using the Converse of the Alternate Exterior Angles Theorem a. From the given figure, y = -3x 2 (2) The angles that are opposite to each other when 2 lines cross are called Vertical angles The converse of the given statement is: Answer: A(- 2, 3), y = \(\frac{1}{2}\)x + 1 We can conclude that option D) is correct because parallel and perpendicular lines have to be lie in the same plane, Question 8. \(\frac{13-4}{2-(-1)}\) Parallel to line a: y=1/4x+1 Perpendicular to line a: y=-4x-3 Neither parallel nor perpendicular to line a: y=4x-8 What is the equation of a line that passes through the point (5, 4) and is parallel to the line whose equation is 2x + 5y = 10? So, In Exercises 13 16. write an equation of the line passing through point P that s parallel to the given line. So, According to the Vertical Angles Theorem, the vertical angles are congruent 6x = 87 REASONING c = 5 \(\frac{1}{2}\) Write an equation for a line perpendicular to y = -5x + 3 through (-5, -4) y = \(\frac{1}{3}\)x 2 -(1) If the line cut by a transversal is parallel, then the corresponding angles are congruent The following summaries about parallel and perpendicular lines maze answer key pdf will help you make more personal choices about more accurate and faster information. So, Explain your reasoning. It is important to have a geometric understanding of this question. c = 1 The given coordinates are: A (-3, 2), and B (5, -4) We can conclude that 8 right angles are formed by two perpendicular lines in spherical geometry. Two lines are termed as parallel if they lie in the same plane, are the same distance apart, and never meet each other. HOW DO YOU SEE IT? The slope is: \(\frac{1}{6}\) We can conclude that When the corresponding angles are congruent, the two parallel lines are cut by a transversal The given equation is: Decide whether there is enough information to prove that m || n. If so, state the theorem you would use. m = 3 and c = 9 Now, Slope (m) = \(\frac{y2 y1}{x2 x1}\) y = \(\frac{1}{3}\)x 4 A(8, 0), B(3, 2); 1 to 4 b is the y-intercept Using X as the center, open the compass so that it is greater than half of XP and draw an arc. The equation of the line that is perpendicular to the given line equation is: Since you are given a point and the slope, use the point-slope form of a line to determine the equation. (A) Corresponding Angles Converse (Thm 3.5) m = \(\frac{1}{4}\) \(\overline{C D}\) and \(\overline{A E}\) are Skew lines because they are not intersecting and are non coplanar The slope of the horizontal line (m) = \(\frac{y2 y2}{x2 x1}\) So, The standard form of the equation is: 3.2). The converse of the Alternate Interior angles Theorem: The coordinates of line c are: (2, 4), and (0, -2) Hence, from the above, The points are: (0, 5), and (2, 4) The lines are named as AB and CD. Answer: Now, From the given figure, 2x = 180 A (x1, y1), B (x2, y2) We can conclude that the distance from point A to \(\overline{X Z}\) is: 4.60. When two lines are cut by a transversal, the pair of angles on one side of the transversal and inside the two lines are called the Consecutive interior angles b. Vertical and horizontal lines are perpendicular. a is perpendicular to d and b isperpendicular to c, Question 22. If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. We know that, So, Slope of LM = \(\frac{0 n}{n n}\) We can conclude that the number of points of intersection of coincident lines is: 0 or 1. Answer: We know that, Hence, x = \(\frac{149}{5}\) The given figure is: 2x = 3 HOW DO YOU SEE IT? Hence, Determine the slope of a line perpendicular to \(3x7y=21\). 5 = 3 (1) + c d = | -2 + 6 |/ \(\sqrt{5}\) The lines that are a straight angle with the given line and are coplanar is called Perpendicular lines 1 = -3 (6) + b x + 2y = -2 We can conclude that the given pair of lines are coincident lines, Question 3. b. The given equation is: y y1 = m (x x1) 3 = 68 and 8 = (2x + 4) These worksheets will produce 6 problems per page. Label its intersection with \(\overline{A B}\) as O. So, The parallel lines do not have any intersecting points So, Answer: Question 26. Now, Compare the above equation with From the given coordinate plane, a. We know that, 2x x = 56 2 Hence, from the above, It is given that m || n By using the Alternate Exterior Angles Theorem, We know that, y = \(\frac{1}{4}\)x + 4, Question 24. Hence, from the above, For example, AB || CD means line AB is parallel to line CD. m1 m2 = -1 Answer: Answer: According to the Converse of the Corresponding angles Theorem, Embedded mathematical practices, exercises provided make it easy for you to understand the concepts quite quickly. Answer: m2 = \(\frac{1}{3}\) = 9.48 The given point is: A (0, 3) Explain. If the sum of the angles of the consecutive interior angles is 180, then the two lines that are cut by a transversal are parallel A triangle has vertices L(0, 6), M(5, 8). Question 12. So, If two lines are intersected by a third line, is the third line necessarily a transversal? y = 3x 5 y = 3x 5 Vertical Angles Theoremstates thatvertical angles,anglesthat are opposite each other and formed by two intersecting straight lines, are congruent Answer: The product of the slopes of the perpendicular lines is equal to -1 Answer: So, 3y = x + 475 The given figure is: c. Use the properties of angles formed by parallel lines cut by a transversal to prove the theorem. m2 = -1 b) Perpendicular line equation: d = \(\sqrt{(x2 x1) + (y2 y1)}\) Now, CONSTRUCTING VIABLE ARGUMENTS THINK AND DISCUSS 1. HOW DO YOU SEE IT? We can conclude that Question 37. 0 = 3 (2) + c Slope of ST = \(\frac{2}{-4}\) When we compare the converses we obtained from the given statement and the actual converse, Question 5. (2) to get the values of x and y y = -3x + c y = 3x 6, Question 20. (11x + 33) and (6x 6) are the interior angles The slope of the given line is: m = \(\frac{1}{4}\) From the given figure, To find the coordinates of P, add slope to AP and PB We get, We know that, We can conclude that 3m2 = -1 THOUGHT-PROVOKING Answer: We can conclude that the length of the field is: 320 feet, b. c = 7 So, It is given that you and your friend walk to school together every day. Slope of AB = \(\frac{1 + 4}{6 + 2}\) Hene, from the given options, Hence, from the given figure, x = \(\frac{84}{7}\) d = | ax + by + c| /\(\sqrt{a + b}\) Hence, from the given figure, Answer: The completed proof of the Alternate Interior Angles Converse using the diagram in Example 2 is: From the given figure, We can conclude that the value of x is: 54, Question 3. Answer: Question 52. According to the Converse of the Interior Angles Theory, m || n is true only when the sum of the interior angles are supplementary If p and q are the parallel lines, then r and s are the transversals y = \(\frac{1}{3}\)x + c The y-intercept is: 9. The given figure is: 3x 5y = 6 Key Question: If x = 115, is it possible for y to equal 115? REASONING Hence, Hence, from the above, = (4, -3) The slopes are equal fot the parallel lines Answer: Question 30. So, From the given figure, consecutive interior Now, Find the distance from the point (6, 4) to the line y = x + 4. Label the ends of the crease as A and B. c = -2 5 = c We can observe that when r || s, CONSTRUCTING VIABLE ARGUMENTS From the given figure, Now, The product of the slopes of the perpendicular lines is equal to -1 Use the steps in the construction to explain how you know that\(\overline{C D}\) is the perpendicular bisector of \(\overline{A B}\). Hence, from the above, What does it mean when two lines are parallel, intersecting, coincident, or skew? We can conclude that, Slope (m) = \(\frac{y2 y1}{x2 x1}\) 5 = -4 + b The lines that do not intersect and are not parallel and are not coplanar are Skew lines Question 3. Why does a horizontal line have a slope of 0, but a vertical line has an undefined slope? The given figure is: From the argument in Exercise 24 on page 153, We can conclude that the value of x is: 133, Question 11. From the given figure, If two straight lines lie in the same plane, and if they never intersect each other, they are called parallel lines. So, What are Parallel and Perpendicular Lines? then they are parallel. Answer: So, Answer: From the given figure, Answer: So, We can observe that the slopes of the opposite sides are equal i.e., the opposite sides are parallel Hence, Hence, from the above, Explain your reasoning. From the given figure, Hence, from the above, Make a conjecture about how to find the coordinates of a point that lies beyond point B along \(\vec{A}\)B. Hence, from the above,