Question 37. \(\frac{1}{2}\)x + 1 = -2x 1 b. Hence,f rom the above, We know that, m is the slope The angle at the intersection of the 2 lines = 90 0 = 90 We can observe that the given angles are corresponding angles So, d = \(\sqrt{(300 200) + (500 150)}\) So, Compare the given points with Answer: Compare the given equation with Answer: Question 36. Answer: if two lines are perpendicular to the same line. In Exploration 2. m1 = 80. We know that, The conjecture about \(\overline{A O}\) and \(\overline{O B}\) is: We know that, d. AB||CD // Converse of the Corresponding Angles Theorem Hence, from the above, So, So, Hence, from the coordinate plane, Perpendicular to \(\frac{1}{2}x\frac{1}{3}y=1\) and passing through \((10, 3)\). We know that, x = 23 y = \(\frac{1}{4}\)x + 4, Question 24. Answer: So, = \(\frac{10}{5}\) 4 = 5 The two lines are vertical lines and therefore parallel. So, Question 15. c = -1 1 Compare the given points with Answer: WHICH ONE did DOESNT BELONG? The given equation is: To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. The product of the slopes of perpendicular lines is equal to -1 She says one is higher than the other. 4.5 Equations of Parallel and Perpendicular Lines Solving word questions The slope of the given line is: m = 4 So, Answer: One way to build stairs is to attach triangular blocks to angled support, as shown. The coordinates of the subway are: (500, 300) The points are: (2, -1), (\(\frac{7}{2}\), \(\frac{1}{2}\)) Hence, from the above, y = \(\frac{2}{3}\) We can rewrite the equation of any horizontal line, \(y=k\), in slope-intercept form as follows: Written in this form, we see that the slope is \(m=0=\frac{0}{1}\). 3 = 53.7 and 4 = 53.7 Answer: Identify all the linear pairs of angles. 2 = \(\frac{1}{4}\) (8) + c Which lines(s) or plane(s) contain point G and appear to fit the description? Solution: We need to know the properties of parallel and perpendicular lines to identify them. The converse of the given statement is: Question 37. P || L1 You meet at the halfway point between your houses first and then walk to school. Let the given points are: Explain. To find the value of b, We know that, The given equation is: What point on the graph represents your school? Hence, from the above, Label the intersection as Z. When two lines are crossed by another line (which is called the Transversal), theanglesin matching corners are calledcorresponding angles. Mark your diagram so that it cannot be proven that any lines are parallel. The equation that is perpendicular to the given line equation is: c = 0 2 0 = 2 + c Hence, from the above, y 175 = \(\frac{1}{3}\) (x -50) \(m_{}=\frac{5}{8}\) and \(m_{}=\frac{8}{5}\), 7. Your classmate decided that based on the diagram. Answer: Question 50. = \(\frac{-3}{-4}\) ERROR ANALYSIS y = \(\frac{1}{2}\)x + c (13, 1), and (9, -4) We can conclude that the distance between the lines y = 2x and y = 2x + 5 is: 2.23. Use the diagram. x || y is proved by the Lines parallel to Transversal Theorem. answer choices Parallel Perpendicular Neither Tags: MGSE9-12.G.GPE.5 Question 7 300 seconds According to the consecutive exterior angles theorem, Alternate Interior Angles are a pair of angleson the inner side of each of those two lines but on opposite sides of the transversal. (-1) (m2) = -1 These worksheets will produce 10 problems per page. \(\begin{aligned} y-y_{1}&=m(x-x_{1}) \\ y-(-2)&=\frac{1}{2}(x-8) \end{aligned}\). Possible answer: 2 and 7 c. Possible answer: 1 and 8 d. Possible answer: 2 and 3 3. \(\frac{1}{2}\) (m2) = -1 = 5.70 \(\overline{C D}\) and \(\overline{E F}\), d. a pair of congruent corresponding angles x1 = x2 = x3 . We can conclude that in order to jump the shortest distance, you have to jump to point C from point A. Answer: You are looking : parallel and perpendicular lines maze answer key pdf Contents 1. The given figure is: Hence, from the above, \(\frac{6 (-4)}{8 3}\) We can conclude that y = -2x + c = 1.67 y = \(\frac{1}{3}\)x \(\frac{8}{3}\). Answer: We know that, Question: What is the difference between perpendicular and parallel? Answer: Hence, from the above, Similarly, observe the intersecting lines in the letters L and T that have perpendicular lines in them. y y1 = m (x x1) The given figure is: y = -2x + 3 Answer: Use an example to support your conjecture. Answer: Question 14. Answer: Question 32. The given figure is: We can conclude that a || b. To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. You and your family are visiting some attractions while on vacation. The equation for another line is: Approximately how far is the gazebo from the nature trail? The equation of a straight line is represented as y = ax + b which defines the slope and the y-intercept. So, Alternate Exterior Angles Theorem: XZ = \(\sqrt{(7) + (1)}\) Now, We can conclude that The given point is: (-5, 2) \(m_{}=10\) and \(m_{}=\frac{1}{10}\), Exercise \(\PageIndex{4}\) Parallel and Perpendicular Lines. We know that, So, In Exploration 2. find more pairs of lines that are different from those given. We can conclude that 4 and 5 are the Vertical angles. So, (- 1, 9), y = \(\frac{1}{3}\)x + 4 The given equation is: m is the slope Hence, from the above, Classify each pair of angles whose measurements are given. P( 4, 3), Q(4, 1) 2m2 = -1 In Example 5, The adjacent angles are: 1 and 2; 2 and 3; 3 and 4; and 4 and 1 If p and q are the parallel lines, then r and s are the transversals We know that, 1. So, y = mx + c 3x 5y = 6 y = mx + c We can conclude that lines intersect at 90. Hence, from the above, Yes, your classmate is correct, Explanation: So, y = mx + c Hence, from the above, 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 Now, So, We know that, Justify your answer with a diagram. Answer: Answer: Hence, from the above, A (-2, 2), and B (-3, -1) We can conclude that y = \(\frac{1}{2}\)x 5, Question 8. What can you conclude about the four angles? The line that passes through point F that appear skew to \(\overline{E H}\) is: \(\overline{F C}\), Question 2. Now, Justify your answer. Answer: Answer: So, So, The representation of the perpendicular lines in the coordinate plane is: In Exercises 21 24, find the distance from point A to the given line. x = n We can conclude that 1 and 5 are the adjacent angles, Question 4. (1) What can you conclude? The given figure is: The given figure is: The Converse of the Corresponding Angles Theorem: The slopes are equal fot the parallel lines Now, Answer: Line c and Line d are perpendicular lines, Question 4. Answer: y = 4x 7 To be proficient in math, you need to communicate precisely with others. Using P as the center and any radius, draw arcs intersecting m and label those intersections as X and Y. Question 3. The given equation is: A(- 6, 5), y = \(\frac{1}{2}\)x 7 The given figure is: From the figure, 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. We can conclude that the given pair of lines are parallel lines. Hence, Substitute (0, -2) in the above equation = \(\frac{1}{3}\) Statement of consecutive Interior angles theorem: Find the equation of the line perpendicular to \(x3y=9\) and passing through \((\frac{1}{2}, 2)\). Answer: The slopes are equal fot the parallel lines a. b = 9 Line 2: (2, 1), (8, 4) You can select different variables to customize these Parallel and Perpendicular Lines Worksheets for your needs. We know that, Answer: From the given figure, 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 A(8, 0), B(3, 2); 1 to 4 m = 2 The lines that have the slopes product -1 and different y-intercepts are Perpendicular lines We can conclude that the perpendicular lines are: So, Explain. USING STRUCTURE The equation of the line that is parallel to the line that represents the train tracks is: c = 5 7 The slopes of the parallel lines are the same Answer: y = \(\frac{1}{2}\) We can conclude that We can conclude that both converses are the same Find the equation of the line passing through \((6, 1)\) and parallel to \(y=\frac{1}{2}x+2\). So, 3 = 60 (Since 4 5 and the triangle is not a right triangle) It is given that 1 = 105 Answer: Question 12. From the given figure, Hence, Given m1 = 115, m2 = 65 For a parallel line, there will be no intersecting point Question 11. If you need more of a review on how to use this form, feel free to go to Tutorial 26: Equations of Lines m is the slope k = 5 XZ = \(\sqrt{(4 + 3) + (3 4)}\) Make a conjecture about how to find the coordinates of a point that lies beyond point B along \(\vec{A}\)B. You are trying to cross a stream from point A. We know that, Hence, The given equation is: The slopes are equal fot the parallel lines Which line(s) or plane(s) contain point B and appear to fit the description? (1) = Eq. In Exercises 3 6. find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. x = 20 9 = 0 + b x = 2 Answer: The given points are A (-1, 2), and B (3, -1) Compare the given points with A (x1, y1), B (x2, y2) m = Substitute A (-1, 2), and B (3, -1) in the formula. Answer: Perpendicular Postulate: Parallel and Perpendicular Lines Maintaining Mathematical Proficiency Find the slope of the line. Answer: So, as corresponding angles formed by a transversal of parallel lines, and so, y = \(\frac{3}{2}\) + 4 and -3x + 2y = -1 19) 5x + y = -4 20) x = -1 21) 7x - 4y = 12 22) x + 2y = 2 = Undefined Answer: The given figure is: Hence, from the above, So, 42 + 6 (2y 3) = 180 Let the given points are: We know that, Now, = | 4 + \(\frac{1}{2}\) | Classify each of the following pairs of lines as parallel, intersecting, coincident, or skew. We can observe that The given point is: A (2, 0) Question 23. x = 12 Hence, The intersection of the line is the y-intercept Answer: The given equation is: AP : PB = 4 : 1 Answer: So, We can conclude that the converse we obtained from the given statement is true The slopes are the same and the y-intercepts are different The distance between the meeting point and the subway is: A new road is being constructed parallel to the train tracks through points V. An equation of the line representing the train tracks is y = 2x. So, Answer: The given equation is: y = \(\frac{1}{2}\)x + c2, Question 3. = \(\sqrt{(9 3) + (9 3)}\) Hence, The parallel lines have the same slope but have different y-intercepts and do not intersect y1 = y2 = y3 How would your We know that, = \(\sqrt{(3 / 2) + (3 / 2)}\) In Exercises 19 and 20, describe and correct the error in the reasoning. Prove c||d From the figure, 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 When two lines are cut by a transversal, the pair ofangles on one side of the transversal and inside the two lines are called the Consecutive interior angles Answer: = \(\frac{5}{6}\) (A) Corresponding Angles Converse (Thm 3.5) Find the distance from point E to 1 = 42 d = \(\sqrt{(x2 x1) + (y2 y1)}\) You decide to meet at the intersection of lines q and p. Each unit in the coordinate plane corresponds to 50 yards. The given point is: A (-9, -3) Then explain how your diagram would need to change in order to prove that lines are parallel. Hence, from the above, Slope (m) = \(\frac{y2 y1}{x2 x1}\) So, When two lines are cut by a transversal, the pair ofangleson one side of the transversal and inside the two lines are called theconsecutive interior angles. = \(\frac{8}{8}\) We can conclude that \(\overline{P R}\) and \(\overline{P O}\) are not perpendicular lines. 12y = 156 The angles are (y + 7) and (3y 17) You can prove that4and6are congruent using the same method. Answer: 1 = 3 (By using the Corresponding angles theorem) The given point is: P (3, 8) x = 12 and y = 7, Question 3. Answer: The given equation in the slope-intercept form is: y = -2x + 8 In Exercise 31 on page 161, a classmate tells you that our answer is incorrect because you should have divided the segment into four congruent pieces. The given points are: m1 m2 = -1 The equation of the line that is perpendicular to the given line equation is: XZ = 7.07 The vertical angles are congruent i.e., the angle measures of the vertical angles are equal So, We can conclude that the given pair of lines are coincident lines, Question 3. To find the value of c, If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. From the above, Verify your formula using a point and a line. m = = So, slope of the given line is Question 2. In geometry, there are three different types of lines, namely, parallel lines, perpendicular lines, and intersecting lines. From the given figure, alternate interior, alternate exterior, or consecutive interior angles. a is both perpendicular to b and c and b is parallel to c, Question 20. Explain why the top rung is parallel to the bottom rung. Show your steps. So, Now, Answer: We can conclude that the given lines are parallel. Part - A Part - B Sheet 1 5) 6) Identify the pair of parallel and perpendicular line segments in each shape. 4x = 24 We can observe that there are 2 pairs of skew lines x = 180 73 Hence, Explain your reasoning. Answer: Answer: We know that, (6, 22); y523 x1 4 13. . Answer: The lines perpendicular to \(\overline{E F}\) are: \(\overline{F B}\) and \(\overline{F G}\), Question 3. Question 38. EG = 7.07 We can observe that The given rectangular prism is: Hence, from the above, it is given that the turf costs $2.69 per square foot The slope is: 3 y = mx + b We can conclude that the theorem student trying to use is the Perpendicular Transversal Theorem. Is your friend correct? So, Answer: Question 28. y = -3x 2 (2) Answer: Each unit in the coordinate plane corresponds to 50 yards. We can conclude that the value of x is: 133, Question 11. The given figure is: From the given figure, Find m1 and m2. ATTENDING TO PRECISION Using X as the center, open the compass so that it is greater than half of XP and draw an arc. According to Contradiction, (11x + 33) and (6x 6) are the interior angles So, 1 + 2 = 180 (By using the consecutive interior angles theorem) To do this, solve for \(y\) to change standard form to slope-intercept form, \(y=mx+b\). 2 and 3 are the congruent alternate interior angles, Question 1. Substitute (1, -2) in the above equation Are the numbered streets parallel to one another? = -1 Parallel lines m = \(\frac{5}{3}\) The equation for another line is: So, m1 and m5 Hence, from the above, We can conclude that the value of x is: 107, Question 10. y = -2x + 2 For which of the theorems involving parallel lines and transversals is the converse true? So, y = \(\frac{1}{2}\)x + 5 So, We know that, \(\overline{D H}\) and \(\overline{F G}\) Answer: Now, 5 = c Answer: We know that, We know that, = \(\frac{1}{3}\) The given figure is: Explain why ABC is a straight angle. The given point is: A(3, 6) The slope of first line (m1) = \(\frac{1}{2}\) We can conclude that the consecutive interior angles of BCG are: FCA and BCA. Answer: All perpendicular lines can be termed as intersecting lines, but all intersecting lines cannot be called perpendicular because they need to intersect at right angles. PROVING A THEOREM y = 4x + 9, Question 7. Answer: (x1, y1), (x2, y2) The parallel line equation that is parallel to the given equation is: So, The equation that is perpendicular to the given line equation is: 5 = \(\frac{1}{2}\) (-6) + c \(m_{}=\frac{3}{2}\) and \(m_{}=\frac{2}{3}\), 19. Your friend claims that lines m and n are parallel. = \(\sqrt{(6) + (6)}\) We can conclude that we can use Perpendicular Postulate to show that \(\overline{A C}\) is not perpendicular to \(\overline{B F}\), Question 3. We can conclude that the top rung is parallel to the bottom rung. 5 (28) 21 = (6x + 32) Corresponding Angles Theorem (Theorem 3.1): If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. We can conclude that Find the equation of the line passing through \((1, 5)\) and perpendicular to \(y=\frac{1}{4}x+2\). The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal, the resulting corresponding anglesare congruent Answer: Answer: The equation of the line along with y-intercept is: y = (5x 17) So, = \(\frac{-3}{-1}\) WRITING Question 43. x = \(\frac{153}{17}\) 2x + 4y = 4 We can conclude that Question 31. d = \(\sqrt{(x2 x1) + (y2 y1)}\) From the given figure, An equation of the line representing the nature trail is y = \(\frac{1}{3}\)x 4. could you still prove the theorem? So, m2 = -2 your friend claims to be able to make the shot Shown in the diagram by hitting the cue ball so that m1 = 25. The equation of the line along with y-intercept is: = \(\frac{45}{15}\) We can conclude that the distance from point A to the given line is: 5.70, Question 5. For perpediclar lines, 1 (m2) = -3 Question 4. Answer: We know that, 4.7 of 5 (20 votes) Fill PDF Online Download PDF. Question 12. The coordinates of line a are: (2, 2), and (-2, 3) = \(\frac{3 2}{-2 2}\) y = mx + c 3 + 8 = 180 We know that, \(\frac{5}{2}\)x = 5 If we try to find the slope of a perpendicular line by finding the opposite reciprocal, we run into a problem: \(m_{}=\frac{1}{0}\), which is undefined. So, Explain your reasoning. Now, We can observe that Step 5: CONSTRUCTION We know that, Question 22. Answer: y = -x + c Answer: -9 = \(\frac{1}{3}\) (-1) + c Hence, from the above, We can conclude that the value of x is: 12, Question 10. We can conclude that the distance from the given point to the given line is: 32, Question 7. Answer: So, The Skew lines are the lines that are non-intersecting, non-parallel and non-coplanar In a plane, if a line is perpendicular to one of the two parallel lines, then it is perpendicular to the other line also Repeat steps 3 and 4 below AB y = 2x + c If the slopes of two distinct nonvertical lines are equal, the lines are parallel. We can conclude that the pair of parallel lines are:
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