The third intersecting line can intersect at the same point that the two lines have intersected as shown below: So, Substitute this slope and the given point into point-slope form. PROVING A THEOREM The slope of first line (m1) = \(\frac{1}{2}\) Hence, from the above, 20 = 3x 2x 4x y = 1 So, x = 6 x = \(\frac{-6}{2}\) Explain your reasoning. CRITICAL THINKING WHAT IF? m1m2 = -1 The equation that is perpendicular to the given equation is: The equation of line p is: c = 2 0 To do this, solve for \(y\) to change standard form to slope-intercept form, \(y=mx+b\). Answer: Question 2. So, d = \(\sqrt{(300 200) + (500 150)}\) d = \(\frac{4}{5}\) So, Once the equation is already in the slope intercept form, you can immediately identify the slope. 1 = 40 and 2 = 140. = \(\sqrt{(3 / 2) + (3 / 2)}\) Why does a horizontal line have a slope of 0, but a vertical line has an undefined slope? y = -2x 1 Then by the Transitive Property of Congruence (Theorem 2.2), _______ . Alternate Interior angles theorem: Hence, from the above, Answer: Using X as the center, open the compass so that it is greater than half of XP and draw an arc. From the figure, ERROR ANALYSIS The theorem we can use to prove that m || n is: Alternate Exterior angles Converse theorem. Question 13. We know that, then they are parallel. -5 2 = b Proof: It is given that E is to \(\overline{F H}\) y = \(\frac{3}{2}\)x + c x = n The equation for another parallel line is: The given point is: (6, 4) y = 0.66 feet (D) Consecutive Interior Angles Converse (Thm 3.8) In Exercises 9 and 10, use a compass and straightedge to construct a line through point P that is parallel to line m. Question 10. = 255 yards = \(\frac{3}{4}\) Substitute (-1, -1) in the above equation Since the given line is in slope-intercept form, we can see that its slope is \(m=5\). So, The standard linear equation is: 7x = 84 We know that, So, These worksheets will produce 6 problems per page. 2 6, c. 1 ________ by the Alternate Exterior Angles Theorem (Thm. We know that, Perpendicular lines are denoted by the symbol . X (-3, 3), Y (3, 1) Now, The plane parallel to plane ADE is: Plane GCB. Answer: Question 28. d = \(\sqrt{(x2 x1) + (y2 y1)}\) Now, (5y 21) = (6x + 32) We know that, So, Proof of the Converse of the Consecutive Exterior angles Theorem: The given equation is: The lines that have the slopes product -1 and different y-intercepts are Perpendicular lines Consider the following two lines: Both lines have a slope \(m=\frac{3}{4}\) and thus are parallel. The equation for another parallel line is: We know that, We can conclude that the distance from point A to the given line is: 1.67. So, Parallel and Perpendicular Lines Perpendicular Lines Two nonvertical lines are perpendicular if their slopes are opposite reciprocals of each other. x = 4 and y = 2 forming a straight line. XY = 6.32 Hence, from the above, that passes through the point (4, 5) and is parallel to the given line. Compare the given points with (x1, y1), and (x2, y2) The letter A has a set of perpendicular lines. We know that, We can observe that there are 2 perpendicular lines Hence, from the above, Slope (m) = \(\frac{y2 y1}{x2 x1}\) Then use a compass and straightedge to construct the perpendicular bisector of \(\overline{A B}\), Question 10. Answer: Bertha Dr. is parallel to Charles St. m = \(\frac{3}{-1.5}\) x = y = 29, Question 8. Hence, from the above, The given equation is: Answer: m2 = \(\frac{2}{3}\) We know that, The given point is: (-1, 5) The lines that have the same slope and different y-intercepts are Parallel lines The given point is: A (2, 0) \(\frac{6-(-4)}{8-3}\) By using the corresponding angles theorem, Eq. Algebra 1 Writing Equations of Parallel and Perpendicular Lines 1) through: (2, 2), parallel to y = x + 4. Decide whether there is enough information to prove that m || n. If so, state the theorem you would use. Answer: Use a square viewing window. Is your classmate correct? For a horizontal line, It is given that 4 5 and \(\overline{S E}\) bisects RSF Explain. So, From Exploration 1, 2x = 3 EG = \(\sqrt{(1 + 4) + (2 + 3)}\) Answer: Hence, from the above, The coordinates of line d are: (0, 6), and (-2, 0) Question 29. Answer: 3.3) = \(\frac{11}{9}\) The angles that have the opposite corners are called Vertical angles Answer: Thus the slope of any line parallel to the given line must be the same, \(m_{}=5\). a. Explain. Question 25. 2 = 140 (By using the Vertical angles theorem) Question 37. Question 4. Given a b To be proficient in math, you need to communicate precisely with others. A(-1, 5), y = \(\frac{1}{7}\)x + 4 justify your answer. It is given that m || n -1 = \(\frac{1}{3}\) (3) + c The theorems involving parallel lines and transversals that the converse is true are: Solution: Using the properties of parallel and perpendicular lines, we can answer the given . The slope of vertical line (m) = \(\frac{y2 y1}{x2 x1}\) Hence, We can observe that, Answer: The angles that have the common side are called Adjacent angles b) Perpendicular line equation: Let the given points are: Sandwich: The highlighted lines in the sandwich are neither parallel nor perpendicular lines. The product of the slopes of perpendicular lines is equal to -1 So, y = -2x + c Now, If so. (1) = Eq. Parallel Curves We can conclue that Two lines, a and b, are perpendicular to line c. Line d is parallel to line c. The distance between lines a and b is x meters. The slope of line a (m) = \(\frac{y2 y1}{x2 x1}\) The sum of the adjacent angles is: 180 For parallel lines, we cant say anything By the Vertical Angles Congruence Theorem (Theorem 2.6). Answer: -5 = 2 (4) + c Answer: Question 28. Answer: Question 30. According to the Alternate Exterior angles Theorem, = \(\sqrt{2500 + 62,500}\) Explain your reasoning. No, your friend is not correct, Explanation: m2 = \(\frac{1}{2}\), b2 = 1 Use the diagram. We can observe that, So, 72 + (7x + 24) = 180 (By using the Consecutive interior angles theory) So, Step 3: Question 1. The slopes are equal fot the parallel lines Now, m2 and m3 Hence, from the above, Answer: So, The given statement is: which ones? Legal. You can select different variables to customize these Parallel and Perpendicular Lines Worksheets for your needs. Hence, from the above, Answer: Question 30. The slopes are equal for the parallel lines ERROR ANALYSIS To find the distance between the two lines, we have to find the intersection point of the line Perpendicular Postulate: y = mx + b It is given that, Simply click on the below available and learn the respective topics in no time. 1 = 2 = 150, Question 6. Enter a statement or reason in each blank to complete the two-column proof. Find an equation of line p. We can conclude that the number of points of intersection of parallel lines is: 0, a. x y = -4 2x y = 4 BCG and __________ are consecutive interior angles. Answer: A(0, 3), y = \(\frac{1}{2}\)x 6 Now, The given coordinates are: A (-2, 1), and B (4, 5) We can observe that when r || s, Proof of the Converse of the Consecutive Interior angles Theorem: Hence, from the above, Question 1. 8 = 65. In Euclidean geometry, the two perpendicular lines form 4 right angles whereas, In spherical geometry, the two perpendicular lines form 8 right angles according to the Parallel lines Postulate in spherical geometry. y = \(\frac{1}{3}\)x 2. What is m1? The given figure is: The perpendicular line equation of y = 2x is: y = 2x + 1 y = \(\frac{1}{7}\)x + 4 The area of the field = 320 140 Hence, y = -2 (-1) + \(\frac{9}{2}\) We were asked to find the equation of a line parallel to another line passing through a certain point. m1=m3 Answer: Compare the given points with (x1, y1), (x2, y2) m1m2 = -1 Perpendicular to \(6x+3y=1\) and passing through \((8, 2)\). First, solve for \(y\) and express the line in slope-intercept form. y = \(\frac{1}{3}\)x + c d = \(\sqrt{(11) + (13)}\) Slope of the line (m) = \(\frac{y2 y1}{x2 x1}\) We know that, The given figure is: Write an equation of a line perpendicular to y = 7x +1 through (-4, 0) Q. Now, So, XZ = \(\sqrt{(7) + (1)}\) Answer: Answer: Each unit in the coordinate plane corresponds to 50 yards. a.) Question 4. Slope of ST = \(\frac{2}{-4}\) Answer: Answer: Question 4. From the given figure, From the given figure, 2 and 7 are vertical angles The equation that is perpendicular to the given equation is: So, A (x1, y1), and B (x2, y2) XY = \(\sqrt{(x2 x1) + (y2 y1)}\) We can conclude that the value of x when p || q is: 54, b. Where, 5x = 149 MAKING AN ARGUMENT y = 13 The equation that is perpendicular to the given line equation is: We know that, The slope of the given line is: m = \(\frac{1}{4}\) Answer: Slope of TQ = 3 Answer: We know that, 3.2). The equation of the line that is perpendicular to the given line equation is: WRITING We can observe that the given lines are parallel lines We can conclude that b.) From the given figure, Parallel to \(2x3y=6\) and passing through \((6, 2)\). Hence, from the above, Now, y = -3x + 650, b. y = mx + c = \(\frac{-2}{9}\) The equation for another line is: c = 3 The lines that have an angle of 90 with each other are called Perpendicular lines The given figure is: line(s) parallel to . So, You started solving the problem by considering the 2 lines parallel and two lines as transversals Alternate exterior anglesare the pair ofanglesthat lie on the outer side of the two parallel lines but on either side of the transversal line The completed proof of the Alternate Interior Angles Converse using the diagram in Example 2 is: From the given figure, Question 9. Write the Given and Prove statements. The coordinates of the school = (400, 300) The angle measures of the vertical angles are congruent Answer: Question 14. The Converse of Corresponding Angles Theorem: If two angles form a linear pair. We can conclude that the value of k is: 5. d = \(\sqrt{(x2 x1) + (y2 y1)}\) From the given figure, What is the length of the field? y = -x -(1) What is the relationship between the slopes? In a plane, if a line is perpendicular to one of the two parallel lines, then it is perpendicular to the other line also The given statement is: This line is called the perpendicular bisector. Parallel to \(x=2\) and passing through (7, 3)\). We can conclude that the school have enough money to purchase new turf for the entire field. 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. 2x y = 4 So, The angle at the intersection of the 2 lines = 90 0 = 90 Compare the given coordinates with (x1, y1), and (x2, y2) y = \(\frac{1}{2}\)x 3, b. Which of the following is true when are skew? Which lines(s) or plane(s) contain point G and appear to fit the description? Given 1 2, 3 4 The given equation is:, a. m5 + m4 = 180 //From the given statement 9 = \(\frac{2}{3}\) (0) + b So, Slope (m) = \(\frac{y2 y1}{x2 x1}\) In Exercises 11 and 12, describe and correct the error in the statement about the diagram. Now, So, The given figure is: c. All the lines containing the balusters. It also shows that a and b are cut by a transversal and they have the same length We know that, m1m2 = -1 If you were to construct a rectangle, Compare the given points with Now, Answer: (x1, y1), (x2, y2) The given equation is: = \(\frac{1}{-4}\) The best editor is directly at your fingertips offering you a range of advantageous instruments for submitting a Algebra 1 Worksheet 3 6 Parallel And Perpendicular Lines. The measure of 1 is 70. It is not always the case that the given line is in slope-intercept form. \(\begin{array}{cc} {\color{Cerulean}{Point}}&{\color{Cerulean}{Slope}}\\{(-1,-5)}&{m_{\perp}=4}\end{array}\). c = 7 We know that, We can conclude that the alternate interior angles are: 3 and 6; 4 and 5, Question 7. The rungs are not intersecting at any point i.e., they have different points x = 29.8 and y = 132, Question 7. x = 54 y = \(\frac{1}{2}\) We can say that they are also parallel The map shows part of Denser, Colorado, Use the markings on the map. c = -2 So, The points are: (2, -1), (\(\frac{7}{2}\), \(\frac{1}{2}\)) 42 + 6 (2y 3) = 180 \(\overline{C D}\) and \(\overline{A E}\) are Skew lines because they are not intersecting and are non coplanar 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. Hence, from the above, Now, Now, Answer: Answer: y = -2x + 2 Find the Equation of a Parallel Line Passing Through a Given Equation and Point Geometrically, we see that the line \(y=4x1\), shown dashed below, passes through \((1, 5)\) and is perpendicular to the given line. Answer: b. Unfold the paper and examine the four angles formed by the two creases. Answer: So, So, y = 2x + c c = -3 + 4 Question 22. Definition of Parallel and Perpendicular Parallel lines are lines in the same plane that never intersect. So, Q (2, 6), R (6, 4), S (5, 1), and T (1, 3) 15) through: (4, -1), parallel to y = - 3 4 x16) through: (4, 5), parallel to y = 1 4 x - 4 17) through: (-2, -5), parallel to y = x + 318) through: (4, -4), parallel to y = 3 19) through . The slope of line l is greater than 0 and less than 1. By using the Corresponding angles Theorem, Question 3. Question 1. According to Euclidean geometry, Chapter 3 Parallel and Perpendicular Lines Key. = 1 y = \(\frac{1}{2}\)x + c x = 2 THOUGHT-PROVOKING The given figure is: P(2, 3), y 4 = 2(x + 3) = \(\frac{325 175}{500 50}\) Hence, from the above, -4 = -3 + c m = -1 [ Since we know that m1m2 = -1] Find the equation of the line passing through \((1, 5)\) and perpendicular to \(y=\frac{1}{4}x+2\). From the given figure, You and your mom visit the shopping mall while your dad and your sister visit the aquarium. From the given figure, The coordinates of line c are: (2, 4), and (0, -2) Write an equation of a line parallel to y = x + 3 through (5, 3) Q. m2 = -3 \(m_{}=\frac{5}{8}\) and \(m_{}=\frac{8}{5}\), 7. These worksheets will produce 6 problems per page. (x1, y1), (x2, y2) Name them. These worksheets will produce 10 problems per page. Now, Find the value of x when a b and b || c. b.) We can observe that the product of the slopes are -1 and the y-intercepts are different Hence, Justify your answer. Substitute (6, 4) in the above equation The product of the slopes of the perpendicular lines is equal to -1 3y = x 50 + 525 c. In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Answer: Since, (1) and eq. (Two lines are skew lines when they do not intersect and are not coplanar.) Now, EG = \(\sqrt{50}\) Proof: Now, i.e., Hence,f rom the above, y = -3 (0) 2 We know that, Answer: Given: 1 2 When two lines are crossed by another line (which is called the Transversal), theanglesin matching corners are calledcorresponding angles. So, The distance between the two parallel lines is: Question 7. 4 ________ b the Alternate Interior Angles Theorem (Thm. m = \(\frac{-2}{7 k}\) 1 = -3 (6) + b 11. Answer: Question 32. 1 = 60 5y = 116 + 21 The given figure is: So, y = \(\frac{3}{2}\)x + 2 y = -x + 8 So, Hence, from the above, Answer: m = \(\frac{1}{4}\) d = \(\sqrt{(x2 x1) + (y2 y1)}\) Compare the given equation with We can conclude that both converses are the same
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