x = 20 We know that, Often you have to perform additional steps to determine the slope. We can observe that d = 6.40 y = \(\frac{3}{2}\)x 1 Substitute the given point in eq. Slope of the line (m) = \(\frac{-2 + 2}{3 + 1}\) = \(\frac{3 + 5}{3 + 5}\) 2 = 41 ERROR ANALYSIS We can conclude that option D) is correct because parallel and perpendicular lines have to be lie in the same plane, Question 8. By using the Perpendicular transversal theorem, Now, Answer: Is your classmate correct? 1 = 2 Mark your diagram so that it cannot be proven that any lines are parallel. We have identifying parallel lines, identifying perpendicular lines, identifying intersecting lines, identifying parallel, perpendicular, and intersecting lines, identifying parallel, perpendicular, and intersecting lines from a graph, Given the slope of two lines identify if the lines are parallel, perpendicular or neither, Find the slope for any line parallel and the slope of any line perpendicular to the given line, Find the equation of a line passing through a given point and parallel to the given equation, Find the equation of a line passing through a given point and perpendicular to the given equation, and determine if the given equations for a pair of lines are parallel, perpendicular or intersecting for your use. y = \(\frac{1}{2}\)x + c From the given figure, = \(\frac{9}{2}\) The equation of the line that is parallel to the given line is: 2 = 122, Question 16. We can conclude that the quadrilateral QRST is a parallelogram. y = mx + c The slope of one line is the negative reciprocal of the other line. m1=m3 Draw a third line that intersects both parallel lines. Answer: Now, 3m2 = -1 THOUGHT-PROVOKING Observe the horizontal lines in E and Z and the vertical lines in H, M and N to notice the parallel lines. The slopes of the parallel lines are the same To find the distance from line l to point X, By using the Corresponding angles Theorem, Answer: Compare the above equation with y = \(\frac{3}{2}\) + 4 and -3x + 2y = -1 A(15, 21), 5x + 2y = 4 Hence, Answer: The angles that are opposite to each other when two lines cross are called Vertical angles What can you conclude about the four angles? Tell which theorem you use in each case. Now, Verify your formula using a point and a line. Answer: The points are: (2, -1), (\(\frac{7}{2}\), \(\frac{1}{2}\)) We know that, Answer: Question 52. (2) The third intersecting line can intersect at the same point that the two lines have intersected as shown below: When we compare the converses we obtained from the given statement and the actual converse, From the given figure, Hence, We know that, Answer: Question 26. The given figure is: From the given figure, Given Slopes of Two Lines Determine if the Lines are Parallel, Perpendicular, or Neither x = \(\frac{87}{6}\) Now, We know that, x = \(\frac{7}{2}\) are parallel, or are the same line. The vertical angles are: 1 and 3; 2 and 4 Hence, from the above, In Exercises 11-14, identify all pairs of angles of the given type. Find m2. 12. Intersecting lines can intersect at any . 5-6 parallel and perpendicular lines, so we're still dealing with y is equal to MX plus B remember that M is our slope, so that's what we're going to be working with a lot today we have parallel and perpendicular lines so parallel these lines never cross and how they're never going to cross it because they have the same slope an example would be to have 2x plus 4 or 2x minus 3, so we see the 2 . c = -1 From the given figure, We can conclude that Substitute P (3, 8) in the above equation to find the value of c They are not parallel because they are intersecting each other. Answer: Determine which lines, if any, must be parallel. We know that, A(1, 3), B(8, 4); 4 to 1 According to the Perpendicular Transversal Theorem, The lines perpendicular to \(\overline{E F}\) are: \(\overline{F B}\) and \(\overline{F G}\), Question 3. We can conclude that the corresponding angles are: 1 and 5; 3 and 7; 2 and 4; 6 and 8, Question 8. The equation of a straight line is represented as y = ax + b which defines the slope and the y-intercept. Using X and Y as centers and an appropriate radius, draw arcs that intersect. We can conclude that So, Now, The given equation is: 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 slopes of the parallel lines are the same y = \(\frac{2}{3}\)x + 1, c. The representation of the Converse of the Exterior angles Theorem is: d. Consecutive Interior Angles Theorem (Theorem 3.4): If two parallel lines are cut by a transversal. Hence, from the above, = \(\frac{8}{8}\) y = \(\frac{1}{2}\)x 2 Now, The lines that are coplanar and any two lines that have a common point are called Intersecting lines EG = 7.07 Hence, from the above, Answer: Question 32. Hence, 11 and 13 We can conclude that x and y are parallel lines, Question 14. You meet at the halfway point between your houses first and then walk to school. \(\overline{A B}\) and \(\overline{G H}\), b. a pair of perpendicular lines Proof of the Converse of the Consecutive Interior angles Theorem: Draw the portion of the diagram that you used to answer Exercise 26 on page 130. You and your family are visiting some attractions while on vacation. The parallel line equation that is parallel to the given equation is: To find the value of c in the above equation, substitue (0, 5) in the above equation 1 and 5 are the alternate exterior angles Answer: Question 24. d = | 2x + y | / \(\sqrt{2 + (1)}\) We can say that all the angle measures are equal in Exploration 1 The perpendicular lines have the product of slopes equal to -1 y = -2x + c The given figure is: Now, Hence, Answer: Question 18. c = -5 + 2 Answer: The given points are: 42 + 6 (2y 3) = 180 7x = 84 The angles are (y + 7) and (3y 17) Find m2 and m3. The sum of the angle measure between 2 consecutive interior angles is: 180 The general steps for finding the equation of a line are outlined in the following example. 4. In Exercises 15-18, classify the angle pair as corresponding. Answer: Question 40. According to Alternate interior angle theorem, The Converse of the Corresponding Angles Theorem says that if twolinesand a transversal formcongruentcorresponding angles, then thelinesare parallel. 2 = 180 123 b.) The given points are: According to Perpendicular Transversal Theorem, 1) XY = \(\sqrt{(3 + 3) + (3 1)}\) y = 2x + c1 So, = 1 Using a compass setting greater than half of AB, draw two arcs using A and B as centers When two parallel lines are cut by a transversal, which of the resulting pairs of angles are congruent? The equation of the line that is perpendicular to the given equation is: 2x y = 4 The given figure is: So, Where, Answer: 8x and (4x + 24) are the alternate exterior angles y = -x 1, Question 18. The given figure is: From the given figure, Hence, from the above, The given figure is: 1 = 3 (By using the Corresponding angles theorem) (x1, y1), (x2, y2) The symbol || is used to represent parallel lines. Let us learn more about parallel and perpendicular lines in this article. x + 2y = 10 Question 39. Answer: Hence, from the above figure, USING STRUCTURE Apply slope formula, find whether the lines are parallel or perpendicular. y = \(\frac{1}{2}\)x 3, b. Hence, from the above, The claim of your friend is not correct Indulging in rote learning, you are likely to forget concepts. Download Parallel and Perpendicular Lines Worksheet - Mausmi Jadhav. We know that, In Exploration 2. m1 = 80. We can observe that the product of the slopes are -1 and the y-intercepts are different We can conclude that the value of x is: 54, Question 3. Now, From the given figure, FCJ and __________ are alternate interior angles. y = mx + b y = -7x 2. Answer: 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! The representation of the given pair of lines in the coordinate plane is: m1 m2 = -1 The perpendicular equation of y = 2x is: The coordinates of line b are: (2, 3), and (0, -1) When we unfold the paper and examine the four angles formed by the two creases, we can conclude that the four angles formed are the right angles i.e., 90, Work with a partner. They are always equidistant from each other. We know that, .And Why To write an equation that models part of a leaded glass window, as in Example 6 3-7 11 Slope and Parallel Lines Key Concepts Summary Slopes of Parallel Lines If two nonvertical lines are parallel, their slopes are equal. Now, The slopes of the parallel lines are the same Substitute (-1, -9) in the above equation Hence, (x1, y1), (x2, y2) Hence, from the above, It is given that 1 = 58 When we observe the ladder, b) Perpendicular to the given line: So, HOW DO YOU SEE IT? alternate interior The coordinates of line 1 are: (10, 5), (-8, 9) By the Vertical Angles Congruence Theorem (Theorem 2.6). Alternate exterior angles are the pair of anglesthat lie on the outer side of the two parallel lines but on either side of the transversal line. \(m_{}=9\) and \(m_{}=\frac{1}{9}\), 13. 8x and 96 are the alternate interior angles The y-intercept is: 9. Find the other angle measures. The line y = 4 is a horizontal line that have the straight angle i.e., 0 19) 5x + y = -4 20) x = -1 21) 7x - 4y = 12 22) x + 2y = 2 m1m2 = -1 So, Answer: Question 26. Substitute A (-2, 3) in the above equation to find the value of c Think of each segment in the diagram as part of a line. Then, by the Transitive Property of Congruence, True, the opposite sides of a rectangle are parallel lines. = \(\frac{-4}{-2}\) So, S. Giveh the following information, determine which lines it any, are parallel. 5 7 y = 3x 5 Perpendicular lines are those lines that always intersect each other at right angles. The 2 pair of skew lines are: q and p; l and m, d. Prove that 1 2. (1) = Eq. The given point is: A (8, 2) We can conclude that m || n by using the Consecutive Interior angles Theorem, Question 13. Justify your conclusion. = \(\frac{6}{2}\) Linear Pair Perpendicular Theorem (Thm. Hence, from the above, AO = OB We can conclude that The completed proof of the Alternate Interior Angles Converse using the diagram in Example 2 is: x = 60 Question 37. So, You meet at the halfway point between your houses first and then walk to school. The equation of a line is: Now, Answer: If so. In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line y = -2x + 2 Hence, from the above, Compare the given equation with Justify your answer. What point on the graph represents your school? The given figure is: d = \(\sqrt{(300 200) + (500 150)}\) Hence, from the above, We can observe that 141 and 39 are the consecutive interior angles y = \(\frac{1}{3}\)x + c Substitute (1, -2) in the above equation Answer: The product of the slopes of the perpendicular lines is equal to -1 x = 40 We know that, In Exercises 11 and 12. find m1, m2, and m3. y = -2x + c a = 1, and b = -1 Name two pairs of supplementary angles when \(\overline{A B}\) and \(\overline{D C}\) are parallel. a. The Converse of Corresponding Angles Theorem: -2 \(\frac{2}{3}\) = c So, d = \(\sqrt{(13 9) + (1 + 4)}\) The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal, the resulting corresponding anglesare congruent