The product of the slopes of the perpendicular lines is equal to -1 By comparing the given pair of lines with THOUGHT-PROVOKING The equation that is perpendicular to the given line equation is: 1 = 3 (By using the Corresponding angles theorem) The equation that is perpendicular to the given line equation is: The given figure is: ERROR ANALYSIS According to the Perpendicular Transversal Theorem, y = \(\frac{1}{2}\)x + 6 We can conclude that the distance from line l to point X is: 6.32. The equation of the line that is parallel to the given line is: Where, So, line(s) parallel to According to the Vertical Angles Theorem, the vertical angles are congruent The parallel lines do not have any intersecting points The given point is: A (8, 2) Parallel to \(6x\frac{3}{2}y=9\) and passing through \((\frac{1}{3}, \frac{2}{3})\). Answer: Question 8. By using the corresponding angles theorem, The given figure is: BCG and __________ are corresponding angles. Name a pair of parallel lines. and N(4, 1), Is the triangle a right triangle? 2x + y = 180 18 In Exercises 21 and 22, write and solve a system of linear equations to find the values of x and y. We can conclude that the value of x is: 14. Answer Keys - These are for all the unlocked materials above. The given point is: A (-\(\frac{1}{4}\), 5) y = -2x + 3 Question 35. The given equation in the slope-intercept form is: Perpendicular lines always intersect at 90. Substitute A (2, 0) in the above equation to find the value of c (2x + 12) + (y + 6) = 180 PROVING A THEOREM For example, the opposite sides of a square and a rectangle have parallel lines in them, and the adjacent lines in the same shapes are perpendicular lines. = \(\frac{4}{-18}\) The representation of the parallel lines in the coordinate plane is: Question 16. The given figure is: So, Answer: = (4, -3) Answer: 90 degrees (a right angle) That's right, when we rotate a perpendicular line by 90 it becomes parallel (but not if it touches!) From the given figure, Let us learn more about parallel and perpendicular lines in this article. The point of intersection = (0, -2) y = -3x + 150 + 500 To find the value of c, Answer: Answer: x = 35 and y = 145, Question 6. 2 = 180 3 4 = 2 (3) + c ATTENDING TO PRECISION m2 = -1 Answer: The given point is: (-1, 6) Given: 1 and 3 are supplementary So, Identify all the linear pairs of angles. The equation of the parallel line that passes through (1, 5) is: We can conclude that We can conclude that the given lines are parallel. Answer: Find an equation of line q. A1.3.1 Write an equation of a line when given the graph of the line, a data set, two points on the line, or the slope and a point of the line; A1.3.2 Describe and calculate the slope of a line given a data set or graph of a line, recognizing that the slope is the rate of change; A1.3.6 . If the slope of AB and CD are the same value, then they are parallel. We can say that w and v are parallel lines by Perpendicular Transversal Theorem Justify your answers. The given figure is: = \(\sqrt{30.25 + 2.25}\) The line through (k, 2) and (7, 0) is perpendicular to the line y = x \(\frac{28}{5}\). The given figure is: 2 and 3 are the consecutive interior angles The standard form of the equation is: We know that, MATHEMATICAL CONNECTIONS invest little times to right of entry this on-line notice Parallel And Perpendicular Lines Answer Key as capably as review them wherever you are now. A group of campers ties up their food between two parallel trees, as shown. One answer is the line that is parallel to the reference line and passing through a given point. Now, Slope of AB = \(\frac{-4 2}{5 + 3}\) Answer: Where, m1 = \(\frac{1}{2}\), b1 = 1 Substitute (2, -2) in the above equation We can conclude that 11 and 13 are the Consecutive interior angles, Question 18. We can conclude that According to the Converse of the Corresponding Angles Theorem, m || n is true only when the corresponding angles are congruent the equation that is perpendicular to the given line equation is: The intersection point is: (0, 5) Answer: Answer: p || q and q || r. Find m8. From the given figure, 11 and 13 We can observe that the given lines are perpendicular lines What can you conclude about the four angles? So, x = \(\frac{153}{17}\) y = mx + c So, Answer: Answer: From the given figure, The conjecture about \(\overline{A B}\) and \(\overline{c D}\) is: The equation that is perpendicular to y = -3 is: (x1, y1), (x2, y2) X (-3, 3), Z (4, 4) We can conclude that the pair of parallel lines are: The coordinates of P are (3.9, 7.6), Question 3. = \(\frac{3 + 5}{3 + 5}\) 3m2 = -1 If two angles form a linear pair. m = \(\frac{-2}{7 k}\) y = mx + c Given: k || l, t k Question 4. Answer: Question 40. According to the Perpendicular Transversal theorem, Proof: The given point is: A (2, -1) The given lines are: These worksheets will produce 6 problems per page. Hence, from the above, We can conclude that the slope of the given line is: 0. Question 23. (1) = Eq. WRITING \(\frac{5}{2}\)x = 5 The line that passes through point F that appear skew to \(\overline{E H}\) is: \(\overline{F C}\), Question 2. y = 180 35 = \(\sqrt{(3 / 2) + (3 / 2)}\) Explain your reasoning. A gazebo is being built near a nature trail. We can conclue that MODELING WITH MATHEMATICS The line through (- 1, k) and (- 7, 2) is parallel to the line y = x + 1. In spherical geometry. b. Since, Make the most out of these preparation resources and stand out from the rest of the crowd. d = \(\frac{4}{5}\) = 2 (320 + 140) Given m1 = 105, find m4, m5, and m8. Proof of Alternate exterior angles Theorem: Now, Prove: l || m Determine the slope of a line parallel to \(y=5x+3\). 3 = 60 (Since 4 5 and the triangle is not a right triangle) Now, y = 2x + 12 x = 97, Question 7. Question 1. x + 2y = 10 Are the numbered streets parallel to one another? 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. Hence, from the above, From the given figure, Answer: Question 16. HOW DO YOU SEE IT? Write the equation of the line that is perpendicular to the graph of 9y = 4x , and whose y-intercept is (0, 3). A(- 3, 2), B(5, 4); 2 to 6 It is given that l || m and l || n, consecutive interior x = 54 Now, From the given figure, So, So, y = \(\frac{2}{3}\)x + b (1) The points of intersection of intersecting lines: Now, Two lines are cut by a transversal. If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent For parallel lines, In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. The representation of the given point in the coordinate plane is: Question 54. Slope (m) = \(\frac{y2 y1}{x2 x1}\) The conjectures about perpendicular lines are: Question 13. Measure the lengths of the midpoint of AB i.e., AD and DB. We know that, We know that, Prove 2 4 The lines that are at 90 are Perpendicular lines We know that, The slope of the given line is: m = -2 We know that, Perpendicular lines are denoted by the symbol . We can observe that The product of the slopes of the perpendicular lines is equal to -1 Now, The equation of the line along with y-intercept is: Question 43. According to the Perpendicular Transversal Theorem, The slope of the given line is: m = \(\frac{1}{4}\) We know that, We know that, i.e., So, 2x + 4y = 4 x = 40 b. y = \(\frac{3}{2}\)x + 2 y = 2x + 1 1 + 138 = 180 Answer: Question 4. Answer: The equation of a line is: The equation of a line is: DRAWING CONCLUSIONS Hence, Question 12. (x1, y1), (x2, y2) (- 1, 5); m = 4 (\(\frac{1}{3}\)) (m2) = -1 EG = \(\sqrt{(1 + 4) + (2 + 3)}\) Hence, 1 and 8 The equation of the line that is perpendicular to the given equation is: To find the value of c, 132 = (5x 17) Question: What is the difference between perpendicular and parallel? Slope (m) = \(\frac{y2 y1}{x2 x1}\) We can observe that Hence, from the above, The area of the field = Length Width The slope of the equation that is parallel t the given equation is: \(\frac{1}{3}\) Answer: Hence, from the above, Hence, from the above, If m1 = 58, then what is m2? We know that, So, lines intersect at 90. = \(\frac{325 175}{500 50}\) XY = \(\sqrt{(3 + 1.5) + (3 2)}\) It is given that In Exercises 11 and 12. prove the theorem. For a pair of lines to be non-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 not be equal to -1 For parallel lines, we cant say anything The given equation is: We know that, (a) parallel to the line y = 3x 5 and We can observe that EG = 7.07 Answer: b) Perpendicular to the given line: Substitute (4, -5) in the above equation c = 3 Question 27. Parallel lines 13) x - y = 0 14) x + 2y = 6 Write the slope-intercept form of the equation of the line described. y = -x -(1) We know that, So, P = (3 + (\(\frac{3}{10}\) 3), 7 + (\(\frac{3}{10}\) 2)) The midpoint of PQ = (\(\frac{x1 + x2}{2}\), \(\frac{y1 + y2}{2}\)) We can conclude that Compare the given points with 10) Expert-Verified Answer The required slope for the lines is given below. Now, Answer: Question 36. (2) The equation for another parallel line is: Hence, The intersection point of y = 2x is: (2, 4) The portion of the diagram that you used to answer Exercise 26 on page 130 is: Question 2. 2x = 18 Slope (m) = \(\frac{y2 y1}{x2 x1}\) So, In Exercises 7-10. find the value of x. Answer: So, (x1, y1), (x2, y2) Substitute (1, -2) in the above equation Hence, from the above, Let the given points are: In Exercises 15 and 16, prove the theorem. parallel Answer: Explanation: In the above image we can observe two parallel lines. XY = \(\sqrt{(3 + 3) + (3 1)}\) Explain. If you need more of a review on how to use this form, feel free to go to Tutorial 26: Equations of Lines Perpendicular to \(\frac{1}{2}x\frac{1}{3}y=1\) and passing through \((10, 3)\). y = mx + c It is given that 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. The given point is: (0, 9) The slope of PQ = \(\frac{y2 y1}{x2 x1}\) We can conclude that Angles Theorem (Theorem 3.3) alike? a. Fro the given figure, P = (4 + (4 / 5) 7, 1 + (4 / 5) 1) Answer: Question 28. Explain your reasoning. We can conclude that the distance that the two of the friends walk together is: 255 yards. We know that, So, Identify all the pairs of vertical angles. From the given figure, Find the equation of the line passing through \((3, 2)\) and perpendicular to \(y=4\). = \(\frac{11}{9}\) Work with a partner: Fold and crease a piece of paper. 2 ________ by the Corresponding Angles Theorem (Thm. x = 6, Question 8. AP : PB = 2 : 6 A(2, 1), y = x + 4 Proof of the Converse of the Consecutive Exterior angles Theorem: Alternate exterior anglesare the pair ofanglesthat lie on the outer side of the two parallel lines but on either side of the transversal line We know that, Answer: The given figure is: By using the consecutive interior angles theorem, Hence, Answer: Slope of the line (m) = \(\frac{y2 y1}{x2 x1}\) The angle measures of the vertical angles are congruent d = 17.02 Use the numbers and symbols to create the equation of a line in slope-intercept form From the given figure, Now, Now, Since k || l,by the Corresponding Angles Postulate, Inverses Tables Table of contents Parallel Lines Example 2 Example 3 Perpendicular Lines Example 1 Example 2 Example 3 Interactive Substitute A (3, 4) in the above equation to find the value of c \(m_{}=9\) and \(m_{}=\frac{1}{9}\), 13. c.) False, parallel lines do not intersect each other at all, only perpendicular lines intersect at 90. The slopes are equal fot the parallel lines The coordinates of line 1 are: (-3, 1), (-7, -2) -5 = 2 + b We have to find 4, 5, and 8 We use this and the point \((\frac{7}{2}, 1)\) in point-slope form. According to Contradiction, In the diagram, how many angles must be given to determine whether j || k? Now, According to the Alternate Interior Angles Theorem, the alternate interior angles are congruent \(m_{}=\frac{5}{8}\) and \(m_{}=\frac{8}{5}\), 7. We can conclude that the alternate exterior angles are: 1 and 8; 7 and 2. Given 1 and 3 are supplementary. Which values of a and b will ensure that the sides of the finished frame are parallel.? When we compare the given equation with the obtained equation, d = 364.5 yards Eq. y = \(\frac{1}{3}\)x + c It is given that 4 5. We know that, An equation of the line representing Washington Boulevard is y = \(\frac{2}{3}\)x. We can conclude that p and q; r and s are the pairs of parallel lines. m2 = -2 Now, Answer: your friend claims to be able to make the shot Shown in the diagram by hitting the cue ball so that m1 = 25. With Cuemath, you will learn visually and be surprised by the outcomes. Is it possible for consecutive interior angles to be congruent? c. In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. The product of the slopes of perpendicular lines is equal to -1 So, d. AB||CD // Converse of the Corresponding Angles Theorem So, Is b || a? A bike path is being constructed perpendicular to Washington Boulevard through point P(2, 2). Prove c||d You started solving the problem by considering the 2 lines parallel and two lines as transversals 2 = 57 To make the top of the step where 1 is present to be parallel to the floor, the angles must be Alternate Interior angles 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? E (-4, -3), G (1, 2) Answer: = \(\frac{-3}{4}\) x = 6 We can observe that d = | -2 + 6 |/ \(\sqrt{5}\) Now, From the given figure, The given point is: (-5, 2) Question 47. a. corresponding angles It is given that 4 5. Answer: Consider the following two lines: Consider their corresponding graphs: Figure 3.6.1 Answer: Question 28. We know that, Answer: Question 26. Find the values of x and y. Since the given line is in slope-intercept form, we can see that its slope is \(m=5\). Compare the given points with (x1, y1), (x2, y2) Hence, from the above, If you go to the zoo, then you will see a tiger. The slope of the parallel equations are the same The two lines are Parallel when they do not intersect each other and are coplanar Explain your reasoning. Substitute (4, 0) in the above equation We can conclude that the distance from point A to the given line is: 8.48. x = \(\frac{4}{5}\) 8 = \(\frac{1}{5}\) (3) + c True, the opposite sides of a rectangle are parallel lines. Hence, from the above, We know that, So, Answer: Name two pairs of supplementary angles when \(\overline{A B}\) and \(\overline{D C}\) are parallel. We know that, Hence, from the above, Draw an arc with center A on each side of AB. It is given that your school has a budget of $1,50,000 but we only need $1,20,512 y = x 6 z x and w z We can conclude that the slope of the given line is: \(\frac{-3}{4}\), Question 2. \(\frac{13-4}{2-(-1)}\) The equation for another line is: Perpendicular to \(y=x\) and passing through \((7, 13)\). From the figure, False, the letter A does not have a set of perpendicular lines because the intersecting lines do not meet each other at right angles. The rungs are not intersecting at any point i.e., they have different points So, y = mx + c The given equation is: From the given figure, Your friend claims that lines m and n are parallel. a.) Possible answer: 1 and 3 b. Answer: Explain your reasoning. We can observe that So, MODELING WITH MATHEMATICS x = \(\frac{3}{2}\) If the slope of two given lines are negative reciprocals of each other, they are identified as perpendicular lines. x + 2y = 10 it is given that the turf costs $2.69 per square foot (180 x) = x We can conclude that the number of points of intersection of intersecting lines is: 1, c. The points of intersection of coincident lines: The equation of the line along with y-intercept is: Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Now, Answer: Question 4. Answer: Slope of line 2 = \(\frac{4 6}{11 2}\) We can conclude that b || a, Question 4. We know that, Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. y = \(\frac{1}{2}\)x 7 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 Example 1: Observe the blue highlighted lines in the following examples and identify them as parallel or perpendicular lines. The equation of the line along with y-intercept is: = \(\frac{-2 2}{-2 0}\) a. 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. F if two coplanar strains are perpendicular to the identical line then the 2 strains are. We know that, Hence, Work with a partner: Write the equations of the parallel or perpendicular lines. You are designing a box like the one shown. -2y = -24 (1) = Eq. (1) and eq. Answer: b. Unfold the paper and examine the four angles formed by the two creases. 3.4) The representation of the given coordinate plane along with parallel lines is: Answer: Compare the given points with (x1, y1), and (x2, y2) (Two lines are skew lines when they do not intersect and are not coplanar.) No, there is no enough information to prove m || n, Question 18. Explain your reasoning. Parallel and Perpendicular Lines Maintaining Mathematical Proficiency Page 123, Parallel and Perpendicular Lines Mathematical Practices Page 124, 3.1 Pairs of Lines and Angles Page(125-130), Lesson 3.1 Pairs of Lines and Angles Page(126-128), Exercise 3.1 Pairs of Lines and Angles Page(129-130), 3.2 Parallel Lines and Transversals Page(131-136), Lesson 3.2 Parallel Lines and Transversals Page(132-134), Exercise 3.2 Parallel Lines and Transversals Page(135-136), 3.3 Proofs with Parallel Lines Page(137-144), Lesson 3.3 Proofs with Parallel Lines Page(138-141), Exercise 3.3 Proofs with Parallel Lines Page(142-144), 3.1 3.3 Study Skills: Analyzing Your Errors Page 145, 3.4 Proofs with Perpendicular Lines Page(147-154), Lesson 3.4 Proofs with Perpendicular Lines Page(148-151), Exercise 3.4 Proofs with Perpendicular Lines Page(152-154), 3.5 Equations of Parallel and Perpendicular Lines Page(155-162), Lesson 3.5 Equations of Parallel and Perpendicular Lines Page(156-159), Exercise 3.5 Equations of Parallel and Perpendicular Lines Page(160-162), 3.4 3.5 Performance Task: Navajo Rugs Page 163, Parallel and Perpendicular Lines Chapter Review Page(164-166), Parallel and Perpendicular Lines Test Page 167, Parallel and Perpendicular Lines Cumulative Assessment Page(168-169), Big Ideas Math Answers Grade 2 Chapter 15 Identify and Partition Shapes, Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors, enVision Math Common Core Grade 7 Answer Key | enVision Math Common Core 7th Grade Answers, Envision Math Common Core Grade 5 Answer Key | Envision Math Common Core 5th Grade Answers, Envision Math Common Core Grade 4 Answer Key | Envision Math Common Core 4th Grade Answers, Envision Math Common Core Grade 3 Answer Key | Envision Math Common Core 3rd Grade Answers, enVision Math Common Core Grade 2 Answer Key | enVision Math Common Core 2nd Grade Answers, enVision Math Common Core Grade 1 Answer Key | enVision Math Common Core 1st Grade Answers, enVision Math Common Core Grade 8 Answer Key | enVision Math Common Core 8th Grade Answers, enVision Math Common Core Kindergarten Answer Key | enVision Math Common Core Grade K Answers, enVision Math Answer Key for Class 8, 7, 6, 5, 4, 3, 2, 1, and K | enVisionmath 2.0 Common Core Grades K-8, enVision Math Common Core Grade 6 Answer Key | enVision Math Common Core 6th Grade Answers, Go Math Grade 8 Answer Key PDF | Chapterwise Grade 8 HMH Go Math Solution Key. Draw a diagram of at least two lines cut by at least one transversal. The given points are: x || y is proved by the Lines parallel to Transversal Theorem. Answer: We can conclude that 1 2. It is given that in spherical geometry, all points are points on the surface of a sphere. Is there enough information in the diagram to conclude that m || n? 1 = 180 138 Question 1. We can conclude that the value of x is: 90, Question 8. y = 3x + 2 Which lines(s) or plane(s) contain point G and appear to fit the description? The slope of perpendicular lines is: -1 These Parallel and Perpendicular Lines Worksheets are great for practicing identifying perpendicular lines from pictures.
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