parallel and perpendicular lines answer key

c. Draw \(\overline{C D}\). Measure the lengths of the midpoint of AB i.e., AD and DB. Answer: MAKING AN ARGUMENT y = -x + 8 1 = 123 From Exploration 1, 1 = 4 a is perpendicular to d and b is perpendicular to c Substitute (2, -3) in the above equation Slope (m) = \(\frac{y2 y1}{x2 x1}\) Proof of Converse of Corresponding Angles Theorem: = \(\frac{11}{9}\) We can conclude that, First, solve for \(y\) and express the line in slope-intercept form. We know that, Hence, 20 = 3x 2x Explain our reasoning. The lines that do not intersect to each other and are coplanar are called Parallel lines . Answer: Question 28. Substitute the given point in eq. Identifying Perpendicular Lines Worksheets = 2 (2) We can conclude that the parallel lines are: The slopes are equal fot the parallel lines m = -2 Question 15. Whereas, if the slopes of two given lines are negative reciprocals of each other, they are considered to be perpendicular lines. Answer: The diagram of the control bar of the kite shows the angles formed between the Control bar and the kite lines. y y1 = m (x x1) Now, line(s) parallel to . Hence, Slope of AB = \(\frac{2}{3}\) REASONING Can you find the distance from a line to a plane? These worksheets will produce 10 problems per page. 5 = \(\frac{1}{2}\) (-6) + c So, PROVING A THEOREM 5 + 4 = b A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. From the given figure, Tell which theorem you use in each case. So, w y and z x y = -2x + c We can observe that 141 and 39 are the consecutive interior angles Answer: The product of the slopes of the perpendicular lines is equal to -1 Which angle pairs must be congruent for the lines to be parallel? It is given that your classmate claims that no two nonvertical parallel lines can have the same y-intercept a is both perpendicular to b and c and b is parallel to c, Question 20. b is the y-intercept One answer is the line that is parallel to the reference line and passing through a given point. m1 and m5 So, So, Compare the given points with could you still prove the theorem? We can conclue that 2 and 11 Step 6: We can conclude that the line that is parallel to the given line equation is: m2 and m4 We can conclude that quadrilateral JKLM is a square. y = \(\frac{2}{3}\)x + 1 Possible answer: 2 and 7 c. Possible answer: 1 and 8 d. Possible answer: 2 and 3 3. We can conclude that the lines that intersect \(\overline{N Q}\) are: \(\overline{N K}\), \(\overline{N M}\), and \(\overline{Q P}\), c. Which lines are skew to ? So, The given point is: A (3, 4) m2 = -2 m1 m2 = \(\frac{1}{2}\) We can conclude that the values of x and y are: 9 and 14 respectively. y = 2x + c2, b. Answer: y = 7 By comparing the slopes, Answer: We know that, Hence, from the given figure, The given point is: (-1, -9) The given figure is: A(- 2, 1), B(4, 5); 3 to 7 x = 29.8 and y = 132, Question 7. If two lines are parallel to the same line, then they are parallel to each other y = -x + 1. P = (4 + (4 / 5) 7, 1 + (4 / 5) 1) ABSTRACT REASONING So, Answer: y = \(\frac{1}{2}\)x 2 Prove: t l. PROOF Compare the given points with d = \(\sqrt{(x2 x1) + (y2 y1)}\) Parallel to \(x=2\) and passing through (7, 3)\). From the given figure, From the given figure, Substitute (-1, -9) in the above equation = \(\frac{3 2}{-2 2}\) c = 7 d. AB||CD // Converse of the Corresponding Angles Theorem. y = 2x + c = \(\frac{4}{-18}\) 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! y = \(\frac{1}{2}\) Hence, from the above, We know that, The given equation is: We know that, The parallel lines have the same slopes The given figure is: The coordinates of line b are: (2, 3), and (0, -1) Parallel lines are those that never intersect and are always the same distance apart. We know that, We can conclude that the alternate exterior angles are: 1 and 8; 7 and 2. a.) Each step is parallel to the step immediately above it. 3: write the equation of a line through a given coordinate point . \(\frac{5}{2}\)x = 5 So, We know that, Use a graphing calculator to graph the pair of lines. The slope is: 3 The lines that do not intersect or not parallel and non-coplanar are called Skew lines So, b. The given figure is: Now, Lines Perpendicular to a Transversal Theorem (Thm. 5x = 132 + 17 Compare the given points with (x1, y1), and (x2, y2) So, m = 2 2 = 133 P(- 7, 0), Q(1, 8) m2 and m3 a. y = -x + c So, The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We can conclude that the Corresponding Angles Converse is the converse of the Corresponding Angles Theorem, Question 3. The slopes are equal fot the parallel lines Answer: a. y = -2x + \(\frac{9}{2}\) (2) Find the equation of the line passing through \((8, 2)\) and perpendicular to \(6x+3y=1\). We know that, 3y = x + 475 The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal, the resulting corresponding anglesare congruent Question 13. The equation of a line is x + 2y = 10. Determine the slope of parallel lines and perpendicular lines. The are outside lines m and n, on . Now, y = -2x + 8 During a game of pool. Hence, Draw a line segment CD by joining the arcs above and below AB P(- 5, 5), Q(3, 3) y = \(\frac{7}{2}\) 3 So, We can conclude that x and y are parallel lines, Question 14. A (x1, y1), and B (x2, y2) Step 2: Substitute the slope you found and the given point into the point-slope form of an equation for a line. \(\begin{array}{cc} {\color{Cerulean}{Point}}&{\color{Cerulean}{Slope}}\\{(-1,-5)}&{m_{\perp}=4}\end{array}\). The Converse of the Alternate Interior Angles Theorem states that if two lines are cut by a transversal and the alternate interior anglesare congruent, then the lines are parallel c. m5=m1 // (1), (2), transitive property of equality x = 107 : n; same-side int. Label the ends of the crease as A and B. Using X and Y as centers and an appropriate radius, draw arcs that intersect. Proof of Alternate exterior angles Theorem: What is m1? Hence, from the above, x = \(\frac{-6}{2}\) So, Embedded mathematical practices, exercises provided make it easy for you to understand the concepts quite quickly. So, ABSTRACT REASONING HOW DO YOU SEE IT? d = | c1 c2 | The product of the slopes of the perpendicular lines is equal to -1 -1 = \(\frac{-2}{7 k}\) \(\frac{8-(-3)}{7-(-2)}\) So, DIFFERENT WORDS, SAME QUESTION We know that, x = 4 We know that, From the given figure, If you need more of a review on how to use this form, feel free to go to Tutorial 26: Equations of Lines The given figure is: y = \(\frac{24}{2}\) 1 = 2 = 123, Question 11. Hence, from the above figure, m2 = 2 c = 3 The map shows part of Denser, Colorado, Use the markings on the map. -x + 2y = 12 (11y + 19) = 96 The given point is: (-8, -5) ax + by + c = 0 So, Hence, from the above, To find the value of b, So, k = 5 = \(\sqrt{31.36 + 7.84}\) ERROR ANALYSIS To find the value of b, The given figure is: Answer: We know that, Given a b The points are: (-9, -3), (-3, -9) The plane parallel to plane ADE is: Plane GCB. Answer: The equation for another line is: Now, We can say that any intersecting line do intersect at 1 point Find the distance from point A to the given line. So, So, Answer: Answer: Question 4. Perpendicular to \(y=x\) and passing through \((7, 13)\). By comparing the given pair of lines with Assume L1 is not parallel to L2 Difference Between Parallel and Perpendicular Lines, Equations of Parallel and Perpendicular Lines, Parallel and Perpendicular Lines Worksheets. 6 + 4 = 180, Question 9. Hence, from the above, So, Maintaining Mathematical Proficiency Perpendicular to \(y=2\) and passing through \((1, 5)\). Parallel and Perpendicular Lines Perpendicular Lines Two nonvertical lines are perpendicular if their slopes are opposite reciprocals of each other. Answer: y = 3x 5 The product of the slopes of perpendicular lines is equal to -1 c = 5 \(\frac{1}{2}\) We can conclude that in order to jump the shortest distance, you have to jump to point C from point A. The given point is: A (-9, -3) Justify your answer. The equation that is perpendicular to the given line equation is: Find equations of parallel and perpendicular lines. MATHEMATICAL CONNECTIONS Lines l and m are parallel. Select the orange Get Form button to start editing. 2x y = 4 = \(\sqrt{2500 + 62,500}\) For a horizontal line, We can conclude that We can conclude that the distance of the gazebo from the nature trail is: 0.66 feet. In Example 2, y = \(\frac{1}{2}\)x 3, b. CONSTRUCTION X (-3, 3), Y (3, 1) (\(\frac{1}{2}\)) (m2) = -1 180 = x + x Use a graphing calculator to verify your answers. So, The slope of the parallel line that passes through (1, 5) is: 3 Vertical Angles Theoremstates thatvertical angles,anglesthat are opposite each other and formed by two intersecting straight lines, are congruent m1 = \(\frac{1}{2}\), b1 = 1 Does either argument use correct reasoning? So, Question 4. Answer: y = 3x 6, Question 20. The points are: (-\(\frac{1}{4}\), 5), (-1, \(\frac{13}{2}\)) Answer: Now, So, According to the Vertical Angles Theorem, the vertical angles are congruent The two lines are Coincident when they lie on each other and are coplanar From Example 1, From the given figure, The coordinates of line 1 are: (-3, 1), (-7, -2) AO = OB Question 3. x = 0 Answer: Question 31. Parallel to \(y=\frac{1}{2}x+2\) and passing through \((6, 1)\). We can conclude that So, So, m = \(\frac{3}{-1.5}\) m2 = \(\frac{1}{2}\) Algebra 1 Writing Equations of Parallel and Perpendicular Lines 1) through: (2, 2), parallel to y = x + 4. It is given that The intersection point of y = 2x is: (2, 4) The painted line segments that brain the path of a crosswalk are usually perpendicular to the crosswalk. y = \(\frac{1}{2}\)x + c Answer: Question 10. = 0 2 = 122 The given points are: 7x = 84 By using the parallel lines property, Alternate Exterior Angles Theorem: Hence,f rom the above, The line parallel to \(\overline{Q R}\) is: \(\overline {L M}\), Question 3. plane(s) parallel to plane LMQ We can observe that 48 and y are the consecutive interior angles and y and (5x 17) are the corresponding angles 3. c.) Parallel lines intersect each other at 90. 1 7 The coordinates of the line of the second equation are: (1, 0), and (0, -2) This contradiction means our assumption (L1 is not parallel to L2) is false, and so L1 must be parallel to L2. 2m2 = -1 We know that, 1 + 138 = 180 The Converse of the consecutive Interior angles Theorem states that if the consecutive interior angles on the same side of a transversal line intersecting two lines are supplementary, then the two lines are parallel. Parallel lines The consecutive interior angles are: 2 and 5; 3 and 8. -4 = \(\frac{1}{2}\) (2) + b From the given graph, We know that, We can conclude that the given pair of lines are parallel lines. Hence, from the above figure, 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. Perpendicular to \(x=\frac{1}{5}\) and passing through \((5, 3)\). Question 35. The "Parallel and Perpendicular Lines Worksheet (+Answer Key)" can help you learn about the different properties and theorems of parallel and perpendicular lines. We know that, Observe the following figure and the properties of parallel and perpendicular lines to identify them and differentiate between them. We have to find the distance between X and Y i.e., XY Hence, If the corresponding angles are congruent, then the lines cut by a transversal are parallel From the given figure, The slope of the line that is aprallle to the given line equation is: So, Where, Answer: We know that, By comparing the given equation with The given figure is: Answer: Question 20. Explain. Answer: Question 28. It is given that m || n y = \(\frac{1}{2}\)x + c Hence, from the above, y = -2x 1 (2) alternate interior y 500 = -3x + 150 So, A (x1, y1), B (x2, y2) From the given figure, k = -2 + 7 Draw an arc with center A on each side of AB. We have seen that the graph of a line is completely determined by two points or one point and its slope. forming a straight line. They are always equidistant from each other. y = \(\frac{1}{2}\)x + 5 Hence, from the above, The equation of a line is: Question 2. So, Parallel to \(x+y=4\) and passing through \((9, 7)\). x 2y = 2 Explain your reasoning. We can conclude that 1 2. So, Proof: Answer: Question 26. Answer: Alternate Interior angles theorem: CONSTRUCTION A(0, 3), y = \(\frac{1}{2}\)x 6 1 = 2 = 133 and 3 = 47. Question 12. The slope of the given line is: m = \(\frac{1}{4}\) 10x + 2y = 12 We can conclude that 4 and 5 are the Vertical angles. PROVING A THEOREM Now, Given: k || l, t k In Exercises 7-10. find the value of x. Now, WHICH ONE did DOESNT BELONG? The given statement is: We can conclude that the value of the given expression is: 2, Question 36. The slopes are equal fot the parallel lines These worksheets will produce 6 problems per page. In Exercises 3-6, find m1 and m2. 4 5, b. A(- 3, 7), y = \(\frac{1}{3}\)x 2 The angles that have the opposite corners are called Vertical angles So, d = \(\sqrt{(x2 x1) + (y2 y1)}\) So, From the Consecutive Exterior angles Converse, 4 6 = c The given pair of lines are: The line that is perpendicular to the given equation is: y = -7x 2. The equation for another perpendicular line is: The given figure is: Answer: 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. c = 4 So, Hence, Answer: Question 20. m2 = \(\frac{1}{2}\), b2 = 1 Answer: Question 28. The representation of the given pair of lines in the coordinate plane is: We know that, From the given figure, = 44,800 square feet Find m2. Hence, from the above, In this case, the negative reciprocal of 1/5 is -5. We can observe that the given angles are the corresponding angles c = 8 \(\frac{3}{5}\) Perpendicular lines are denoted by the symbol . The slope of perpendicular lines is: -1 Then by the Transitive Property of Congruence (Theorem 2.2), _______ . 2x + y = 180 18 EG = \(\sqrt{50}\) Parallel & perpendicular lines from equation Writing equations of perpendicular lines Writing equations of perpendicular lines (example 2) Write equations of parallel & perpendicular lines Proof: parallel lines have the same slope Proof: perpendicular lines have opposite reciprocal slopes Analytic geometry FAQ Math > High school geometry > By comparing eq. By using the Consecutive Interior Angles Theorem, = (\(\frac{8 + 0}{2}\), \(\frac{-7 + 1}{2}\)) 3 = 180 133 So, Lines Perpendicular to a Transversal Theorem (Theorem 3.12): In a plane. We can conclude that a line equation that is perpendicular to the given line equation is: x = 12 We can conclude that the distance from point X to \(\overline{W Z}\) is: 6.32, Find XZ y = 3x + 9 Compare the given equation with y = mx + b The representation of the Converse of Corresponding Angles Theorem is: b. Alternate Interior Angles Theorem (Theorem 3.2): If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Hence, from the given figure, So, Answer: Once the equation is already in the slope intercept form, you can immediately identify the slope. 6 (2y) 6(3) = 180 42 x + 2y = 10 So, With Cuemath, you will learn visually and be surprised by the outcomes. We can conclude that m || n, Question 15. Is your classmate correct? For a square, Vertical Angles are the anglesopposite each other when two lines cross