For the proofs of the theorems that you found to be true, refer to Exploration 1. MODELING WITH MATHEMATICS We know that, Hence, Hence, Explain why the Corresponding Angles Converse is the converse of the Corresponding Angles Theorem (Theorem 3.1). Compare the given points with Find the slope of a line perpendicular to each given line. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. y = \(\frac{3}{2}\) + 4 and -3x + 2y = -1 lines intersect at 90. So, Now, b = -7 The equation that is perpendicular to the given line equation is: The line that is perpendicular to y=n is: We can conclude that 1 and 5 are the adjacent angles, Question 4. Prove the statement: If two lines are horizontal, then they are parallel. Hence, from the above, To find the value of b, Question: What is the difference between perpendicular and parallel? HOW DO YOU SEE IT? 3y + 4x = 16 (D) Consecutive Interior Angles Converse (Thm 3.8) The alternate interior angles are: 3 and 5; 2 and 8, c. alternate exterior angles So, Question 39. So, Compare the effectiveness of the argument in Exercise 24 on page 153 with the argument You can find the distance between any two parallel lines What flaw(s) exist in the argument(s)? 140 21 32 = 6x Answer/Step-by-step Explanation: To determine if segment AB and CD are parallel, perpendicular, or neither, calculate the slope of each. Perpendicular lines are those that always intersect each other at right angles. From the given figure, We can conclude that the slope of the given line is: \(\frac{-3}{4}\), Question 2. Step 4: 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. So, We can conclude that in order to jump the shortest distance, you have to jump to point C from point A. P(0, 1), y = 2x + 3 Enter a statement or reason in each blank to complete the two-column proof. In Exercises 43 and 44, find a value for k based on the given description. The corresponding angles are: and 5; 4 and 8, b. alternate interior angles Hence, The equation of a line is: 2 = 2 (-5) + c b) Perpendicular to the given line: Hence, from the above, so they cannot be on the same plane. Answer: So, Statement of consecutive Interior angles theorem: The coordinates of the quadrilateral QRST is: To use the "Parallel and Perpendicular Lines Worksheet (with Answer Key)" use the clues in identifying whether two lines are parallel or perpendicular with each other using the slope. x = 60 Answer: Hence, from the given figure, Now, So, y = \(\frac{1}{3}\)x 2 -(1) y = \(\frac{1}{2}\)x 6 All the angles are right angles. Compare the given equations with b is the y-intercept Hence, from the above, Compare the given points with y = -x + c justify your answer. From the given figure, From the given figure, Answer: -2 = \(\frac{1}{2}\) (2) + c 4 and 5 are adjacent angles m = \(\frac{3}{-1.5}\) Now, We can say that they are also parallel 1 = 32 Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 3m2 = -1 So, m = -2 From the above figure, y = -2x + c x = 9 Find the equation of the line passing through \((\frac{7}{2}, 1)\) and parallel to \(2x+14y=7\). The given diagram is: = \(\frac{2}{-6}\) = \(\frac{8 0}{1 + 7}\) m = \(\frac{5}{3}\) The y-intercept is: 9. x = \(\frac{40}{8}\) A(- \(\frac{1}{4}\), 5), x + 2y = 14 Hence, from the above figure, From the given figure, 1 = 2 (By using the Vertical Angles theorem) x 2y = 2 x = \(\frac{-6}{2}\) _____ lines are always equidistant from each other. We can conclude that 2 and 7 are the Vertical angles, Question 5. The equation of the line that is parallel to the given line equation is: Equations of vertical lines look like \(x=k\). = 44,800 square feet Question 11. 2m2 = -1 Question 17. If the pairs of alternate interior angles are, Answer: The given table is: Your classmate claims that no two nonvertical parallel lines can have the same y-intercept. It is given that m || n The midpoint of PQ = (\(\frac{x1 + x2}{2}\), \(\frac{y1 + y2}{2}\)) 13) y = -5x - 2 14) y = -1 G P2l0E1Q6O GKouHttad wSwoXfptiwlaer`eU yLELgCH.r C DAYlblQ wrMiWgdhstTsF wr_eNsVetrnv[eDd\.x B kMYa`dCeL nwHirtmhI KILnqfSisnBiRt`ep IGAeJokmEeCtPr[yY. Identify two pairs of parallel lines so that each pair is in a different plane. Answer: Question 50. Use the numbers and symbols to create the equation of a line in slope-intercept form It is given that 1 = 105 We know that, Exercise \(\PageIndex{5}\) Equations in Point-Slope Form. We can observe that The given figure is: The given coordinates are: A (-2, 1), and B (4, 5) We know that, V = (-2, 3) Now, Section 6.3 Equations in Parallel/Perpendicular Form. So, Question 23. Answer: Perpendicular to \(\frac{1}{2}x\frac{1}{3}y=1\) and passing through \((10, 3)\). We can observe that the given angles are corresponding angles Hence, from the above, We can conclude that The equation of the line that is perpendicular to the given line equation is: We can conclude that the distance from line l to point X is: 6.32. For the Converse of the alternate exterior angles Theorem, We can conclude that we can use Perpendicular Postulate to show that \(\overline{A C}\) is not perpendicular to \(\overline{B F}\), Question 3. y = mx + b y = -2x x + 2y = 2 Any fraction that contains 0 in the denominator has its value undefined BCG and __________ are consecutive interior angles. From the given figure, So, So, Find the slope of a line perpendicular to each given line. So, = \(\sqrt{30.25 + 2.25}\) For parallel lines, From the given figure, MAKING AN ARGUMENT Hence, from the coordinate plane, To find the y-intercept of the equation that is parallel to the given equation, substitute the given point and find the value of c Answer: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent The Alternate Interior angles are congruent 3.4) From the given figure, We know that, The angles that have the opposite corners are called Vertical angles The equation that is perpendicular to the given line equation is: Compare the given equation with -2 = \(\frac{1}{3}\) (-2) + c Explain your reasoning. \(\frac{1}{3}\)x + 3x = -2 + 2 corresponding Converse: The parallel lines have the same slope Hence, from the above, We get Question 12. Answer: We can conclude that the tallest bar is parallel to the shortest bar, b. The given lines are: 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. If the line cut by a transversal is parallel, then the corresponding angles are congruent answer choices Parallel Perpendicular Neither Tags: MGSE9-12.G.GPE.5 Question 7 300 seconds y 500 = -3x + 150 The equation of the line that is parallel to the line that represents the train tracks is: Compare the given coordinates with We can conclude that = \(\frac{-4}{-2}\) From the given figure, 2m2 = -1 So, how many right angles are formed by two perpendicular lines? From the given figure, Explain your reasoning. EG = \(\sqrt{50}\) Write a conjecture about the resulting diagram. Question 1. Now, Enter your answer in the box y=2/5x2 We have to keep the lengths of the length of the rectangles the same and the widths of the rectangle also the same, Question 3. Then, according to the parallel line axiom, there is a different line than L2 that passes through the intersection point of L2 and L3 (point A in the drawing), which is parallel to L1. The Alternate Interior Angles Theorem states that, when two parallel lines are cut by a transversal, the resultingalternate interior anglesare congruent y = mx + b You can prove that4and6are congruent using the same method. By using the Consecutive Interior angles Converse, Select all that apply. y = 3x 5 The two lines are Intersecting when they intersect each other and are coplanar Grade: Date: Parallel and Perpendicular Lines. Now, Graph the equations of the lines to check that they are parallel. The equation that is perpendicular to the given line equation is: Now, We know that, Compare the given coordinates with The given point is: A (-6, 5) Hence, from the above, Given: k || l, t k Slope of QR = \(\frac{4 6}{6 2}\) So, x = \(\frac{18}{2}\) y = 145 Which pair of angle measures does not belong with the other three? Let the given points are: Identifying Parallel, Perpendicular, and Intersecting Lines Worksheets No, the third line does not necessarily be a transversal, Explanation: c = -2 According to the Perpendicular Transversal Theorem, So, 2 and 4 are the alternate interior angles y = \(\frac{1}{2}\)x + c2, Question 3. We know that, = (\(\frac{8}{2}\), \(\frac{-6}{2}\)) 5 = c The rungs are not intersecting at any point i.e., they have different points -2 . Name two pairs of supplementary angles when \(\overline{A B}\) and \(\overline{D C}\) are parallel. This can be expressed mathematically as m1 m2 = -1, where m1 and m2 are the slopes of two lines that are perpendicular. So, We know that, From the given figure, So, 2 and 3 are the consecutive interior angles The diagram shows lines formed on a tennis court. 1 = 2 = 3 = 4 = 5 = 6 = 7 = 53.7, Work with a partner. Examples of perpendicular lines: the letter L, the joining walls of a room. So, Which line(s) or plane(s) contain point B and appear to fit the description? The lines that do not have any intersection points are called Parallel lines y = mx + c c = 1 -x + 2y = 14 We know that, Answer: Now, 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 > (2) 5x = 132 + 17 Vertical Angles Theoremstates thatvertical angles,anglesthat are opposite each other and formed by two intersecting straight lines, are congruent The width of the field is: 140 feet Answer: \(\overline{A B}\) and \(\overline{G H}\), b. a pair of perpendicular lines Hence, from the above, According to the Consecutive Exterior angles Theorem, Then by the Transitive Property of Congruence (Theorem 2.2), _______ . The intersection point of y = 2x is: (2, 4) Decide whether there is enough information to prove that m || n. If so, state the theorem you would use. Perpendicular to \(5x+y=1\) and passing through \((4, 0)\). Can you find the distance from a line to a plane? Answer: You and your mom visit the shopping mall while your dad and your sister visit the aquarium. 1 5 The slope of line a (m) = \(\frac{y2 y1}{x2 x1}\) = 4 y = \(\frac{1}{2}\)x + c y = \(\frac{3}{2}\)x + c Given: m5 + m4 = 180 = \(\frac{1}{-4}\) The given point is:A (6, -1) When we observe the ladder, So, Draw \(\overline{P Z}\), CONSTRUCTION x1 = x2 = x3 . x = 20 Hence, from the above, The given point is: (-1, -9) -4 = \(\frac{1}{2}\) (2) + b Solution: We need to know the properties of parallel and perpendicular lines to identify them. We know that, Answer: We were asked to find the equation of a line parallel to another line passing through a certain point. Hence, Draw a line segment of any length and name that line segment as AB The given point is: A (3, -1) The product of the slopes of the perpendicular lines is equal to -1 The distance from point C to AB is the distance between point C and A i.e., AC Label its intersection with \(\overline{A B}\) as O. Answer: Question 40. Substitute the given point in eq. m1 = 76 Quick Link for All Parallel and Perpendicular Lines Worksheets, Detailed Description for All Parallel and Perpendicular Lines Worksheets. c = -3 From the figure, Write the equation of the line that is perpendicular to the graph of 53x y = , and MODELING WITH MATHEMATICS The equation that is perpendicular to the given equation is: Question 15. Answer: 1 = 40 and 2 = 140. The slope of second line (m2) = 1 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 80, Question 1. So, The postulates and theorems in this book represent Euclidean geometry. Hence, from the above, Consider the following two lines: Consider their corresponding graphs: Figure 4.6.1 Substitute (0, -2) in the above equation Solve eq. Answer: Question 32. line(s) parallel to m1m2 = -1 The two pairs of parallel lines so that each pair is in a different plane are: q and p; k and m, b. y = 3x 5 So, Name the line(s) through point F that appear skew to . y = -9 To find the value of c, substitute (1, 5) in the above equation Answer: Prove m||n The given figure is: Solving the concepts from the Big Ideas Math Book Geometry Ch 3 Parallel and Perpendicular Lines Answers on a regular basis boosts the problem-solving ability in you. m2 and m3 To prove: l || k. Question 4. \(\begin{aligned} y-y_{1}&=m(x-x_{1}) \\ y-1&=-\frac{1}{7}\left(x-\frac{7}{2} \right) \\ y-1&=-\frac{1}{7}x+\frac{1}{2} \\ y-1\color{Cerulean}{+1}&=-\frac{1}{7}x+\frac{1}{2}\color{Cerulean}{+1} \\ y&=-\frac{1}{7}x+\frac{1}{2}+\color{Cerulean}{\frac{2}{2}} \\ y&=-\frac{1}{7}x+\frac{3}{2} \end{aligned}\). Slope of AB = \(\frac{5}{8}\) Given Slope of a Line Find Slopes for Parallel and Perpendicular Lines Worksheets XY = 4.60 Answer: Question 44. ax + by + c = 0 Answer: The given equation is: We can conclude that the distance from point E to \(\overline{F H}\) is: 7.07. We know that, So, = \(\frac{-2}{9}\) We want to prove L1 and L2 are parallel and we will prove this by using Proof of Contradiction The coordinates of P are (22.4, 1.8), Question 2. 9. 2 = 180 3 1 and 3 are the vertical angles From Exploration 2, y = -x + 4 -(1) Answer: Hence, from the above, -4 = 1 + b = \(\frac{3 2}{-2 2}\) We can observe that, We can conclude that the equation of the line that is perpendicular bisector is: Then explain how your diagram would need to change in order to prove that lines are parallel. Perpendicular to \(y=\frac{1}{3}x+2\) and passing through \((4, 3)\). 2x = 180 We know that, From the given figure, -1 = -1 + c From the given figure, Draw an arc with center A on each side of AB. Slope of ST = \(\frac{1}{2}\), Slope of TQ = \(\frac{3 6}{1 2}\) = | 4 + \(\frac{1}{2}\) | We can conclude that c2= \(\frac{1}{2}\) So, \(\overline{C D}\) and \(\overline{A E}\) are Skew lines because they are not intersecting and are non coplanar Answer: We know that, The lines that do not intersect and are not parallel and are not coplanar are Skew lines a. We know that, m1m2 = -1 Then use a compass and straightedge to construct the perpendicular bisector of \(\overline{A B}\), Question 10. Answer: Question 40. 11y = 96 19 Assume L1 is not parallel to L2 Perpendicular lines are those lines that always intersect each other at right angles. The lines perpendicular to \(\overline{Q R}\) are: \(\overline{R M}\) and \(\overline{Q L}\), Question 2. Suppose point P divides the directed line segment XY So that the ratio 0f XP to PY is 3 to 5. We can conclude that if you use the third statement before the second statement, you could still prove the theorem, Question 4. What shape is formed by the intersections of the four lines? From the given figure, we know that, y y1 = m (x x1) M = (150, 250), b. A(3, 6) Answer: Parallel Lines - Lines that move in their specific direction without ever intersecting or meeting each other at a point are known as the parallel lines. m2 = \(\frac{1}{2}\) Hence, from the above, x = \(\frac{153}{17}\) Answer the questions related to the road map. So, y = -2x 2, f. We get Slope (m) = \(\frac{y2 y1}{x2 x1}\) ERROR ANALYSIS The given figure is: 1. Explain your reasoning. NAME _____ DATE _____ PERIOD _____ Chapter 4 26 Glencoe Algebra 1 4-4 Skills Practice Parallel and Perpendicular Lines -5 = \(\frac{1}{4}\) (-8) + b y = x \(\frac{28}{5}\) m2 = -1 The coordinates of P are (4, 4.5). The slopes of perpendicular lines are undefined and 0 respectively We can conclude that Answer: From the given figure, m = -7 Let the congruent angle be P So, We can observe that Given: 1 2 d = \(\frac{4}{5}\) In Exercise 31 on page 161, from the coordinate plane, m = 2 Answer: Question 18. Horizontal and vertical lines are perpendicular to each other. For a square, 2 = 180 58 y y1 = m (x x1) (x1, y1), (x2, y2) Now, i.e., (\(\frac{1}{2}\)) (m2) = -1 x = c By comparing the given pair of lines with The lines that are a straight angle with the given line and are coplanar is called Perpendicular lines So, The angles that are opposite to each other when two lines cross are called Vertical angles Now, The given figure is: We can observe that the given lines are perpendicular lines We can conclude that the distance from point A to the given line is: 9.48, Question 6. In other words, if \(m=\frac{a}{b}\), then \(m_{}=\frac{b}{a}\). We know that, = 3, The slope of line d (m) = \(\frac{y2 y1}{x2 x1}\) y = -3x + 19, Question 5. Answer: 0 = \(\frac{5}{3}\) ( -8) + c We know that, So, Tell which theorem you use in each case. AP : PB = 3 : 7 (6, 22); y523 x1 4 13. We can conclude that Hence, 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 \(m=0=\frac{0}{1}\). We have to divide AB into 10 parts 3 = 76 and 4 = 104 Answer: The given equation is: Your school has a $1,50,000 budget. Answer: Answer: The equation that is perpendicular to the given equation is: Answer: We know that, Eq. Answer: Answer: The parallel line equation that is parallel to the given equation is: For example, if the equations of two lines are given as, y = -3x + 6 and y = -3x - 4, we can see that the slope of both the lines is the same (-3).