parallel and perpendicular lines answer key

So, c is the y-intercept 3 + 4 = c We know that, Find m2 and m3. View Notes - 4.5 Equations of Parallel and Perpendicular Lines.pdf from BIO 187 at Beach High School. Hence, Answer: Question 14. A(1, 6), B(- 2, 3); 5 to 1 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 coordinates of P are (3.9, 7.6), Question 3. The midpoint of PQ = ($\frac{x1 + x2}{2}$, $\frac{y1 + y2}{2}$) According to the Alternate Exterior angles Theorem, If so. Slope of TQ = $\frac{-3}{-1}$ We know that, y = $\frac{13}{2}$ Hence, from the above, We know that, Question 12. X (3, 3), Y (2, -1.5) a. We can conclude that we can not find the distance between any two parallel lines if a point and a line is given to find the distance, Question 2. The given equation is: FCA and __________ are alternate exterior angles. So, b. Hence, from the above, If two parallel lines are cut by a transversal, then the pairs of Corresponding angles are congruent. Substitute the given point in eq. Answer: Question 6. From the given figure, The coordinates of the line of the first equation are: (-1.5, 0), and (0, 3) The plane containing the floor of the treehouse is parallel to the ground. It is given that your friend claims that because you can find the distance from a point to a line, you should be able to find the distance between any two lines 3 = 2 (-2) + x (- 1, 5); m = 4 In spherical geometry, all points are points on the surface of a sphere. To find an equation of a line, first use the given information to determine the slope. We know that, y = 3x + 2 Answer: We know that, Answer: The given equation is: If you use the diagram below to prove the Alternate Exterior Angles Converse. Hence, from the above, So, Hence, from the above, 1 = 60 m is the slope We know that, Verticle angle theorem: Perpendicular to $y=2$ and passing through $(1, 5)$. X (-3, 3), Y (3, 1) m || n is true only when x and 73 are the consecutive interior angles according to the Converse of Consecutive Interior angles Theorem 2x y = 18 So, b. Click here for More Geometry Worksheets Answer: We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. m = 2 y= 2x 3 y = -2x + $\frac{9}{2}$ (2) $m_{}=9$ and $m_{}=\frac{1}{9}$, 13. c.) Parallel lines intersect each other at 90. Question 27. The opposite sides are parallel and the intersecting lines are perpendicular. y = 4x + b (1) 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. When we compare the given equation with the obtained equation, The pair of lines that are different from the given pair of lines in Exploration 2 are: Question 39. By comparing the given pair of lines with We know that, The given point is: A (2, -1) From the given figure, So, Hence, from the above, Since you are given a point and the slope, use the point-slope form of a line to determine the equation. We can observe that there are a total of 5 lines. Find m2. For a square, So, we can conclude that the converse we obtained from the given statement is false, c. Alternate Exterior Angles Theorem (Theorem 3.3): If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Answer: The line that is perpendicular to y=n is: Graph the equations of the lines to check that they are perpendicular. If the angle measure of the angles is a supplementary angle, then the lines cut by a transversal are parallel Explain your reasoning. We have to divide AB into 5 parts lines intersect at 90. Hence, from the above, Identify all pairs of angles of the given type. The coordinates of a quadrilateral are: The lines skew to $\overline{Q R}$ are: $\overline{J N}$, $\overline{J K}$, $\overline{K L}$, and $\overline{L M}$, Question 4. Hence, from the above, We can conclude that The product of the slopes of perpendicular lines is equal to -1 We can conclude that In Exploration 2. find more pairs of lines that are different from those given. Explain Your reasoning. X (-3, 3), Y (3, 1) The slopes of parallel lines, on the other hand, are exactly equal. 2y + 4x = 180 Converse: Describe the point that divides the directed line segment YX so that the ratio of YP Lo PX is 5 to 3. Answer: In Exercises 27-30. find the midpoint of $\overline{P Q}$. x = $\frac{40}{8}$ Now, So, x + 2y = -2 ($\frac{1}{3}$) (m2) = -1 Definition of Parallel and Perpendicular Parallel lines are lines in the same plane that never intersect. Simply click on the below available and learn the respective topics in no time. The intersecting lines intersect each other and have different slopes and have the same y-intercept 1 and 8 are vertical angles 6 (2y) 6(3) = 180 42 Explain your reasoning. We know that, A(2, 0), y = 3x 5 P(- 5, 5), Q(3, 3) c = 8 $\frac{3}{5}$ So, Answer: In Exercises 15 and 16, use the diagram to write a proof of the statement. The conjecture about $\overline{A B}$ and $\overline{c D}$ is: The equation that is perpendicular to the given line equation is: = 44,800 square feet Given: 1 and 3 are supplementary Find the slope $m$ by solving for $y$. The representation of the parallel lines in the coordinate plane is: In Exercises 17 20. write an equation of the line passing through point P that is perpendicular to the given line. The slope of perpendicular lines is: -1 1 + 2 = 180 For example, the letter H, in which the vertical lines are parallel and the horizontal line is perpendicular to both the vertical lines. So, Answer: y = 4x + 9, Question 7. The slope of the parallel line is 0 and the slope of the perpendicular line is undefined. The representation of the perpendicular lines in the coordinate plane is: Question 19. then they are parallel. y = $\frac{1}{2}$x + 5 We know that, b. The line through (k, 2) and (7, 0) is perpendicular to the line y = x $\frac{28}{5}$. We know that, The equation that is perpendicular to the given line equation is: To make the top of the step where 1 is present to be parallel to the floor, the angles must be Alternate Interior angles Answer: Answer: PROVING A THEOREM Compare the given equation with 2x x = 56 2 Perpendicular lines always intersect at 90. These worksheets will produce 6 problems per page. Now, Answer: Use the diagram to find the measure of all the angles. Parallel lines are always equidistant from each other. The given figure is: The 2 pair of skew lines are: q and p; l and m, d. Prove that 1 2. Compare the given points with (x1, y1), and (x2, y2) The representation of the given point in the coordinate plane is: Question 56. ABSTRACT REASONING These worksheets will produce 6 problems per page. To find the value of c, Proof: Question 17. m1m2 = -1 3.1 Lines and Angles 3.2 Properties of Parallel Lines 3.3 Proving Lines Parallel 3.4 Parallel Lines and Triangles 3.5 Equations of Lines in the Coordinate Plane 3.6 Slopes of Parallel and Perpendicular Lines Unit 3 Review Answer: The Skew lines are the lines that are non-intersecting, non-parallel and non-coplanar c.) Parallel lines intersect each other at 90. Hence, from the above, Answer: Hence, MODELING WITH MATHEMATICS Answer: Question 36. According to the consecutive exterior angles theorem, We have to find the point of intersection x + 2y = 2 m2 = -1 y = -2x + 8 AP : PB = 3 : 7 We can observe that 141 and 39 are the consecutive interior angles Find the equation of the line perpendicular to $x3y=9$ and passing through $(\frac{1}{2}, 2)$. Two nonvertical lines in the same plane, with slopes $m_{1}$ and $m_{2}$, are perpendicular if the product of their slopes is $1: m1m2=1$. If we want to find the distance from the point to a given line, we need the perpendicular distance of a point and a line By using the Corresponding Angles Theorem, Slope of MJ = $\frac{0 0}{n 0}$ Download Parallel and Perpendicular Lines Worksheet - Mausmi Jadhav. A(- 6, 5), y = $\frac{1}{2}$x 7 Label points on the two creases. So, Exercise $\PageIndex{5}$ Equations in Point-Slope Form. So, The product of the slopes of the perpendicular lines is equal to -1 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. If a || b and b || c, then a || c The given figure is: Hence, The standard linear equation is: A(- 3, 7), y = $\frac{1}{3}$x 2 To find the distance between E and $\overline{F H}$, we need to find the distance between E and G i.e., EG = 0 y = 132 a. corresponding angles y = $\frac{1}{2}$x + c x = n We can conclude that if you use the third statement before the second statement, you could still prove the theorem, Question 4. The distance from the perpendicular to the line is given as the distance between the point and the non-perpendicular line Prove: m || n Hence, from the above, So, by the Corresponding Angles Converse, g || h. Question 5. So, = 1 Answer: Question 12. XY = $\sqrt{(x2 x1) + (y2 y1)}$ If you need more of a review on how to use this form, feel free to go to Tutorial 26: Equations of Lines We can conclude that the consecutive interior angles are: 3 and 5; 4 and 6, Question 6. We can conclude that Hence, from the above, $m_{}=10$ and $m_{}=\frac{1}{10}$, Exercise $\PageIndex{4}$ Parallel and Perpendicular Lines. Compare the above equation with y = $\frac{1}{2}$x + 6 y = $\frac{77}{11}$ -x x = -3 The lines that have the same slope and different y-intercepts are Parallel lines Substitute (-1, 6) in the above equation Answer: From the given figure, The are outside lines m and n, on . Does the school have enough money to purchase new turf for the entire field? We know that, Decide whether there is enough information to prove that m || n. If so, state the theorem you would use. y = -x -(1) = $\frac{8}{8}$ y = $\frac{1}{4}$x + b (1) Answer: Question 44. $m\cdot m_{\perp}=-\frac{5}{8}\cdot\frac{8}{5}=-\frac{40}{40}=-1\quad\color{Cerulean}{\checkmark}$. We can solve for $m_{1}$ and obtain $m_{1}=\frac{1}{m_{2}}$. The given figure is: It is given that your school has a budget of $1,50,000 but we only need $1,20,512 The given line has the slope $m=\frac{1}{7}$, and so $m_{}=\frac{1}{7}$. The representation of the parallel lines in the coordinate plane is: Question 16. y = $\frac{1}{3}$x $\frac{8}{3}$. We know that, In Exercises 13 and 14, prove the theorem. The equation that is perpendicular to the given equation is: We know that, Hence, from the above, y1 = y2 = y3 We can conclude that the values of x and y are: 9 and 14 respectively. 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. Substitute A (-$\frac{1}{4}$, 5) in the above equation to find the value of c According to the Converse of the Corresponding angles Theorem, We can observe that The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal, the resulting corresponding anglesare congruent So, In the parallel lines, Slope of line 2 = $\frac{4 6}{11 2}$ Parallel lines y = -2x + 2, Question 6. Slope of RS = 3, Slope of ST = $\frac{3 1}{1 5}$ These guidelines, with the editor will assist you with the whole process. Parallel to $y=\frac{1}{4}x5$ and passing through $(2, 1)$. = $\frac{-2}{9}$ The given equation is: The sides of the angled support are parallel. A(3, 4),y = x + 8 The coordinates of line 1 are: (-3, 1), (-7, -2) Answer: Fold the paper again so that point A coincides with point B. Crease the paper on that fold. How do you know? Hence, from the above, Answer: The given figure is: To find the value of b, c1 = 4 We know that, (1) = Eq. So, By using the Consecutive Interior Angles Theorem, PDF Name: Unit 3: Parallel & Perpendicular Lines Bell: Homework 5: Linear. Compare the given points with Eq. Substitute A (-3, 7) in the above equation to find the value of c The slopes are the same but the y-intercepts are different We know that, 6x = 140 53 According to the consecutive Interior Angles Theorem, The postulates and theorems in this book represent Euclidean geometry. We know that, We can observe that the given angles are the consecutive exterior angles What can you conclude? We can conclude that the value of the given expression is: $\frac{11}{9}$. The given equation is: Answer: To find the value of b, Hence, 0 = $\frac{1}{2}$ (4) + c Explain why or why not. 42 and (8x + 2) are the vertical angles Now, Hence, from the above, In Exploration 1, explain how you would prove any of the theorems that you found to be true. The given line that is perpendicular to the given points is: m = $\frac{-30}{15}$ Mark your diagram so that it cannot be proven that any lines are parallel. The conjecture about $\overline{A O}$ and $\overline{O B}$ is: The standard linear equation is: x = 54 We can conclude that On the other hand, when two lines intersect each other at an angle of 90, they are known as perpendicular lines. The given figure is: Answer: We know that, We can observe that the given lines are perpendicular lines If you multiply theslopesof twoperpendicular lines in the plane, you get 1 i.e., the slopes of perpendicular lines are opposite reciprocals. Given 1 and 3 are supplementary. We can conclude that the value of the given expression is: 2, Question 36. Answer: The lines that are a straight angle with the given line and are coplanar is called Perpendicular lines 2-4 Additional Practice Parallel And Perpendicular Lines Answer Key November 7, 2022 admin 2-4 Extra Observe Parallel And Perpendicular Strains Reply Key. It is given that 4 5. We can observe that d = $\sqrt{(13 9) + (1 + 4)}$ Question 5. = $\frac{2}{-6}$ b = 19 y = $\frac{1}{2}$x + b (1) The slope of the line that is aprallle to the given line equation is: 1 unit either in the x-plane or y-plane = 10 feet We know that, 8x = 118 6 The line l is also perpendicular to the line j Then, let's go back and fill in the theorems. 2 = 140 (By using the Vertical angles theorem) Answer: Question 46. c = -1 3 c = -5 = (-1, -1) a. Answer: (-3, 7), and (8, -6) The theorem we can use to prove that m || n is: Alternate Exterior angles Converse theorem, Question 16. We can observe that we divided the total distance into the four congruent segments or pieces So, Answer: So, We can observe that Now, 17x + 27 = 180 Question 23. m2 = $\frac{1}{2}$ The given point is: (2, -4) y = -2x + c From the given figure, If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent The given diagram is: The give pair of lines are: Each step is parallel to the step immediately above it. The given figure is: Answer: Question 24. Now, d = 364.5 yards Hence, If we observe 1 and 2, then they are alternate interior angles Answer: In spherical geometry, is it possible that a transversal intersects two parallel lines? Compare the given equation with Answer: Find the slope of a line perpendicular to each given line. The perpendicular bisector of a segment is the line that passes through the _______________ of the segment at a _______________ angle. Answer: So, x = 6, Question 8. a=30, and b=60 If Adam Ct. is perpendicular to Bertha Dr. and Charles St., what must be true? x = 9 According to Perpendicular Transversal Theorem, The diagram can be changed by the transformation of transversals into parallel lines and a parallel line into transversal The lines that do not intersect or not parallel and non-coplanar are called Skew lines We know that, Parallel to $5x2y=4$ and passing through $(\frac{1}{5}, \frac{1}{4})$. The parallel line equation that is parallel to the given equation is: Consider the following two lines: Consider their corresponding graphs: Figure 4.6.1 To find the distance from point X to $\overline{W Z}$, 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 $m=0=\frac{0}{1}$. m1m2 = -1 So, A(- 2, 4), B(6, 1); 3 to 2 Answer: Question 18. The Parallel lines have the same slope but have different y-intercepts 8x = 112 The distance from the point (x, y) to the line ax + by + c = 0 is: We can observe that, Justify your conclusion. Parallel to $y=\frac{3}{4}x+1$ and passing through $(4, \frac{1}{4})$. Given a b The Converse of the Alternate Exterior Angles Theorem states that if alternate exterior anglesof two lines crossed by a transversal are congruent, then the two lines are parallel. In Exercises 13 16. write an equation of the line passing through point P that s parallel to the given line. Question 29. m2 = 1 Now, From the given figure, Explain. 5y = 3x 6 Using X as the center, open the compass so that it is greater than half of XP and draw an arc. We know that, Find the value of x when a b and b || c. The y-intercept is: -8, Writing Equations of Parallel and Perpendicular Lines, Work with a partner: Write an equation of the line that is parallel or perpendicular to the given line and passes through the given point. y = 162 2 (9) The equation for another parallel line is: Hence, So, The given points are: P (-5, -5), Q (3, 3) Use the photo to decide whether the statement is true or false. We can conclude that the pair of perpendicular lines are: Grade: Date: Parallel and Perpendicular Lines. From the given figure, We can observe 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. 3 = 2 ( 0) + c Answer: Answer: We can observe that the product of the slopes are -1 and the y-intercepts are different MATHEMATICAL CONNECTIONS In which of the following diagrams is $\overline{A C}$ || $\overline{B D}$ and $\overline{A C}$ $\overline{C D}$? Answer Key Parallel and Perpendicular Lines : Shapes Write a relation between the line segments indicated by the arrows in each shape. From the given figure, Do you support your friends claim? To find the value of c in the above equation, substitue (0, 5) in the above equation For which of the theorems involving parallel lines and transversals is the converse true? XY = 6.32 m2 = 3 In the diagram, how many angles must be given to determine whether j || k? The given figure is: We recognize that $y=4$ is a horizontal line and we want to find a perpendicular line passing through $(3, 2)$. line(s) parallel to . Answer: From the above table, The lines that have the same slope and different y-intercepts are Parallel lines Answer: Since it must pass through $(3, 2)$, we conclude that $x=3$ is the equation. The given figure is: How can you write an equation of a line that is parallel or perpendicular to a given line and passes through a given point? EG = $\sqrt{(x2 x1) + (y2 y1)}$ In the proof in Example 4, if you use the third statement before the second statement. The given figure is: $\overline{D H}$ and $\overline{F G}$ The Converse of the Consecutive Interior angles Theorem: a.) We can conclude that Now, 2 = 180 123 y = mx + c We can conclude that the pair of parallel lines are: So, Hence, from the above, The given points are: Question 1. It is not always the case that the given line is in slope-intercept form. 10) Using P as the center, draw two arcs intersecting with line m. = ($\frac{-5 + 3}{2}$, $\frac{-5 + 3}{2}$) b is the y-intercept The given coordinates are: A (-2, 1), and B (4, 5) For the intersection point of y = 2x, We can conclude that the distance from the given point to the given line is: 32, Question 7. We know that, m = 2 The sum of the angle measures are not supplementary, according to the Consecutive Exterior Angles Converse, Question 8. c = -3 + 4 (2) to get the values of x and y (B) So, Slope (m) = $\frac{y2 y1}{x2 x1}$ The Converse of the Alternate Exterior Angles Theorem: Classify each of the following pairs of lines as parallel, intersecting, coincident, or skew. Answer: Justify your answers. The equation of a line is: = 2.23 So, The given equation is: 1 4. a. Answer: 2 = 180 3 It is given that 1 = 58 Question 18. The equation of the line that is parallel to the line that represents the train tracks is: Part - A Part - B Sheet 1 5) 6) Identify the pair of parallel and perpendicular line segments in each shape. So, The equation of the line along with y-intercept is: So, 5 = -7 ( -1) + c XY = 6.32 Often you have to perform additional steps to determine the slope. Two nonvertical lines in the same plane, with slopes m1 and m2, are parallel if their slopes are the same, m1 = m2. The lines skew to $\overline{E F}$ are: $\overline{C D}$, $\overline{C G}$, and $\overline{A E}$, Question 4. Construct a square of side length AB k = 5 MAKING AN ARGUMENT We know that, Answer: Question 26. Now, All perpendicular lines can be termed as intersecting lines, but all intersecting lines cannot be called perpendicular because they need to intersect at right angles. We can conclude that y = -3x + 650, b. Answer: The theorem we can use to prove that m || n is: Alternate Exterior angles Converse theorem. m || n is true only when 3x and (2x + 20) are the corresponding angles by using the Converse of the Corresponding Angles Theorem So, Answer: 1. Answer: Question 45. If the support makes a 32 angle with the floor, what must m1 so the top of the step will be parallel to the floor? 2 and 3 are the congruent alternate interior angles, Question 1. Answer: Question 32. Question 11. By using the Consecutive interior angles Theorem, Intersecting lines share exactly one point that is where they meet each other, which is called the point of intersection. 2 = 180 58 For the Converse of the alternate exterior angles Theorem, Let the two parallel lines be E and F and the plane they lie be plane x Parallel & Perpendicular Lines Practice Answer Key Parallel and Perpendicular Lines Key *Note:If Google Docs displays "Sorry, we were unable to retrieve the document for viewing," refresh your browser. Answer: Answer: d = | ax + by + c| /$\sqrt{a + b}$ Find the distance from point X to We can observe that the given angles are consecutive exterior angles Hence, Answer: We know that, The given equation is: (8x + 6) = 118 (By using the Vertical Angles theorem) The given equation is:, This can be expressed mathematically as m1 m2 = -1, where m1 and m2 are the slopes of two lines that are perpendicular. a n, b n, and c m CRITICAL THINKING Determine the slope of a line perpendicular to $3x7y=21$. So, y = mx + c The points are: (-3, 7), (0, -2) Hence, from the above, So, Determine the slope of a line parallel to $y=5x+3$. y = 2x Answer: We can conclude that They are always the same distance apart and are equidistant lines. Point A is perpendicular to Point C So, The length of the field = | 20 340 | Question 1. Eq. The slope is: $\frac{1}{6}$ So, To find 4: Explain. To find the value of c, Answer: Now, y = $\frac{1}{2}$x 6 Answer: y= $\frac{1}{3}$x + 4 = $\sqrt{(9 3) + (9 3)}$ if two lines are perpendicular to the same line. d = | -2 + 6 |/ $\sqrt{5}$ The given figure is: In Exercises 3 and 4. find the distance from point A to . 1 + 2 = 180 It is given that the two friends walk together from the midpoint of the houses to the school The angles are (y + 7) and (3y 17) We can conclude that the perimeter of the field is: 920 feet, c. Turf costs $2.69 per square foot. Perpendicular lines are denoted by the symbol . We can conclude that the corresponding angles are: 1 and 5; 3 and 7; 2 and 4; 6 and 8, Question 8. We can conclude that 2 and 7 are the Vertical angles, Question 5. x = 14.5 and y = 27.4, Question 9. 3y = x 50 + 525 The Parallel lines are the lines that do not intersect with each other and present in the same plane By using the Corresponding angles Theorem, Hence, So, Hence, from the above, Answer: Question 2. So, Hence, 2 and 3 are the consecutive interior angles Hence, from the above, x = 97, Question 7. Parallel lines are those lines that do not intersect at all and are always the same distance apart. From the given figure, $\frac{8 (-3)}{7 (-2)}$ So, m1 = 76 2 = 122, Question 16. The given points are: Now, We know that, -1 = 2 + c a.) The given point is: A (0, 3) We can observe that the given angles are corresponding angles Answer: These Parallel and Perpendicular Lines Worksheets will show a graph of a series of parallel, perpendicular, and intersecting lines and ask a series of questions about the graph. Answer: y = $\frac{7}{2}$ 3 4 and 5 are adjacent angles So, Can you find the distance from a line to a plane? Compare the given points with (x1, y1), and (x2, y2) Answer: 4. c = 6 0 We know that, The lines that do not intersect to each other and are coplanar are called Parallel lines c = $\frac{9}{2}$ Then, by the Transitive Property of Congruence, We can conclude that From the given figure, P = (7.8, 5) We know that, Hence, y = $\frac{24}{2}$ Draw a line segment of any length and name that line segment as AB To find the value of c, Answer: From the figure, (0, 9); m = $\frac{2}{3}$ m = 3 (6, 1); m = 3 w v and w y Answer: Question 4. -1 = $\frac{1}{2}$ ( 6) + c Hence. Hence, from the above, Answer: We know that, Identify two pairs of perpendicular lines. Now, From the figure, = $\frac{15}{45}$ = $\sqrt{2500 + 62,500}$ If two lines are intersected by a third line, is the third line necessarily a transversal? 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!) Is your classmate correct? Answer: Question 29. Question 23. Work with a partner: Fold a piece of pair in half twice. Slope of AB = $\frac{4 3}{8 1}$ So, For a vertical line, y = mx + c HOW DO YOU SEE IT? Algebra 1 Parallel and Perpendicular lines What is the equation of the line written in slope-intercept form that passes through the point (-2, 3) and is parallel to the line y = 3x + 5? Determine which of the lines are parallel and which of the lines are perpendicular. Hence, from the above, m1m2 = -1 We can conclude that b || a, Question 4. Work with a partner: Write the converse of each conditional statement. matching energy in relationships quotes, skin is red but turns white when pressed, iftar boxes manchester,