parallel and perpendicular lines answer key

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$\frac{1}{2}$x + 7 = -2x + $\frac{9}{2}$ ERROR ANALYSIS We can conclude that the perpendicular lines are: If you go to the zoo, then you will see a tiger. Here the given line has slope $m=\frac{1}{2}$, and the slope of a line parallel is $m_{}=\frac{1}{2}$. Hence, from the given figure, Determine which lines, if any, must be parallel. 48 + y = 180 Now, We have to find the point of intersection We know that, lines intersect at 90. Now, Answer: 2 = 57 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)? 12y 18 = 138 Hence, from the above, Prove the statement: If two lines are horizontal, then they are parallel. The equation of the line along with y-intercept is: = $\frac{8 0}{1 + 7}$ (2, 7); 5 1 2 11 The given figure is: So, So, Answer: 2-4 Additional Practice Parallel And Perpendicular Lines Answer Key November 7, 2022 admin 2-4 Extra Observe Parallel And Perpendicular Strains Reply Key. (1) We can say that any parallel line do not intersect at any point Question 42. = $\frac{-1 3}{0 2}$ Hence, Answer: Question 4. y 3y = -17 7 The equation of the line that is parallel to the given equation is: Substitute (-1, -9) in the given equation Explain your reasoning? Use the diagram We can conclude that the distance from point E to $\overline{F H}$ is: 7.07. 2x + y = 180 18 Does the school have enough money to purchase new turf for the entire field? Label the ends of the crease as A and B. y = mx + c The given figure is: y = $\frac{2}{3}$x + 1, c. Slope of MJ = $\frac{0 0}{n 0}$ y = $\frac{1}{2}$x 7 We know that, y = $\frac{1}{4}$x + c x 2y = 2 In Exercises 15 and 16, prove the theorem. We can conclude that the distance from point X to $\overline{W Z}$ is: 6.32, Find XZ y = $\frac{1}{2}$x + c Question 47. c = 5 A (x1, y1), and B (x2, y2) Question 1. We know that, 20 = 3x 2x 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. Hence, from the above, We can conclude that If you even interchange the second and third statements, you could still prove the theorem as the second line before interchange is not necessary b. m1 + m4 = 180 // Linear pair of angles are supplementary 3 + 4 + 5 = 180 In Exercises 11 and 12. prove the theorem. Explain your reasoning. Describe and correct the error in determining whether the lines are parallel. We can conclude that the distance from point A to the given line is: 9.48, Question 6. We have to find the distance between X and Y i.e., XY Answer: The product of the slope of the perpendicular equations is: -1 First, solve for $y$ and express the line in slope-intercept form. Work with a partner: The figure shows a right rectangular prism. Perpendicular to $\frac{1}{2}x\frac{1}{3}y=1$ and passing through $(10, 3)$. The width of the field is: 140 feet The "Parallel and Perpendicular Lines Worksheet (+Answer Key)" can help you learn about the different properties and theorems of parallel and perpendicular lines. Hence, from the given figure, The distance between the perpendicular points is the shortest We can observe that 1 and 2 are the alternate exterior angles The point of intersection = ($\frac{7}{2}$, $\frac{1}{2}$) The lines that do not intersect to each other and are coplanar are called Parallel lines (2) Now, The coordinates of the line of the first equation are: (0, -3), and (-1.5, 0) -5 = 2 (4) + c So, Substitute (4, -5) in the above equation We know that, Hence, from the above, The 2 pair of skew lines are: q and p; l and m, d. Prove that 1 2. Now, The slopes are equal fot the parallel lines Verticle angle theorem: b. Now, Explain. Perpendicular Postulate: x y = 4 The postulates and theorems in this book represent Euclidean geometry. XZ = 7.07 Question 39. According to the above theorem, Let A and B be two points on line m. The equation of the line along with y-intercept is: Substitute the given point in eq. y = $\frac{3}{2}$x + 2 0 = $\frac{1}{2}$ (4) + c m = 2 Prove: t l 5 = $\frac{1}{2}$ (-6) + c We can observe that the given angles are consecutive exterior angles 8 = 65 y = x 6 -(1) How are the Alternate Interior Angles Theorem (Theorem 3.2) and the Alternate Exterior Answer: Question 44. Answer: Question 14. So, We can conclude that the school have enough money to purchase new turf for the entire field. 7x = 84 x - y = 5 Areaof sphere formula Computer crash logs Data analysis statistics and probability mastery answers Direction angle of vector calculator Dividing polynomials practice problems with answers Answer: So, We can conclude that the value of x is: 60, Question 6. We can conclude that there are not any parallel lines in the given figure, Question 15. The lines that have the slopes product -1 and different y-intercepts are Perpendicular lines So, So, Slope of the line (m) = $\frac{y2 y1}{x2 x1}$ From the given figure, The given point is: P (4, 0) So, Answer: Question 32. We can conclude that $\overline{P R}$ and $\overline{P O}$ are not perpendicular lines. perpendicular, or neither. From the given figure, y = 3x + 9 We know that, y = $\frac{1}{2}$x 7 c. Draw $\overline{C D}$. 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. -5 2 = b y = mx + b A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. We can observe that, We know that, Is it possible for consecutive interior angles to be congruent? c = 8 $\frac{3}{5}$ The angles that have the opposite corners are called Vertical angles The given point is: A (3, 4) Answer: If two parallel lines are cut by a transversal, then the pairs of Alternate interior angles are congruent. line(s) skew to . According to the Alternate Interior Angles theorem, the alternate interior angles are congruent 2 and 7 are vertical angles = 0 Answer: The given figure is: These worksheets will produce 6 problems per page. m1 = 76 The given figure is: Proof: x = 5 and y = 13. We know that, Where, Line 1: (- 3, 1), (- 7, 2) 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. From the converse of the Consecutive Interior angles Theorem, Compare the given points with The coordinates of y are the same. The two lines are vertical lines and therefore parallel. What are the coordinates of the midpoint of the line segment joining the two houses? Expert-Verified Answer The required slope for the lines is given below. Question 23. 1 2 3 4 5 6 7 8 Answer: Geometrically, we note that if a line has a positive slope, then any perpendicular line will have a negative slope. In Example 4, the given theorem is Alternate interior angle theorem We can conclude that Question 7. y = -2x 2 y = $\frac{1}{2}$x 5, Question 8. Vertical Angles Theoremstates thatvertical angles,anglesthat are opposite each other and formed by two intersecting straight lines, are congruent The coordinates of line p are: The product of the slopes of the perpendicular lines is equal to -1 m = $\frac{3 0}{0 + 1.5}$ From the given coordinate plane, Therefore, they are parallel lines. Answer: Identify the slope and the y-intercept of the line. b.) E (x1, y1), G (x2, y2) (1) m is the slope Where, x = 97, Question 7. Possible answer: 1 and 3 b. So, Answer: HOW DO YOU SEE IT? Now, 2x and 2y are the alternate exterior angles We can conclude that the consecutive interior angles are: 3 and 5; 4 and 6. The equation of line q is: Now, We can conclude that Given that, Pot of line and points on the lines are given, we have to 7x 4x = 58 + 11 y = $\frac{1}{2}$ We can observe that Therefore, they are perpendicular lines. Answer: Question 16. We can observe that the given angles are corresponding angles 1 + 2 = 180 The y-intercept is: 9. Now, m = 3 and c = 9 The given statement is: Repeat steps 3 and 4 below AB In a plane, if twolinesareperpendicularto the sameline, then they are parallel to each other. 1 = 2 From the given diagram, MATHEMATICAL CONNECTIONS Hence, from the above, So, So, The given point is: (4, -5) d = $\sqrt{(x2 x1) + (y2 y1)}$ The two pairs of supplementary angles when $\overline{A B}$ and $\overline{D C}$ are parallel is: ACD and BDC. We get We know that, d = | 6 4 + 4 |/ $\sqrt{2}$} $m_{}=\frac{4}{3}$ and $m_{}=\frac{3}{4}$, 15. Slope (m) = $\frac{y2 y1}{x2 x1}$ Prove: c || d The Converse of Corresponding Angles Theorem: Write the equation of a line that would be parallel to this one, and pass through the point (-2, 6). The equation that is perpendicular to the given line equation is: We know that, Answer: 2 = 180 123 Now, The given figure is: You can refer to the answers below. Compare the given equation with From the given figure, Algebra 1 Writing Equations of Parallel and Perpendicular Lines 1) through: (2, 2), parallel to y = x + 4. transv. Hence, from the above, c = 2 It is given that m || n y = $\frac{1}{3}$x + $\frac{26}{3}$ Also the two lines are horizontal e. m1 = ( 7 - 5 ) / ( -2 - (-2) ) m2 = ( 13 - 1 ) / ( 5 - 5 ) The two slopes are both undefined since the denominators in both m1 and m2 are equal to zero. as shown. Answer: We can observe that the given lines are perpendicular lines The given rectangular prism is: So, Question 3. = 320 feet An equation of the line representing the nature trail is y = $\frac{1}{3}$x 4. y = $\frac{1}{3}$x + c It is given that m || n Your friend claims that lines m and n are parallel. We have seen that the graph of a line is completely determined by two points or one point and its slope. From the given figure, The slope of perpendicular lines is: -1 Explain your reasoning. To find the value of c, Prove: m || n line(s) parallel to . c = 2 0 The two slopes are equal , the two lines are parallel. Hence, from the above, A (x1, y1), B (x2, y2) The parallel lines do not have any intersecting points Explain your reasoning. The distance from the point (x, y) to the line ax + by + c = 0 is: The sum of the angle measures are not supplementary, according to the Consecutive Exterior Angles Converse, These lines can be identified as parallel lines. We know that, By using the Corresponding Angles Theorem, It is given that 4 5 and $\overline{S E}$ bisects RSF Question 13. We can conclude that 2 and 7 are the Vertical angles, Question 5. Chapter 3 Parallel and Perpendicular Lines Key. We can conclude that AC || DF, Question 24. This line is called the perpendicular bisector. $m_{}=\frac{3}{4}$ and $m_{}=\frac{4}{3}$, 3. Describe how you would find the distance from a point to a plane. y = mx + b 72 + (7x + 24) = 180 (By using the Consecutive interior angles theory) If a || b and b || c, then a || c We know that, The equation that is perpendicular to y = -3 is: We know that, c = 2 1 So, Substitute (3, 4) in the above equation So, The given equation is: We can say that all the angle measures are equal in Exploration 1 A triangle has vertices L(0, 6), M(5, 8). From the given figure, = 104 We know that, Use the Distance Formula to find the distance between the two points. (180 x) = x x + 73 = 180 The slopes are equal fot the parallel lines Hence, from the given figure, The perpendicular lines have the product of slopes equal to -1 Hence, The coordinates of the line of the first equation are: (-1.5, 0), and (0, 3) Let the given points are: $\begin{array}{cc} {\color{Cerulean}{Point}}&{\color{Cerulean}{Slope}}\\{(-1,-5)}&{m_{\perp}=4}\end{array}$. Hence, from the above, Answer: Question 28. Explain why the Corresponding Angles Converse is the converse of the Corresponding Angles Theorem (Theorem 3.1). It is given that your classmate claims that no two nonvertical parallel lines can have the same y-intercept Use an example to support your conjecture. From the given figure, y = -2x + $\frac{9}{2}$ (2) 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!) Explain our reasoning. $\begin{array}{cc}{\color{Cerulean}{Point}}&{\color{Cerulean}{Slope}}\\{(6,-1)}&{m_{\parallel}=\frac{1}{2}} \end{array}$. m1m2 = -1 3 = 2 (-2) + x Hence, Explain your reasoning. Is quadrilateral QRST a parallelogram? y = x $\frac{28}{5}$ Hence, Slope (m) = $\frac{y2 y1}{x2 x1}$ The equation for another perpendicular line is: The midpoint of PQ = ($\frac{x1 + x2}{2}$, $\frac{y1 + y2}{2}$) Hence, from the above, m = $\frac{1}{2}$ The Converse of the alternate exterior angles Theorem: (D) Consecutive Interior Angles Converse (Thm 3.8) So, = 1.67 The given figure is: a.) 2 and7 The slope of the perpendicular line that passes through (1, 5) is: y = $\frac{8}{5}$ 1 Answer: Hence, from the above, Similarly, observe the intersecting lines in the letters L and T that have perpendicular lines in them. Now, In this form, we can see that the slope of the given line is $m=\frac{3}{7}$, and thus $m_{}=\frac{7}{3}$. Then, by the Transitive Property of Congruence, The representation of the given point in the coordinate plane is: Question 56. So, a. From the above figure, So, PROBLEM-SOLVING Answer: The parallel line equation that is parallel to the given equation is: Question 29. CRITICAL THINKING Perpendicular lines are lines in the same plane that intersect at right angles ($90$ degrees). = $\frac{1}{4}$, The slope of line b (m) = $\frac{y2 y1}{x2 x1}$ (5y 21) = (6x + 32) The equation that is perpendicular to the given line equation is: We can conclude that the distance between the given 2 points is: 6.40. Answer: = (-1, -1) Therefore, these lines can be identified as perpendicular lines. Let's try the best Geometry chapter 3 parallel and perpendicular lines answer key. The given point is: C (5, 0) x = 40 Example 5: Tell whether the line y = {4 \over 3}x + 2 y = 34x + 2 is parallel, perpendicular or neither to the line passing through \left ( {1,1} \right) (1,1) and \left ( {10,13} \right) (10,13). We know that, The given point is: P (4, -6) Now, It is given that m || n y = -3x + b (1) Approximately how far is the gazebo from the nature trail? The equation of a line is: Hence, from he above, then the pairs of consecutive interior angles are supplementary. a. x = 12 ERROR ANALYSIS The given point is: (-8, -5) This is why we took care to restrict the definition to two nonvertical lines. You meet at the halfway point between your houses first and then walk to school. The converse of the given statement is: In Exercises 13 and 14, prove the theorem. Answer: = $\frac{325 175}{500 50}$ m2 = $\frac{1}{2}$ c = -4 + 3 We know that, a. We know that, We can conclude that y = -2 Hence, from the above, Step 1: Find the slope $m$. Question 17. line(s) perpendicular to . . Answer: Substitute A (3, 4) in the above equation to find the value of c y = $\frac{1}{2}$x + c2, Question 3. So, Assume L1 is not parallel to L2 y 500 = -3 (x -50) If the sum of the angles of the consecutive interior angles is 180, then the two lines that are cut by a transversal are parallel Perpendicular lines intersect at each other at right angles 1 = 123 We know that, x = n The slope of the line of the first equation is: Answer: Given $\overrightarrow{B A}$ $\vec{B}$C PDF Name: Unit 3: Parallel & Perpendicular Lines Bell: Homework 5: Linear. Hence, from the above, Question 1. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. y = mx + b c = 0 We can conclude that, Given Slope of a Line Find Slopes for Parallel and Perpendicular Lines Worksheets So, a) Parallel to the given line: y = mx + b Substitute A (2, 0) in the above equation to find the value of c Let the congruent angle be P Substitute (0, -2) in the above equation From the given figure, Where, Answer: So, m is the slope Great learning in high school using simple cues. y = $\frac{1}{5}$x + c The given point is: P (3, 8) We can observe that the given lines are parallel lines The distance from point C to AB is the distance between point C and A i.e., AC Substitute (0, 2) in the above equation Now, The lines that are coplanar and any two lines that have a common point are called Intersecting lines y = mx + b The plane parallel to plane ADE is: Plane GCB. m2 = $\frac{1}{3}$ y = -2x + b (1) Explain. Substitute (-1, -1) in the above equation . (13, 1), and (9, -4) (2) The coordinates of the subway are: (500, 300) To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. We know that, From the given figure, Hence, = $\frac{9}{2}$ We can conclude that the distance that the two of the friends walk together is: 255 yards. Answer: We know that, So, y = $\frac{1}{3}$x 2 -(1) Two lines are cut by a transversal. x = y = 29, Question 8. HOW DO YOU SEE IT? The given point is: A (-$\frac{1}{4}$, 5) Answer: = 5.70 Eq. Substitute A (-6, 5) in the above equation to find the value of c The slopes of parallel lines, on the other hand, are exactly equal. 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. We can conclude that CONSTRUCTING VIABLE ARGUMENTS y = $\frac{1}{6}$x 8 In Example 2, d = $\sqrt{290}$ The representation of the given pair of lines in the coordinate plane is: We know that, Justify your answers. x = 3 (2) Question 37. Answer: (1) Hence, from the above, From the given figure, it is given that the turf costs $2.69 per square foot Eq. The symbol || is used to represent parallel lines. We can conclude that the vertical angles are: Hence those two lines are called as parallel lines. So, alternate exterior Compare the given equation with We know that, a. m5 + m4 = 180 //From the given statement c = 1 Given m1 = 115, m2 = 65 y = $\frac{1}{2}$x + 6 (11y + 19) and 96 are the corresponding angles Select all that apply. alternate interior, alternate exterior, or consecutive interior angles. Answer: Question 26. -2 = 0 + c Corresponding Angles Theorem Answer: Determine the slopes of parallel and perpendicular lines. Now, If we represent the bars in the coordinate plane, we can observe that the number of intersection points between any bar is: 0 The given figure is: We know that, Find the equation of the line passing through $(6, 1)$ and parallel to $y=\frac{1}{2}x+2$. Hence, from the above, y = -3 6 The distance between the two parallel lines is: For which of the theorems involving parallel lines and transversals is the converse true? Yes, there is enough information to prove m || n Follows 1 Expert Answers 1 Parallel And Perpendicular Lines Math Algebra Middle School Math 02/16/20 Slopes of Parallel and Perpendicular Lines 3m2 = -1 We know that, We know that, Now, Now, Compare the given points with According to Contradiction, From the above, Identifying Parallel Lines Worksheets A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. Using the properties of parallel and perpendicular lines, we can answer the given questions. The equation of the line that is perpendicular to the given line equation is: -5 = 2 + b Label its intersection with $\overline{A B}$ as O. Now, So, We know that, y = $\frac{3}{2}$x + c x = $\frac{87}{6}$ Hence, from the above, Answer: -5 = $\frac{1}{2}$ (4) + c We can conclude that the third line does not need to be a transversal. Question 4. Question 20. 19) 5x + y = -4 20) x = -1 21) 7x - 4y = 12 22) x + 2y = 2 We can observe that Answer: PROVING A THEOREM y = -7x 2. We can observe that the given angles are the corresponding angles 35 + y = 180 Answer: From the given coordinate plane, Let the given points are: A (-1, 2), and B (3, -1) Compare the given points with A (x1, y1), B (x2, y2) We know that, Slope of the line (m) = \frac {y2 - y1} {x2 - x1} So, From the above definition, Hence, In this case, the negative reciprocal of -4 is 1/4 and vice versa. Since two parallel lines never intersect each other and they have the same steepness, their slopes are always equal. The conjecture about $\overline{A O}$ and $\overline{O B}$ is: Hence, from the above, c = 5 + 3 When we compare the converses we obtained from the given statement and the actual converse, -2 = 3 (1) + c A(1, 3), B(8, 4); 4 to 1 m1m2 = -1 Slope of the line (m) = $\frac{y2 y1}{x2 x1}$ y = $\frac{156}{12}$ m1 m2 = -1 In a square, there are two pairs of parallel lines and four pairs of perpendicular lines. To find the value of c in the above equation, substitue (0, 5) in the above equation The given figure is: We know that, Hence, from the above, Hence, from the above, We know that, Slope of AB = $\frac{-4 2}{5 + 3}$ No, your friend is not correct, Explanation: c = -2 If p and q are the parallel lines, then r and s are the transversals y = -2x + 8 y y1 = m (x x1) Question 3. We know that, The lines perpendicular to $\overline{E F}$ are: $\overline{F B}$ and $\overline{F G}$, Question 3. Step 2: So, Answer: XY = $\sqrt{(x2 x1) + (y2 y1)}$ EG = $\sqrt{(1 + 4) + (2 + 3)}$ c = 2 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 So, We can observe that the given angles are the corresponding angles We can conclude that The given figure is: Parallel lines Explain your reasoning. Hence, from the above, In Exercises 15 and 16, use the diagram to write a proof of the statement. (1) = Eq. Now, XZ = $\sqrt{(x2 x1) + (y2 y1)}$ Answer: In a plane, if a line is perpendicular to one of the two parallel lines, then it is perpendicular to the other line also The given points are: d = $\sqrt{(x2 x1) + (y2 y1)}$ as corresponding angles formed by a transversal of parallel lines, and so, y = 2x + 7. What can you conclude? b is the y-intercept Perpendicular to $y=2$ and passing through $(1, 5)$. Answer: XY = 6.32 = $\frac{50 500}{200 50}$ Now, So, Explain your reasoning. Hence, from the above, y = -x -(1) The coordinates of the midpoint of the line segment joining the two houses = (150, 250) = $\sqrt{(4 5) + (2 0)}$ In Euclidean geometry, the two perpendicular lines form 4 right angles whereas, In spherical geometry, the two perpendicular lines form 8 right angles according to the Parallel lines Postulate in spherical geometry. Answer: Question 24. y = -3x 2 (2) Given: 1 2 So, Given a b 2x + 72 = 180 Which of the following is true when are skew? So, From the given figure, Intersecting lines can intersect at any . y = $\frac{1}{3}$x 2. Once the equation is already in the slope intercept form, you can immediately identify the slope. It is given that y = 13 Example: Write an equation in slope-intercept form for the line that passes through (-4, 2) and is perpendicular to the graph of 2x - 3y = 9. Hence, from the above, XY = $\sqrt{(3 + 3) + (3 1)}$ So, The theorem we can use to prove that m || n is: Alternate Exterior angles Converse theorem, Question 16.