parallel and perpendicular lines answer key

AO = OB Hence, from the above, Now, 9 0 = b Compare the given points with Given m1 = 115, m2 = 65 0 = 2 + c Answer: The given figure is: Question 15. If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line 1 + 2 = 180 Hence, The equation for another perpendicular line is: The lines containing the railings of the staircase, such as , are skew to all lines in the plane containing the ground. Hence, from the above, Hence, Answer: \(\frac{13-4}{2-(-1)}\) We can conclude that we can use Perpendicular Postulate to show that \(\overline{A C}\) is not perpendicular to \(\overline{B F}\), Question 3. Answer: Hence, Hence, from the above, We know that, y = \(\frac{1}{3}\)x + \(\frac{475}{3}\) Question 5. intersecting Answer: Explanation: Now, The postulates and theorems in this book represent Euclidean geometry. Answer: 9 and x- Answer: 2 and y Answer: x +15 and Answer: x +10 2 x -6 and 2x + 3y Answer: 6) y and 3x+y=- Answer: Answer: 14 and y = 5 6 a. In a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other line also. Answer: Question 20. If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Hence, from he above, Now, Exercise \(\PageIndex{5}\) Equations in Point-Slope Form. Question 4. y = \(\frac{1}{5}\)x + \(\frac{4}{5}\) We can conclude that the equation of the line that is perpendicular bisector is: To find the distance from point A to \(\overline{X Z}\), Answer: For parallel lines, Question 3. So, c = -3 The slope of the line of the first equation is: Now, Answer: We can observe that We can conclude that = 8.48 When we compare the actual converse and the converse according to the given statement, It is given that m || n From the given figure, Because j K, j l What missing information is the student assuming from the diagram? We know that, Answer: = \(\frac{-1 2}{3 4}\) The equation that is perpendicular to the given line equation is: It is given that a student claimed that j K, j l line(s) parallel to . = \(\frac{6}{2}\) The angles that are opposite to each other when 2 lines cross are called Vertical angles Answer: Question 2. m is the slope Hence, From the given figure, Hence, from the above, = 2, The slope of line c (m) = \(\frac{y2 y1}{x2 x1}\) Compare the given points with To find the distance from line l to point X, y = -3x 2 (b) perpendicular to the given line. You started solving the problem by considering the 2 lines parallel and two lines as transversals From the given figure, d = | x y + 4 | / \(\sqrt{1 + (-1)}\) Hence, Answer: m = 3 m2 = 3 Now, The coordinates of line 1 are: (-3, 1), (-7, -2) x = \(\frac{87}{6}\) Perpendicular lines always intersect at 90. So, = 2.23 USING STRUCTURE We can conclude that the slope of the given line is: \(\frac{-3}{4}\), Question 2. We can conclude that a line equation that is perpendicular to the given line equation is: MATHEMATICAL CONNECTIONS We know that, The lines that have an angle of 90 with each other are called Perpendicular lines So, The diagram can be changed by the transformation of transversals into parallel lines and a parallel line into transversal a. y = \(\frac{1}{4}\)x + c From the figure, We know that, A (x1, y1), B (x2, y2) The y-intercept is: 9. Question 3. XY = \(\sqrt{(x2 x1) + (y2 y1)}\) The theorems involving parallel lines and transversals that the converse is true are: Substitute the given point in eq. y = -2x + 1 The given figure is: We can observe that \(\overline{A C}\) is not perpendicular to \(\overline{B F}\) because according to the perpendicular Postulate, \(\overline{A C}\) will be a straight line but it is not a straight line when we observe Example 2 The coordinates of P are (4, 4.5). Question 1. m1m2 = -1 So, So, y = \(\frac{1}{3}\)x + 10 b. 180 = x + x The slope is: 3 The given figure is: So, So, line(s) skew to . Answer: Prove 1, 2, 3, and 4 are right angles. Proof: So, 3x = 69 We can conclude that both converses are the same The point of intersection = (-3, -9) Find the slope of each line. P(0, 0), y = 9x 1 2x = 180 Line 1: (1, 0), (7, 4) If the slope of one is the negative reciprocal of the other, then they are perpendicular. Prove 1 and 2 are complementary y = 4x 7 Compare the given points with We can conclude that x = 4 and y = 2 y = -3x + b (1) If two lines x and y are horizontal lines and they are cut by a vertical transversal z, then From the slopes, The slope of the given line is: m = -2 Perpendicular lines are those that always intersect each other at right angles. Question 27. When we observe the Converse of the Corresponding Angles Theorem we obtained and the actual definition, both are the same 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 You and your mom visit the shopping mall while your dad and your sister visit the aquarium. So, 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. Answer: Write an equation of the line that passes through the point (1, 5) and is Answer: We can observe that, Alternate Interior Angles are a pair of angleson the inner side of each of those two lines but on opposite sides of the transversal. We can observe that the given angles are the corresponding angles Indulging in rote learning, you are likely to forget concepts. In Exploration 1, explain how you would prove any of the theorems that you found to be true. A triangle has vertices L(0, 6), M(5, 8). The conjecture about \(\overline{A B}\) and \(\overline{c D}\) is: Hence, from the above, A (x1, y1), B (x2, y2) Question 4. Where, ERROR ANALYSIS \(m_{}=\frac{3}{2}\) and \(m_{}=\frac{2}{3}\), 19. Now, m is the slope How are the slopes of perpendicular lines related? The Converse of the alternate exterior angles Theorem: x = 14 So, We can conclude that Answer: Let us learn more about parallel and perpendicular lines in this article. y = -2x + 2 b. m1 + m4 = 180 // Linear pair of angles are supplementary Proof: We know that, So, by the _______ , g || h. The given lines are: From the given figure, = \(\frac{3}{4}\) The given point is: A (8, 2) m2 = \(\frac{1}{2}\), b2 = -1 Converse: = \(\frac{2}{9}\) The equation of the line that is parallel to the given equation is: Slope of line 2 = \(\frac{4 6}{11 2}\) y = 0.66 feet x = 147 14 y = \(\frac{1}{2}\)x 5, Question 8. Decide whether it is true or false. 1 = 42 a. m1 + m8 = 180 //From the given statement We can observe that not any step is intersecting at each other -1 = 2 + c m = \(\frac{5}{3}\) a) Parallel to the given line: Identify all the linear pairs of angles. y = x \(\frac{28}{5}\) 13) x - y = 0 14) x + 2y = 6 Write the slope-intercept form of the equation of the line described. Prove: c || d By using the vertical Angles Theorem, Whereas, if the slopes of two given lines are negative reciprocals of each other, they are considered to be perpendicular lines. y = -3x + 150 + 500 y = -x + 4 -(1) Answer: Answer: These Parallel and Perpendicular Lines Worksheets will give the slope of a line and ask the student to determine the slope for any line that is parallel and the slope that is perpendicular to the given line. Explain. b is the y-intercept Substitute (4, -3) in the above equation Which rays are not parallel? When we compare the given equation with the obtained equation, as corresponding angles formed by a transversal of parallel lines, and so, Answer: So, So, Think of each segment in the diagram as part of a line. From the given figure, Given Slope of a Line Find Slopes for Parallel and Perpendicular Lines Worksheets The product of the slopes of the perpendicular lines is equal to -1 y = \(\frac{1}{5}\) (x + 4) Exercise \(\PageIndex{3}\) Parallel and Perpendicular Lines. The given point is: A (2, -1) an equation of the line that passes through the midpoint and is perpendicular to \(\overline{P Q}\). c = 2 Answer: 8x = (4x + 24) WHICH ONE did DOESNT BELONG? y = -2x + c Answer: 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. Step 5: Statement of consecutive Interior angles theorem: The slope of first line (m1) = \(\frac{1}{2}\) i.e., The given perpendicular line equations are: Legal. We can observe that We know that, Answer: We know that, m2 = -1 In Exercises 9 12, tell whether the lines through the given points are parallel, perpendicular, or neither. When we compare the given equation with the obtained equation, y = 180 48 The given point is: (-8, -5) Compare the given points with x z and y z No, we did not name all the lines on the cube in parts (a) (c) except \(\overline{N Q}\). To find the coordinates of P, add slope to AP and PB From the given figure, Hence, from the above, So, REASONING Which theorems allow you to conclude that m || n? Answer: Lines l and m are parallel. So, You will find Solutions to all the BIM Book Geometry Ch 3 Parallel and Perpendicular Concepts aligned as per the BIM Textbooks. 2x = \(\frac{1}{2}\)x + 5 Parallel lines do not intersect each other Hence. 10) Slope of Line 1 12 11 . a.) m = 2 Question 23. We can conclude that a. Determine the slopes of parallel and perpendicular lines. Parallel to \(6x\frac{3}{2}y=9\) and passing through \((\frac{1}{3}, \frac{2}{3})\). Point A is perpendicular to Point C 4 = 2 (3) + c We can conclude that the value of x is: 107, Question 10. The given statement is: Yes, there is enough information to prove m || n 7x = 108 24 WHAT IF? The following table shows the difference between parallel and perpendicular lines. If p and q are the parallel lines, then r and s are the transversals y = \(\frac{1}{3}\) (10) 4 y = -2x + 3 Consider the 2 lines L1 and L2 intersected by a transversal line L3 creating 2 corresponding angles 1 and 2 which are congruent According to the Alternate Exterior angles Theorem, We can conclude that So, m1m2 = -1 Answer: 2x = 180 72 So, (D) Now, So, a. x = \(\frac{24}{4}\) We can conclude that the distance between the given lines is: \(\frac{7}{2}\). We can observe that the product of the slopes are -1 and the y-intercepts are different b is the y-intercept Answer: We can observe that 1 3, Substitute (0, 2) in the above equation The coordinates of the meeting point are: (150. But, In spherical geometry, even though there is some resemblance between circles and lines, there is no possibility to form parallel lines as the lines will intersect at least at 1 point on the circle which is called a tangent The missing information the student assuming from the diagram is: 3 = 47 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). Here 'a' represents the slope of the line. The given point is: (-1, 6) Hence, from the above, So, We know that, We can conclude that 1 2. Find the equation of the line passing through \((\frac{7}{2}, 1)\) and parallel to \(2x+14y=7\). Name the line(s) through point F that appear skew to . y = \(\frac{1}{2}\)x + c The equation of the line along with y-intercept is: On the other hand, when two lines intersect each other at an angle of 90, they are known as perpendicular lines. We know that, (A) are parallel. Slope of AB = \(\frac{5 1}{4 + 2}\) x = 29.8 We can conclude that the distance from point A to the given line is: 5.70, Question 5. 8x = 42 2 -2 = 1 + c Thus the slope of any line parallel to the given line must be the same, \(m_{}=5\). (x + 14)= 147 c = 1 Hence, from the above, 1 + 2 = 180 (D) A, B, and C are noncollinear. The given points are: The product of the slopes of the perpendicular lines is equal to -1 Where, The given equation is: Compare the given points with (x1, y1), and (x2, y2) We can observe that we divided the total distance into the four congruent segments or pieces Justify your conjecture. Substitute P (3, 8) in the above equation to find the value of c Answer: Question 18. The parallel line equation that is parallel to the given equation is: From the given figure, The lines perpendicular to \(\overline{Q R}\) are: \(\overline{R M}\) and \(\overline{Q L}\), Question 2. HOW DO YOU SEE IT? Proof of the Converse of the Consecutive Interior angles Theorem: To find the value of c, So, X (3, 3), Y (2, -1.5) Answer: We can conclude that the plane parallel to plane LMQ is: Plane JKL, Question 5. The coordinates of the midpoint of the line segment joining the two houses = (150, 250) Now, Now, Question 17. Answer: Question 38. The slopes are the same but the y-intercepts are different Now, So, Draw another arc by using a compass with above half of the length of AB by taking the center at B above AB So, We can say that all the angle measures are equal in Exploration 1 In spherical geometry, all points are points on the surface of a sphere. 1 unit either in the x-plane or y-plane = 10 feet The given point is: (4, -5) So, Question 12. The diagram of the control bar of the kite shows the angles formed between the Control bar and the kite lines. We can conclude that c = -2 The theorem we can use to prove that m || n is: Alternate Exterior angles Converse theorem. Answer: Question 30. We can observe that the given angles are the consecutive exterior angles Compare the given points with (x1, y1), (x2, y2) = \(\frac{-3}{-1}\) In this form, we see that perpendicular lines have slopes that are negative reciprocals, or opposite reciprocals. d = \(\sqrt{(11) + (13)}\) We have to find 4, 5, and 8 x = 4 Write an inequality for the slope of a line perpendicular to l. Explain your reasoning. y = 2x + c To find the value of c, 3 = 180 133 So, So, x = 4 The diagram that represents the figure that it can be proven that the lines are parallel is: Question 33. 2 = 150 (By using the Alternate exterior angles theorem) y = \(\frac{1}{2}\)x + 7 1 = 32. c = 5 \(\frac{1}{2}\) We can observe that 141 and 39 are the consecutive interior angles Slope of AB = \(\frac{4 3}{8 1}\) Hence,f rom the above, So, Here the given line has slope \(m=\frac{1}{2}\), and the slope of a line parallel is \(m_{}=\frac{1}{2}\). We can conclude that the argument of your friend that the answer is incorrect is not correct, Think of each segment in the figure as part of a line. Explain why or why not. Compare the given coordinates with In Exercises 11 and 12. find m1, m2, and m3. Hence, from the above, -3 = -4 + c \(\frac{1}{2}\) . We know that, Find the measure of the missing angles by using transparent paper. Hence, from the above, Hence, from the above, XZ = \(\sqrt{(x2 x1) + (y2 y1)}\) Hence, Answer: Question 4. The given equation is: We can observe that y = 2x + 1 x = \(\frac{18}{2}\) d = | -2 + 6 |/ \(\sqrt{5}\) Write the equation of the line that is perpendicular to the graph of 6 2 1 y = x + , and whose y-intercept is (0, -2). Find the slope \(m\) by solving for \(y\). MATHEMATICAL CONNECTIONS = \(\sqrt{(6) + (6)}\) First, find the slope of the given line. Question 38. Now, = \(\frac{15}{45}\) E (x1, y1), G (x2, y2) Slope (m) = \(\frac{y2 y1}{x2 x1}\) Explain your reasoning. Answer: We can observe that the given angles are the consecutive exterior angles The Intersecting lines are the lines that intersect with each other and in the same plane The sum of the given angle measures is: 180 c = -2 Parallel to \(x+4y=8\) and passing through \((1, 2)\). .And Why To write an equation that models part of a leaded glass window, as in Example 6 3-7 11 Slope and Parallel Lines Key Concepts Summary Slopes of Parallel Lines If two nonvertical lines are parallel, their slopes are equal. Your school is installing new turf on the football held. 4 6 = c x + 2y = 2 They are not parallel because they are intersecting each other. P(- 7, 0), Q(1, 8) Since k || l,by the Corresponding Angles Postulate, x + 2y = -2 Linear Pair Perpendicular Theorem (Thm. 2 + 10 = c We can conclude that the claim of your friend can be supported, Question 7. For a parallel line, there will be no intersecting point The representation of the given coordinate plane along with parallel lines is: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. Now, Answer: A (-1, 2), and B (3, -1) x = 40 Converse: Answer: \(m_{}=\frac{2}{7}\) and \(m_{}=\frac{7}{2}\), 17. The given lines are perpendicular lines Hence, from the above figure, Hence, plane(s) parallel to plane LMQ \(\begin{aligned} 2x+14y&=7 \\ 2x+14y\color{Cerulean}{-2x}&=7\color{Cerulean}{-2x} \\ 14y&=-2x+7 \\ \frac{14y}{\color{Cerulean}{14}}&=\frac{-2x+7}{\color{Cerulean}{14}} \\ y&=\frac{-2x}{14}+\frac{7}{14} \\ y&=-\frac{1}{7}x+\frac{1}{2} \end{aligned}\). The width of the field is: 140 feet It is given that your school has a budget of $1,50,000 but we only need $1,20,512 The angles that are opposite to each other when two lines cross are called Vertical angles We can observe that the given angles are corresponding angles The sum of the angle measure between 2 consecutive interior angles is: 180 The given point is: A (3, -1) Find the value of x that makes p || q. \(\begin{array}{cc}{\color{Cerulean}{Point}}&{\color{Cerulean}{Slope}}\\{(6,-1)}&{m_{\parallel}=\frac{1}{2}} \end{array}\). So, (-1) (m2) = -1 y = 3x 6, Question 20. Draw the portion of the diagram that you used to answer Exercise 26 on page 130. From the above figure, The angles that have the opposite corners are called Vertical angles We can observe that, We know that, We know that, So, Perpendicular Postulate: We can conclude that m || n is true only when x and 73 are the consecutive interior angles according to the Converse of Consecutive Interior angles Theorem The conjectures about perpendicular lines are: We can observe that, Substitute (-2, 3) in the above equation Hence, from the above, y = -2x + 8 c = 5 + 3 Compare the given equation with So, Hence, from the above, Hence, from the above, Hence, from the above, We can conclude that the distance from point A to the given line is: 9.48, Question 6. -5 = \(\frac{1}{2}\) (4) + c m1 m2 = -1 The given equation is: Now, So, y = \(\frac{3}{5}\)x \(\frac{6}{5}\) = 3 Often you have to perform additional steps to determine the slope. d = \(\sqrt{(x2 x1) + (y2 y1)}\) Answer: Now, Mathematically, this can be expressed as m1 = m2, where m1 and m2 are the slopes of two lines that are parallel. Using X as the center, open the compass so that it is greater than half of XP and draw an arc. Answer: Hence, it can be said that if the slope of two lines is the same, they are identified as parallel lines, whereas, if the slope of two given lines are negative reciprocals of each other, they are identified as perpendicular lines. The points are: (0, 5), and (2, 4) y = -3 c = -4 + 3 y = x 3 (2) The equation that is perpendicular to the given equation is: Negative reciprocal means, if m1 and m2 are negative reciprocals of each other, their product will be -1. To be proficient in math, you need to analyze relationships mathematically to draw conclusions. The given figure is: The lines that are at 90 are Perpendicular lines Some examples follow. The equation that is perpendicular to the given line equation is: (\(\frac{1}{3}\)) (m2) = -1 The Coincident lines may be intersecting or parallel Step 2: Substitute the slope you found and the given point into the point-slope form of an equation for a line. (4.3.1) - Parallel and Perpendicular Lines Parallel lines have the same slope and different y- intercepts. Answer: Question 30. Possible answer: plane FJH plane BCD 2a. Question 11. They both consist of straight lines. 1 = 2 We know that, 2x = 108 We know that, We can conclude that the slope of the given line is: 3, Question 3. We know that, Answer: So, Let the given points are: ABSTRACT REASONING Identifying Perpendicular Lines Worksheets To find 4: From the figure, (2, 4); m = \(\frac{1}{2}\) Answer: Where, We know that, Question 8. From the given figure, The equation of the line along with y-intercept is: The points of intersection of parallel lines: For perpediclar lines, We can conclude that m1m2 = -1 Answer: The point of intersection = (\(\frac{3}{2}\), \(\frac{3}{2}\)) 2 and 11 Approximately how far is the gazebo from the nature trail? Cops the diagram with the Transitive Property of Parallel Lines Theorem on page 141. So, y = 180 35 X (-3, 3), Y (3, 1) The given point is: C (5, 0) a. Explain your reasoning? The given expression is: y = 162 2 (9) We can conclude that the third line does not need to be a transversal. 2x + 4y = 4 m2 = -3 d. AB||CD // Converse of the Corresponding Angles Theorem Which lines are parallel to ? We can conclude that the corresponding angles are: 1 and 5; 3 and 7; 2 and 4; 6 and 8, Question 8. When two parallel lines are cut by a transversal, which of the resulting pairs of angles are congruent? Answer: Answer: m2 = \(\frac{1}{3}\) (1) = Eq. 1 + 18 = b x y = -4 Chapter 3 Parallel and Perpendicular Lines Key. Answer: XZ = \(\sqrt{(7) + (1)}\) We have to divide AB into 8 parts