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ability in you. Embedded mathematical practices, exercises provided make it easy for you to understand the concepts quite quickly. Begin your preparation right away and clear the exams with utmost confidence. Simply click on the below available and learn the respective topics in no time. Find the slope of the line. Question 1. Question 2. Answer: From the given coordinate plane, Let the given points are: A (-2, 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, Slope of the line (m) = \(\frac{-1 – 2}{-3 + 2}\) = \(\frac{-3}{-1}\) = 3 Hence, from the above, We can conclude that the slope of the given line is: 3 Question 3. Answer: From the given coordinate plane, Let the given points are: A (-3, -2), and B (1, -2) 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, Slope of the line (m) = \(\frac{-2 + 2}{3 + 1}\) = \(\frac{0}{4}\) = 0 Hence, from the above, We can conclude that the slope of the given line is: 0 Write an equation of the line that passes through the given point and has the given slope. Question 4. Question 5. Question 6. Question 7. Question 8. Question 9. Question 10. Parallel and Perpendicular Lines Mathematical PracticesUse a graphing calculator to graph the pair of lines. Use a square viewing window. Classify the lines as parallel, perpendicular, coincident, or non-perpendicular intersecting lines. Justify your answer. Question 1. We know that, For a pair of lines to be perpendicular, the product of the slopes i.e., the product of the slope of the first line and the slope of the second line will be equal to -1 So, By comparing the given pair of lines with y = mx + b We get The slope of first line (m1) = –\(\frac{1}{2}\) The slope of second line (m2) = 2 So, m1 ×m2 = –\(\frac{1}{2}\) × 2 = -1 Hence, from the above, We can conclude that the given pair of lines are perpendicular lines Question 2. We know that, For a pair of lines to be coincident, the pair of lines have the same slope and the same y-intercept So, By comparing the given pair of lines with y = mx + b We get m1 = –\(\frac{1}{2}\), b1 = 1 m2 = –\(\frac{1}{2}\), b2 = 1 Hence, from the above, We can conclude that the given pair of lines are coincident lines Question 3. We know that, For a pair of lines to be parallel, the pair of lines have the same slope but different y-intercepts So, By comparing the given pair of lines with y = mx + b We get m1 = –\(\frac{1}{2}\), b1 = 1 m2 = \(\frac{1}{2}\), b2 = -1 Hence, from the above, We can conclude that the given pair of lines are parallel lines Question 4. We know that, 3.1 Pairs of Lines and AnglesExploration 1 Points of intersection work with a partner: Write the number of points of intersection of each pair of coplanar lines. Answer: The given coplanar lines are: a. The points of intersection of parallel lines: We know that, The “Parallel lines” have the same slope but have different y-intercepts So, We can say that any parallel line do not intersect at any point Hence, from the above, We can conclude that the number of points of intersection of parallel lines is: 0 a. The points of intersection of intersecting lines: c. The points of intersection of coincident lines: Exploration 2 Classifying Pairs of Lines Work with a partner: The figure shows a right rectangular prism. All its angles are right angles. Classify each of the following pairs of lines as parallel, intersecting, coincident, or skew. Justify your answers. (Two lines are skew lines when they do not intersect and are not coplanar.) Answer: The given rectangular prism is: We know that, The “Parallel lines” are the lines that do not intersect with each other and present in the same plane The “Intersecting lines” are the lines that intersect with each other and in the same plane The “Coincident lines” are the lines that lie on one another and in the same plane The “Skew lines” are the lines that do not present in the same plane and do not intersect Hence, The completed table of the nature of the given pair of lines is: Exploration 3 Identifying Pairs of Angles Work with a partner: In the figure, two parallel lines are intersected by a third line called a transversal. a. Identify all the pairs of vertical angles. Explain your reasoning. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results. Answer: We know that, The angles that are opposite to each other when two lines cross are called “Vertical angles” Hence, from the given figure, We can conclude that the vertical angles are: ∠1 and ∠3; ∠2 and ∠4; ∠5 and ∠7; ∠6 and ∠8 b. Identify all the linear pairs of angles. Explain your reasoning. Communicate Your Answer Question 4. Question 5. The pair of lines that are different from the given pair of lines in Exploration 2 are: a. \(\overline{C D}\) and \(\overline{A E}\) b. \(\overline{D H}\) and \(\overline{F G}\) Hence, from the above, We can conclude that a. \(\overline{C D}\) and \(\overline{A E}\) are “Skew lines” because they are not intersecting and are non coplanar b. \(\overline{D H}\) and \(\overline{F G}\) are “Skew lines” because they are not intersecting and are non coplanar Lesson 3.1 Pairs of Lines and AnglesMonitoring Progress Question 1. Answer: From Example 1, We can observe that The line that passes through point F that appear skew to \(\overline{E H}\) is: \(\overline{F C}\) Question 2. Answer: Perpendicular Postulate: According to this Postulate, 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 Now, In Example 2, 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 Hence, from the above, We can conclude that we can use “Perpendicular Postulate” to show that \(\overline{A C}\) is not perpendicular to \(\overline{B F}\) Classify the pair of numbered angles. Question 3. Answer: The given figure is: We know that, The angles that have the same corner are called “Adjacent angles” Hence, from the above, We can conclude that ∠1 and ∠5 are the adjacent angles Question 4. Answer: The given figure is: We know that, The angles that have the opposite corners are called “Vertical angles” Hence, from the above, We can conclude that ∠2 and ∠7 are the “Vertical angles” Question 5. Answer: The given figure is: We know that, The angles that have the opposite corners are called “Vertical angles” Hence, from the above, We can conclude that ∠4 and ∠5 are the “Vertical angles” Exercise 3.1 Pairs of Lines and AnglesVocabulary and Core Concept Check Question 1. Question 2. ∠2 and ∠3 ∠4 and ∠5 ∠1 and ∠8 ∠2 and∠7 Answer: The given figure is: We know that, The angles that have the common side are called “Adjacent angles” The angles that are opposite to each other when 2 lines cross are called “Vertical angles” So, ∠2 and ∠3 are vertical angles ∠4 and ∠5 are adjacent angles ∠1 and ∠8 are vertical angles ∠2 and ∠7 are vertical angles Hence, from the above, We can conclude that ∠4 and ∠5 angle-pair do not belong with the other three Monitoring Progress and Modeling with Mathematics In Exercises 3 – 6, think of each segment in the diagram as part of a line. All the angles are right angles. Which line(s) or plane(s) contain point B and appear to fit the description? Question 3. line(s) parallel to . Answer: Question 4. Answer: We know that, The lines that are a straight angle with the given line and are coplanar is called “Perpendicular lines” So, From the given figure, We can conclude that the line that is perpendicular to \(\overline{C D}\) is: \(\overline{A D}\) and \(\overline{C B}\) Question 5. Answer: Question
6. In Exercises 7-10, Use the diagram. Question 7. Question 8. Question 9. Answer: Question 10. Answer: We know that, The lines that have an angle of 90° with each other are called “Perpendicular lines” Hence, From the figure, We can conclude that \(\overline{P R}\) and \(\overline{P O}\) are not perpendicular lines In Exercises 11-14, identify all pairs of angles of the given type. Question 11. corresponding Answer: Question
12. Question 13. Question
14. USING STRUCTURE Question 15. Question 16. Question 17. Question 18. ERROR ANALYSIS Question 19. Answer: Question
20. Answer: We know that, The “Perpendicular Postulate” states that 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. Hence, from the above, We can conclude that the given statement is not correct Question 21. a. The plane containing the floor of the treehouse is parallel to the ground. b. The lines containing the railings of the staircase, such as , are skew to all lines in the plane containing the ground. c. All the lines containing the balusters. such as , are perpendicular to the plane containing the floor of the treehouse. Answer: Question
22. Explanation: Hence, from the above, We can conclude that the third line does not need to be a transversal Question 23. Question 24. a. Which lines are parallel to ? Answer: We know that, The lines that do not intersect to each other and are coplanar are called “Parallel lines” Hence, from the above figure, We can conclude that the line parallel to \(\overline{N Q}\) is: \(\overline{M P}\) b. Which lines intersect ?Answer: We know that, The lines that are coplanar and any two lines that have a common point are called “Intersecting lines” Hence, from the above figure, 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 ?Answer: We know that, The lines that do not intersect or not parallel and non-coplanar are called “Skew lines” Hence, from the above figure, We can conclude that \(\overline{K L}\), \(\overline{L M}\), and \(\overline{L S}\) d. Should you have named all the lines on the cube in parts (a)-(c) except \(\overline{N Q}\)? Explain. In exercises 25-28. copy and complete the statement. List all possible correct answers. Question 25. Question
26. Question 27. Question
28. Question 29. Answer: Maintaining Mathematical Proficiency Use the diagram to find the measure of all the angles. Question 30. Question 31. 3.2 Parallel Lines and TransversalsExploration 1 Exploring parallel Lines Work with a partner: Use dynamic geometry software to draw two parallel lines. Draw a third line that intersects both
parallel lines. Find the measures of the eight angles that are formed. What can you conclude? Answer: By using the dynamic geometry, The representation of the given coordinate plane along with parallel lines is: Hence, from the coordinate plane, We can observe that, ∠3 = 53.7° and ∠4 = 53.7° We know that, The angle measures of the vertical angles are congruent So, ∠1 = 53.7° and ∠5 = 53.7° We know that, In the parallel lines, All the angle measures are equal Hence, from the above, We can conclude that ∠1 = ∠2 = ∠3 = ∠4 = ∠5 = ∠6 = ∠7 = 53.7° Exploration 2 Writing conjectures Work with a partner. Use the results of Exploration 1 to write conjectures about the following pairs of angles formed by two parallel lines and a transversal. Answer: We know that, When two lines are crossed by another line (which is called the Transversal), the angles in matching corners are called “Corresponding angles” Hence, from the given figure, We can conclude that The corresponding angles are: ∠ and ∠5; ∠4 and ∠8 b. alternate interior angles Answer: We know that, “Alternate Interior Angles” are a pair of angles on the inner side of each of those two lines but on opposite sides of the transversal. Hence, from the above figure, We can conclude that The alternate interior angles are: ∠3 and ∠5; ∠2 and ∠8 c. alternate exterior angles Answer: We know that, “Alternate exterior angles” are the pair of angles that lie on the outer side of the two parallel lines but on either side of the transversal line. Hence, from the above figure, We can conclude that The alternate exterior angles are: ∠1 and ∠7; ∠6 and ∠4 d. consecutive interior angles Answer: We know that, When two lines are cut by a transversal, the pair of angles on one side of the transversal and inside the two lines are called the “Consecutive interior angles” Hence, from the above figure, We can conclude that The consecutive interior angles are: ∠2 and ∠5; ∠3 and ∠8 Communicate Your Answer Question 3. Question 4. Lesson 3.2 Parallel Lines and TransversalsMonitoring Progress Use the diagram Question 1. To find ∠5: To find ∠8: Question 2. Question 3. Question
4. Exercise 3.2 Parallel Lines and TransversalsVocabulary and Core Concept Check Question 1. Question 2. m∠1 and m∠3 m∠2 and m∠4 m∠2 and m∠3 m∠1 and m∠5 Answer: The given figure is: From the given figure, ∠1 and ∠3 are the vertical angles ∠2 and ∠4 are the alternate interior angles ∠2 and ∠3 are the consecutive interior angles ∠1 and ∠5 are the alternate exterior angles So, From the above, We can observe that all the angles except ∠1 and ∠3 are the interior and exterior angles Hence, from the above, We can conclude that ∠1 and ∠3 pair does not belong with the other three Monitoring Progress and Modeling with Mathematics In Exercises 3-6, find m∠1 and m∠2. Tell which theorem you use in each case. Question 3. Answer: Question 4. Answer: The given figure is: From the given figure, We can observe that, ∠1 = ∠2 (By using the Vertical Angles theorem) ∠2 = 150° (By using the Alternate exterior angles theorem) Hence, from the above, We can conclude that ∠1 = ∠2 = 150° Question 5. Answer: Question 6. Answer: The given figure is: From the given figure, We can observe that, ∠1 + ∠2 = 180° (By using the consecutive interior angles theorem) ∠2 = 140° (By using the Vertical angles theorem) So, ∠1 = 180° – 140° ∠1 = 40° Hence, from the above, We can conclude that ∠1 = 40° and ∠2 = 140° In Exercises 7-10. find the value of x. Show your steps. Question 7. Answer: Question 8. Answer: The given figure is: From the given figure, We can observe that 72° + (7x + 24)° = 180° (By using the Consecutive interior angles theory) (7x + 24)° = 180° – 72° (7x + 24)° = 108° 7x° = 108° – 24° 7x° = 84° x° = \(\frac{84}{7}\) x° = 12° Hence, from the above, We can conclude that the value of x is: 12° Question 9. Answer: Question 10. Answer: The given figure is: From the given figure, We can observe that (8x + 6)° = 118° (By using the Vertical Angles theorem) 8x° = 118° – 6° 8x° = 112° x° = \(\frac{112}{8}\) x° = 14° Hence, from the above, We can conclude that the value of x is: 14° In Exercises 11 and 12. find m∠1, m∠2, and m∠3. Explain our reasoning. Question 11. Answer: Question 12. Answer: The given figure is: From the given figure, We can observe that ∠3 + 133° = 180° (By using the Consecutive Interior angles theorem) ∠3 = 180° – 133° ∠3 = 47° Now, We can observe that ∠2 + ∠3 = 180° ∠2 = 180° – ∠3 ∠2 = 180° – 47° ∠2 = 133° Now, We can observe that ∠1 = ∠2 Hence, from the above, We can conclude that ∠1 = ∠2 = 133° and ∠3 = 47° Question 13. Answer: Question 14. a. Name two pairs of congruent angles when \(\overline{A D}\) and \(\overline{B C}\) are parallel? Explain your reasoning? Answer: Let the congruent angle be ∠P So, From the figure, We can observe that the pair of angle when \(\overline{A D}\) and \(\overline{B C}\) are parallel is: ∠APB and ∠DPB b. Name two pairs of supplementary angles when \(\overline{A B}\) and \(\overline{D C}\) are parallel. Explain your reasoning. PROVING A THEOREM Question 15. Question 16. Question 17. Answer: Question 18. a. The measure of ∠1 is 70°. Find m∠2 and m∠3. b. Explain why ∠ABC is a straight angle. c. If m∠1 is 60°, will ∠ABC still he a straight angle? Will the opening of the box be more steep or less steep? Explain. Answer: Question 19. Question 20. MATHEMATICAL CONNECTIONS Question 21. Answer: Question 22. Answer: The given figure is: From the given figure, We can observe that the given pairs of angles are consecutive interior angles So, 2y° + 4x° = 180° (2x + 12)° + (y + 6)° = 180° 2x° + y° + 18° = 180° 2x° + y° = 180° – 18° 2x° + y° = 162°———(1) 4x° + 2y° = 180°——–(2) Solve eq. (1) and eq. (2) to get the values of x and y 2x° = 18° x° = \(\frac{18}{2}\) x° = 9° Now, y° = 162° – 2 (9°) y° = 162° – 18° y° = 144° Hence, from the above, We can conclude that the values of x and y are: 9° and 14° respectively Question 23. Answer: Question 24. Answer: It is given that ∠4 ≅∠5 and \(\overline{S E}\) bisects ∠RSF So, ∠FSE = ∠ESR From ΔESR, We know that, The sum of the angle measures of a triangle is: 180° So, ∠3 + ∠4 + ∠5 = 180° So, ∠3 = 60° (Since ∠4 ≅ ∠5 and the triangle is not a right triangle) From the given figure, We can observe that, ∠1 = ∠3 (By using the Corresponding angles theorem) So, ∠1 = 60° Hence, from the above, We can conclude that ∠1 = 60° Maintaining Mathematical Proficiency Write the converse of the conditional statement. Decide whether it is true or false. Question 25. Question 26. Question
27. Question 28. 3.3 Proofs with Parallel LinesExploration 1 Exploring Converses Work with a partner: Write the converse of each conditional statement. Draw a diagram to represent the converse. Determine whether the converse is true. Justify your conclusion. a. Corresponding Angles Theorem (Theorem 3.1): If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Converse: If the pairs of corresponding angles are congruent, then the two parallel lines are cut by a transversal. Answer: 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. Converse: If the pairs of alternate interior angles are congruent, then the two parallel lines are cut by a transversal. Answer: 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. Converse: If the pairs of alternate exterior angles are congruent, then the two parallel lines are cut by a transversal. Answer: 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. then the pairs of consecutive interior angles are supplementary. Converse: If the pairs of consecutive interior angles are supplementary, then the two parallel lines are cut by a transversal Answer: Communicate Your Answer Question 2. Question 3. Lesson 3.3 Proofs with Parallel LinesMonitoring Progress Question 1. Answer: Yes, there is enough information in the diagram to conclude m || n. Explanation: From the given figure, We can observe that the given angles are the consecutive exterior angles Now, We have to prove that m || n So, We will use “Converse of Consecutive Exterior angles Theorem” to prove m || n Proof of the Converse of the Consecutive Exterior angles Theorem: a. m∠1 + m∠8 = 180° //From the given statement b. m∠1 + m∠4 = 180° // Linear pair of angles are supplementary c. m∠5=m∠1 // (1), (2), transitive property of equality d. AB||CD // Converse of the Corresponding Angles Theorem The representation of the Converse of the Consecutive Interior angles Theorem is: Question 2. Question 3. Answer: The given figure is: It is given that the given angles are the alternate exterior angles Now, Alternate Exterior angle Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent The Converse of the Alternate Exterior Angles Theorem: The “Converse of the Alternate Exterior Angles Theorem” states that if alternate exterior angles of two lines crossed by a transversal are congruent, then the two lines are parallel. Hence, from the above, We can conclude that For the Converse of the alternate exterior angles Theorem, The given statement is: ∠1 ≅ 8 To prove: l || k Question 4. It is given that ∠4 ≅∠5. By the _______ . ∠1 ≅ ∠4. Then by the Transitive Property of Congruence (Theorem 2.2), _______ . So, by the _______ , g || h. Question 5. Answer: From the given figure, We can observe that not any step is intersecting at each other In the same way, when we observe the floor from any step, We can say that they are also parallel Hence, from the above, 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. Answer: The given figure is: From the figure, We can observe that the given angles are the consecutive exterior angles We know that, According to the Consecutive Exterior angles Theorem, ∠8 + 115° = 180° ∠8 = 180° – 115° ∠8 = 65° Hence, from the above, We can conclude that ∠8 = 65° Exercise 3.3 Proofs with Parallel LinesVocabulary and Core Concept Check Question 1. Question
2. Monitoring Progress and Modeling with Mathematics In Exercises 3-8. find the value of x that makes m || n. Explain your reasoning. Question 3. Answer: Question 4. Answer: The given figure is: From the given figure, We can observe that the given angles are the corresponding angles Now, According to Corresponding Angles Theorem, (2x + 15)° = 135° 2x° = 135° – 15° 2x° = 120° x° = \(\frac{120}{2}\) x° = 60° Hence, from the above, We can conclude that the value of x is: 60° Question 5. Answer: Question 6. Answer: The given figure is: From the given figure, We can observe that the given angles are the corresponding angles Now, According to Corresponding Angles Theorem, (180 – x)° = x° 180° = x° + x° 2x° = 180° x° = \(\frac{180}{2}\) x° = 90° Hence, from the above, We can conclude that the value of x is: 90° Question 7. Answer: Question 8. Answer: The given figure is: From the given figure, We can observe that the given angles are the corresponding angles Now, According to Corresponding Angles Theorem, (2x + 20)° = 3x° 20° = 3x° – 2x° x° = 20° Hence, from the above, We can conclude that the value of x is: 20° In Exercises 9 and 10, use a compass and straightedge to construct a line through point P that is parallel to line m. Question 9. Answer: Question 10. Answer: Let A and B be two points on line m. Draw \(\overline{A P}\) and construct an angle ∠1 on n at P so that ∠PAB and ∠1 are corresponding angles Hence, The representation of the complete figure is: PROVING A THEOREM Question 12. Now, a. m∠5 + m∠4 = 180° //From the given statement b. m∠1 + m∠4 = 180° // Linear pair of angles are supplementary c. m∠5=m∠1 // (1), (2), transitive property of equality d. AB||CD // Converse of the Corresponding Angles Theorem In Exercises 13-18. decide whether there is enough information to prove that m || n. If so, state the theorem you would use. Question 13. Answer: Question 14. Answer: Yes, there is enough information to prove m || n The theorem we can use to prove that m || n is: Alternate Exterior angles Converse theorem Question 15. Answer: Question 16. Answer: No, there is no enough information to prove m || n Question 17. Answer: Question 18. Answer: Yes, there is enough information to prove m || n The theorem we can use to prove that m || n is: Alternate Exterior angles Converse theorem ERROR ANALYSIS Question 19. Answer: Question 20. Answer: The given figure shows that angles 1 and 2 are Consecutive Interior angles It also shows that a and b are cut by a transversal and they have the same length So, From the converse of the Consecutive Interior angles Theorem, We can conclude that a || b In Exercises 21-24. are and parallel? Explain your reasoning.Question 21. Answer: Question 22. Answer: The given figure is: From the given figure, We can observe that The sum of the given angle measures is: 180° From the given figure, We can observe that the given angles are consecutive exterior angles So, From the Consecutive Exterior angles Converse, We can conclude that AC || DF Question 23. Answer: Question
24. Answer: The given figure is: From the given figure, We can observe that the sum of the angle measures of all the pairs i.e., (115 + 65)°, (115 + 65)°, and (65 + 65)° is not 180° Since, The sum of the angle measures are not supplementary, according to the Consecutive Exterior Angles Converse, AC is not parallel to DF Question 25. Answer: Question
26. Answer: When we observe the ladder, The rungs are not intersecting at any point i.e., they have different points We know that, The parallel lines do not have any intersecting points Hence, from the above, We can conclude that the top rung is parallel to the bottom rung Question 27. Answer: Question 28. Answer: m∠1 + m∠2 61 + 59 = 120 degrees m∠3 = 180 – (m∠1 + m∠2) m∠3 = 180 – 120 m∠3 = 60 m∠4 + m∠5 58 + 62 = 120 m∠6 = 180 – (m∠4 + m∠5) m∠6 = 180 – 120 m∠6 = 60 m∠3 = m∠6, this shows the corresponding angle between the lines AE and CH. EB is not parallel to line HD. Question 29. (A) Corresponding Angles Converse (Thm 3.5) (B) Alternate Interior Angles Converse (Thm 3.6) (C) Alternate Exterior Angles Converse (Thm 3.7) (D) Consecutive Interior Angles Converse (Thm 3.8) Answer: Question 30. Answer: It is given that the sides of the angled support are parallel and the support makes a 32° angle with the floor So, To make the top of the step where ∠1 is present to be parallel to the floor, the angles must be “Alternate Interior angles” We know that, The “Alternate Interior angles” are congruent So, ∠1 = 32° Hence, from the above, We can conclude that ∠1 = 32° Question 31. Answer: Question
32. From the above, The diagram can be changed by the transformation of transversals into parallel lines and a parallel line into transversal Hence, The diagram that represents the figure that it can be proven that the lines are parallel is: PROOF Question 33. Answer: Question 34. Answer: Given: ∠1 and ∠3 are supplementary Prove: m || n Hence, Question 35. Answer: Question 36. Answer: Given: a || b, ∠2 ≅ ∠3 Prove: c || d Hence, Question 37. Answer: Question 38. Answer: The given diagram is: From the given diagram, We can observe that ∠1 and ∠4; ∠2 and ∠3 are the pairs of corresponding angles We know that, According to the Converse of the Corresponding angles Theorem, If the corresponding angles are congruent, then the two lines that cut by a transversal are parallel lines Hence, We can conclude that p and q; r and s are the pairs of parallel lines Question 39. Question 40. a. Find the value of x that makes p || q. Answer: From the given figure, We can observe that when p || q, The angles are: (2x + 2)° and (x + 56)° We can observe that the given angles are corresponding angles Hence, (2x + 2)° = (x + 56)° 2x – x = 56° – 2° x° = 54° Hence, from the above, We can conclude that the value of x when p || q is: 54° b. Find the value of y that makes r || s. c. Can r be parallel to s and can p, be parallel to q at the same time? Explain your reasoning. Maintaining Mathematical Proficiency Question
41. Question 42. Question
43. Question 44. 3.1 – 3.3 Study Skills: Analyzing Your ErrorsMathematical Practices Question 1. Question 2. 3.1 – 3.3 QuizThink of each segment in the diagram as part of a line. Which lines(s) or plane(s) contain point G and appear to fit the description? Question 1. Answer: The line parallel to \(\overline{E F}\) is: \(\overline{D H}\) Question 2. Answer: The lines perpendicular to \(\overline{E F}\) are: \(\overline{F B}\) and \(\overline{F G}\) Question 3. Answer: The lines skew to \(\overline{E F}\) are: \(\overline{C D}\), \(\overline{C G}\), and \(\overline{A E}\) Question 4. Identify all pairs of angles of the given type. Question 5. Question 6. Question 7. Question 8. Find m∠1 and m∠2. Tell which theorem you use in each case. Question 9. Answer: The given figure is: From the given figure, By using the linear pair theorem, ∠1 + 138° = 180° ∠1 = 180° – 138° ∠1 = 42° Now, By using the Alternate Exterior Angles Theorem, ∠1 = ∠2 Hence, from the above, We can conclude that ∠1 = ∠2 = 42° Question 10. Answer: The given figure is: From the given figure, We can observe that By using the Vertical Angles Theorem, ∠2 = 123° Now, By using the vertical Angles Theorem, ∠1 = ∠2 Hence, from the above, We can conclude that ∠1 = ∠2 = 123° Question 11. Answer: The given figure is: From the given figure, By using the linear pair theorem, ∠1 + 57° = 180° ∠1 = 180° – 57° ∠1 = 123° Now, By using the consecutive interior angles theorem, ∠1 + ∠2 = 180° ∠2 = 180° – 123° ∠2 = 57° Hence, from the above, We can conclude that ∠1 = 123° and ∠2 = 57° Decide whether there is enough information to prove that m || n. If so, state the theorem you would use. Question 12. Answer: The given figure is: We know that, By using the “Consecutive Interior angles Converse”, If the angle measure of the angles is a supplementary angle, then the lines cut by a transversal are parallel Now, 69° + 111° = 180° Hence, from the above, We can conclude that m || n by using the Consecutive Interior angles Theorem Question 13. Answer: The given figure is: We know that, By using the Corresponding Angles Theorem, If the corresponding angles are congruent, then the lines cut by a transversal are parallel Hence, from the above, We can conclude that m || n by using the Corresponding Angles Theorem Question 14. Answer: The given figure is: From the given figure, It is given that l || m and l || n, So, We know that, By using the parallel lines property, If a || b and b || c, then a || c Hence, from the above, We can conclude that m || n Question 15. a. Explain why the tallest bar is parallel to the shortest bar. Answer: From the given bars, We can observe that there is no intersection between any bars If we represent the bars in the coordinate plane, we can observe that the number of intersection points between any bar is: 0 We know that, The number of intersection points for parallel lines is: 0 Hence, from the above, We can conclude that the tallest bar is parallel to the shortest bar b. Imagine that the left side of each bar extends infinitely as a line. Question 16. a. Identify two pairs of parallel lines so that each pair is in a different plane. Answer: From the given figure, We can observe that there are a total of 5 lines. Hence, The two pairs of parallel lines so that each pair is in a different plane are: q and p; k and m b. Identify two pairs of perpendicular lines. c.
Identify two pairs of skew line d. Prove that ∠1 ≅ ∠2. 3.4 Proofs with Perpendicular LinesExploration 1 Writing Conjectures Work with a partner: Fold a piece of pair in half twice. Label points on the two creases. as shown. a. Write a conjecture about \(\overline{A B}\) and \(\overline{C D}\). Justify your conjecture. Answer: The conjecture about \(\overline{A B}\) and \(\overline{c D}\) is: If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. b. Write a conjecture about \(\overline{A O}\) and \(\overline{O B}\) Justify your conjecture. Exploration 2 Exploring a segment Bisector Work with a
partner: Fold and crease a piece of paper. as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. Answer: b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? Exploration 3 Writing a conjecture Work with a partner. a. Draw \(\overline{A B}\), as shown. b. Draw an arc with center A on each side of AB. Using the same compass selling, draw an arc with center B on each side \(\overline{A B}\). Label the intersections of arcs C and D. c. Draw \(\overline{C D}\). Label its intersection with \(\overline{A B}\) as O. Write a conjecture about the resulting diagram. Justify your conjecture. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. Answer: The resultant diagram is: From the above diagram, We can conclude that The angles formed at all the intersection points are: 90° The lengths of the line segments are equal i.e., AO = OB and CO = OD Communicate Your Answer Question 4. Question 5. Lesson 3.4 Proofs with Perpendicular LinesMonitoring Progress Question 1. Answer: The given figure is: It is given that E is ⊥ to \(\overline{F H}\) So, To find the distance between E and \(\overline{F H}\), we need to find the distance between E and G i.e., EG Now, From the coordinate plane, E (-4, -3), G (1, 2) Compare the given points with E (x1, y1), G (x2, y2) So, EG = \(\sqrt{(x2 – x1)² + (y2 – y1)²}\) EG = \(\sqrt{(1 + 4)² + (2 + 3)²}\) EG = \(\sqrt{(5)² + (5)²}\) EG = \(\sqrt{50}\) EG = 7.07 Hence, from the above, We can conclude that the distance from point E to \(\overline{F H}\) is: 7.07 Question 2. Proof: Given: k || l, t ⊥ k Prove: t ⊥ l Alternate Exterior Angles Theorem: The Alternate Exterior Angles Theorem states that, when two parallel lines are cut by a transversal, the resulting alternate exterior angles are congruent Proof: Given: k || l Prove: ∠1 ≅ ∠7 and ∠4 ≅ ∠6 Since k || l, by the Corresponding Angles Postulate, ∠1 ≅ ∠5 Also, by the Vertical Angles Theorem, ∠5 ≅ ∠7 Then, by the Transitive Property of Congruence, ∠1 ≅ ∠7 You can prove that ∠4 and ∠6 are congruent using the same method. Use the lines marked in the photo. Question 3. Question 4. Exercise 3.4 Proofs with Perpendicular LinesVocabulary and core Concept Check Question 1. Question 2. Find the distance from point X to Answer: The given figure is: To find the distance from point X to \(\overline{W Z}\), We have to find the distance between X and Y i.e., XY Now, The given points are: X (-3, 3), Y (3, 1) Compare the given points with (x1, y1), (x2, y2) Now, XY = \(\sqrt{(x2 – x1)² + (y2 – y1)²}\) XY = \(\sqrt{(3 + 3)² + (3 – 1)²}\) XY = \(\sqrt{(6)² + (2)²}\) XY = 6.32 Hence, from the above, We can conclude that the distance from point X to \(\overline{W Z}\) is: 6.32 Find XZ Find the length of \(\overline{X Y}\) Find
the distance from line l to point X. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4. find the distance from point A to .Question 3. Answer: Question 4. Answer: To find the distance from point A to \(\overline{X Z}\), We have to find the distance between A and Y i.e., AY Now, The given points are: X (3, 3), Y (2, -1.5) Compare the given points with (x1, y1), (x2, y2) Now, XY = \(\sqrt{(x2 – x1)² + (y2 – y1)²}\) XY = \(\sqrt{(3 + 1.5)² + (3 – 2)²}\) XY = \(\sqrt{(4.5)² + (1)²}\) XY = 4.60 Hence, from the above, We can conclude that the distance from point A to \(\overline{X Z}\) is: 4.60 CONSTRUCTION Question 5. Answer: Question 6. Answer: The given figure is: Now, Using P as the center, draw two arcs intersecting with line m. Label the intersections as points X and Y. Using X and Y as centers and an appropriate radius, draw arcs that intersect. Label the intersection as Z. Draw \(\overline{P Z}\) Question
7. Answer: Question 8. Answer: The given figure is: Now, Using P as the center and any radius, draw arcs intersecting m and label those intersections as X and Y. Using X as the center, open the compass so that it is greater than half of XP and draw an arc. Using Y as the center and retaining the same compass setting, draw an arc that intersects with the first Label the point of intersection as Z. Draw \(\overline{P Z}\) CONSTRUCTION Question 9. Answer: Question 10. Answer: The given figure is: Now, Using a compass setting greater than half of AB, draw two arcs using A and B as centers Connect the points of intersection of the arcs with a straight line ERROR ANALYSIS Answer: Question 12. Answer: We know that, According to the Perpendicular Transversal theorem, The distance from the perpendicular to the line is given as the distance between the point and the non-perpendicular line So, From the given figure, The distance from point C to AB is the distance between point C and A i.e., AC Hence, from the above, We can conclude that the distance from point C to AB is: 12 cm PROVING A THEOREM Question 14. Proof: Given: k || l, t ⊥ k Prove: t ⊥ l PROOF Question 15. Answer: Question
16. Answer: We have to find the theorem that satisfies the given statement. We know that the given angles∠4, ∠5 are complementary thus, the theorem 3.10 justifies the statement which states that-If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. In Exercises 17-22, determine which lines, if any, must be parallel. Explain your reasoning. Question 17. Answer: Question 18. Answer: The given figure is: From the given figure, We can observe that a is perpendicular to both the lines b and c Hence, from the above, We can conclude that By using the Perpendicular transversal theorem, a is both perpendicular to b and c and b is parallel to c Question 19. Answer: Question 20. Answer: The given figure is; From the given figure, We can observe that a is perpendicular to d and b is perpendicular to c For parallel lines, we can’t say anything Hence, from the above, By using the Perpendicular transversal theorem, a is perpendicular to d and b isperpendicular to c Question 21. Answer: Question 22. Answer: The given figure is: From the given figure, We can observe that w ⊥ v and w⊥ y So, We can say that w and v are parallel lines by “Perpendicular Transversal Theorem” We can observe that z ⊥ x and w ⊥ z So, We can say that w and x are parallel lines by “Perpendicular Transversal theorem” Question 23. Answer: Question 24. Explanation: Hence, from the above, We can conclude that your friend is not correct Question 25. Answer: Question 26. Answer: From the given figure, We can observe that Point A is perpendicular to Point C We know that, According to Perpendicular Transversal Theorem, The distance between the perpendicular points is the shortest Hence, from the above, We can conclude that in order to jump the shortest distance, you have to jump to point C from point A Question 27. (B) (C) (D) (E) Answer: Question 28. Question 29. Answer: Question 30. Answer: Question 31. Question 32. Question 33. Maintaining Mathematical Proficiency Simplify the ratio. Question 34. Question 35. Question 36. Question 37. Identify the slope and the y-intercept of the line. Question 38. Question 39. Question 40. Question 41. 3.5 Equations of Parallel and Perpendicular LinesExploration 1 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. Use a graphing calculator to verify your answer. What is the relationship between the slopes? Answer: The given figure is: We know that, From the given figure, We can observe that the given lines are parallel lines Now, The equation for another line is: y = \(\frac{3}{2}\)x + c Substitute (0, 2) in the above equation So, 2 = 0 + c c = 2 So, The equation for another parallel line is: y = \(\frac{3}{2}\)x + 2 When we compare the given equation with the obtained equation, We can observe that the slopes are the same and the y-intercepts are different We know that, The lines that have the same slope and different y-intercepts are “Parallel lines” Hence, from the above, We can conclude that the parallel lines are: y = \(\frac{3}{2}\)x – 1 y = \(\frac{3}{2}\)x + 2 b. Answer: The given figure is: We know that, From the given figure, We can observe that the given lines are perpendicular lines The equation for another line is: y = \(\frac{3}{2}\)x + c We know that, The slope of perpendicular lines is: -1 So, m1m2 = -1 \(\frac{3}{2}\) . m2 = -1 So, m2 = –\(\frac{2}{3}\) So, y = –\(\frac{2}{3}\)x + c Substitute (0, 1) in the above equation So, 1 = 0 + c c = 1 So, The equation for another perpendicular line is: y = –\(\frac{2}{3}\)x + 1 When we compare the given equation with the obtained equation, We can observe that the product of the slopes are -1 and the y-intercepts are different We know that, The lines that have the slopes product -1 and different y-intercepts are “Perpendicular lines” Hence, from the above, We can conclude that the perpendicular lines are: y = \(\frac{3}{2}\)x – 1 y = –\(\frac{2}{3}\)x + 1 c. Answer: The given figure is: We know that, From the given figure, We can observe that the given lines are parallel lines Now, The equation for another line is: y = \(\frac{1}{2}\)x + c Substitute (2, -2) in the above equation So, -2 = \(\frac{1}{2}\) (2) + c -2 = 1 + c c = 2 – 1 c = -3 So, The equation for another parallel line is: y = \(\frac{1}{2}\)x – 3 When we compare the given equation with the obtained equation, We can observe that the slopes are the same and the y-intercepts are different We know that, The lines that have the same slope and different y-intercepts are “Parallel lines” Hence, from the above, We can conclude that the parallel lines are: y = \(\frac{1}{2}\)x + 2 y = \(\frac{1}{2}\)x – 3 d. Answer: The given figure is: We know that, From the given figure, We can observe that the given lines are perpendicular lines The equation for another line is: y = \(\frac{1}{2}\)x + c We know that, The slope of perpendicular lines is: -1 So, m1m2 = -1 \(\frac{1}{2}\) . m2 = -1 So, m2 = -2 So, y = -2x + c Substitute (2, -3) in the above equation So, -3 = -2 (2) + c -3 = -4 + c c = 4 – 3 c = 1 So, The equation for another perpendicular line is: y = -2x + 1 When we compare the given equation with the obtained equation, We can observe that the product of the slopes are -1 and the y-intercepts are different We know that, The lines that have the slopes product -1 and different y-intercepts are “Perpendicular lines” Hence, from the above, We can conclude that the perpendicular lines are: y = \(\frac{1}{2}\)x + 2 y = -2x + 1 e. Answer: The given figure is: We know that, From the given figure, We can observe that the given lines are parallel lines Now, The equation for another line is: y = -2x + c Substitute (0, -2) in the above equation So, -2 = 0 + c c = -2 So, The equation for another parallel line is: y = -2x – 2 When we compare the given equation with the obtained equation, We can observe that the slopes are the same and the y-intercepts are different We know that, The lines that have the same slope and different y-intercepts are “Parallel lines” Hence, from the above, We can conclude that the parallel lines are: y = -2x + 2 y = -2x – 2 f. Answer: The given figure is: We know that, From the given figure, We can observe that the given lines are perpendicular lines The equation for another line is: y = -2x + c We know that, The slope of perpendicular lines is: -1 So, m1m2 = -1 -2 . m2 = -1 So, m2 = \(\frac{1}{2}\) So, y = \(\frac{1}{2}\)x + c Substitute (4, 0) in the above equation So, 0 = \(\frac{1}{2}\) (4) + c 0 = 2 + c c = 0 – 2 c = -2 So, The equation for another perpendicular line is: y = \(\frac{1}{2}\)x – 2 When we compare the given equation with the obtained equation, We can observe that the product of the slopes are -1 and the y-intercepts are different We know that, The lines that have the slopes product -1 and different y-intercepts are “Perpendicular lines” Hence, from the above, We can conclude that the perpendicular lines are: y = \(\frac{1}{2}\)x – 2 y = -2x + 2 Exploration 2 Writing Equations of Parallel and Perpendicular Lines Work with a partner: Write the equations of the parallel or perpendicular lines. Use a graphing calculator to verify your answers. a. Answer: The given figure is: From the given graph, We can observe that The given lines are the parallel lines Now, The coordinates of the line of the first equation are: (-1.5, 0), and (0, 3) The coordinates of the line of the second equation are: (1, 0), and (0, -2) Compare the given coordinates with A (x1, y1), and B (x2, y2) We know that, Slope (m) = \(\frac{y2 – y1}{x2 – x1}\) Now, The slope of the line of the first equation is: m = \(\frac{3 – 0}{0 + 1.5}\) m = \(\frac{3}{1.5}\) m = 2 Now, We know that, The standard linear equation is: y = mx + c So, y = 2x + c We know that, For parallel lines, The slopes are the same but the y-intercepts are different Hence, The given parallel line equations are: y = 2x + c1 y = 2x + c2 b. Answer: The given figure is: From the given figure, We can observe that The given lines are perpendicular lines So, The coordinates of the line of the first equation are: (0, -3), and (-1.5, 0) The coordinates of the line of the second equation are: (-4, 0), and (0, 2) Compare the given coordinates with A (x1, y1), and B (x2, y2) We know that, Slope (m) = \(\frac{y2 – y1}{x2 – x1}\) Now, The slope of the line of the first equation is: m = \(\frac{0 + 3}{0 – 1.5}\) m = \(\frac{3}{-1.5}\) m = \(\frac{-30}{15}\) m = -2 Now, We know that, The standard linear equation is: y = mx + c So, y = -2x + c We know that, For perpediclar lines, The product of the slopes is -1 and the y-intercepts are different So, m1 × m2 = -1 -2 × m2 = -1 m2 = \(\frac{1}{2}\) Hence, The given perpendicular line equations are: y = -2x + c1 y = \(\frac{1}{2}\)x + c2 Communicate Your Answer Question 3. Question 4. Lesson 3.5 Equations of Parallel and Perpendicular LinesMonitoring Progress Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. Question 1. Question 2. Question 3. Answer: The given figure is: From the given figure, The coordinates of line a are: (0, 2), and (-2, -2) The coordinates of line b are: (2, 3), and (0, -1) The coordinates of line c are: (4, 2), and (3, -1) The coordinates of line d are: (-3, 0), and (0, -1) Now, Compare the given coordinates with (x1, y1), and (x2, y2) So, The slope of line a (m) = \(\frac{y2 – y1}{x2 – x1}\) = \(\frac{-2 – 2}{-2 – 0}\) = \(\frac{-4}{-2}\) = 2 The slope of line b (m) = \(\frac{y2 – y1}{x2 – x1}\) The slope of line c (m) = \(\frac{y2 – y1}{x2 – x1}\) The slope of
line d (m) = \(\frac{y2 – y1}{x2 – x1}\) Question 4. (b) perpendicular to the line y = 3x – 5. Question 5. Question 6. Question 7. Exercise 3.5 Equations of Parallel and Perpendicular LinesVocabulary and Core Concept Check Question 1. Question 2. Monitoring Progress and Modeling with Mathematics In Exercises 3 – 6. find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. Question 3. Question 4. Question 5. Question 6. In Exercises 7 and 8, determine which of the lines are parallel and which of the lines are perpendicular. Question 7. Answer: Question 8. Answer: The given figure is: From the given figure, The coordinates of line a are: (2, 2), and (-2, 3) The coordinates of line b are: (3, -2), and (-3, 0) The coordinates of line c are: (2, 4), and (0, -2) The coordinates of line d are: (0, 6), and (-2, 0) Now, Compare the given coordinates with (x1, y1), and (x2, y2) So, The slope of line a (m) = \(\frac{y2 – y1}{x2 – x1}\) = \(\frac{3 – 2}{-2 – 2}\) = \(\frac{1}{-4}\) = –\(\frac{1}{4}\) The slope of line b (m) = \(\frac{y2 – y1}{x2 – x1}\) The slope of line c (m) = \(\frac{y2 – y1}{x2 – x1}\) The slope of line d (m) = \(\frac{y2 – y1}{x2 – x1}\) In Exercises 9 – 12, tell whether the lines through the given points are parallel, perpendicular, or neither. justify your answer. Question 9. Question
10. Question 11. Question 12. In Exercises 13 – 16. write an equation of the line passing through point P that ¡s parallel to the given line. Graph the equations of the lines to check that they are parallel. Question 13. Question 14. Question 15. Question 16. In Exercises 17 – 20. write an equation of the line passing through point P that is perpendicular to the given line. Graph the equations of the lines to check that they are perpendicular. Question 17. Question
18. Question 19. Question 20. In Exercises 21 – 24, find the distance from point A to the given line. Question 21. Question 22. Question 23. Question 24. Question 25. Answer: Question 26. Answer: The given equation of the line is: y = 2x + 1 The given point is: (3, 4) We know that, The slopes of the parallel lines are the same Now, Compare the given equation with y = mx + c So, m = 2 So, The slope of the line that is parallel to the given line equation is: m = 2 So, The equation of the line that is parallel to the given equation is: y = 2x + c To find the value of c, Substitute (3, 4) in the above equation So, 4 = 2 (3) + c 4 – 6 = c c = -2 Hence, from the above, We can conclude that the equation of the line that is parallel to the given line is: y = 2x – 2 In Exercises 27-30. find the
midpoint of \(\overline{P Q}\). Then write Question 27. Question 28. Question 29. Question
30. Question 31. Answer: Question 32. Answer: The given figure is: From the above figure, The coordinates of the quadrilateral QRST is: Q (2, 6), R (6, 4), S (5, 1), and T (1, 3) Compare the given points with (x1, y1), and (x2, y2) Now, We know that, If both pairs of opposite sides of a quadrilateral are parallel, then it is a parallelogram So, If the slopes of the opposite sides of the quadrilateral are equal, then it is called as “Parallelogram” We know that, Slope (m) = \(\frac{y2 – y1}{x2 – x1}\) So, Slope of QR = \(\frac{4 – 6}{6 – 2}\) Slope of QR = \(\frac{-2}{4}\) Slope of QR = –\(\frac{1}{2}\) Slope of RS =
\(\frac{1 – 4}{5 – 6}\) Slope of ST = \(\frac{3 – 1}{1 – 5}\) Slope of TQ = \(\frac{3 – 6}{1 – 2}\) Question 33. REASONING Question 34. Answer: The given figure is: 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. So, From the figure, V = (-2, 3) We know that, The slopes of the parallel lines are the same So, By comparing the given equation with y = mx + c We get, m = 2 So, The equation of the line that is parallel to the line that represents the train tracks is: y = 2x + c Now, To find the value of c, Substitute (-2, 3) in the above equation So, 3 = 2 (-2) + x 3 + 4 = c c = 7 Hence, from the above, We can conclude that the equation of the line that is parallel to the line representing railway tracks is: y = 2x + 7 Question 35. Answer: Question 36. Answer: The given figure is: It is given that a gazebo is being built near a nature trail. An equation of the line representing the nature trail is y = \(\frac{1}{3}\)x – 4. Each unit in the coordinate plane corresponds to 10 feet So, It can be observed that 1 unit either in the x-plane or y-plane = 10 feet So, y = \(\frac{1}{3}\)x – 4 y = \(\frac{1}{3}\) (10) – 4 y = \(\frac{10 – 12}{3}\) y = –\(\frac{2}{3}\) We know that, The distance won’t be in negative value, y = \(\frac{2}{3}\) y = 0.66 feet Hence, from the above, We can conclude that the distance of the gazebo from the nature trail is: 0.66 feet Question 37. Question 38. Answer: The given figure is: From the given figure, The coordinates of a quadrilateral are: J (0 0), K (0, n), L (n, n), M (n, 0) Compare the given points with (x1, y1), and (x2, y2) We know that, Slope (m) = \(\frac{y2 – y1}{x2 – x1}\) So, Slope of JK = \(\frac{n – 0}{0 – 0}\) = Undefined Slope of KL = \(\frac{n – n}{n – 0}\) = 0 Slope of LM = \(\frac{0 – n}{n – n}\) = Undefined Slope of MJ = \(\frac{0 – 0}{n – 0}\) = 0 We know that, For a square, The slopes of perpendicular lines are undefined and 0 respectively Hence, from the above, We can conclude that quadrilateral JKLM is a square Question 39. Question 40. Explanation: Question 41. Question
42. MATHEMATICAL CONNECTIONS In Exercises 43 and 44, find a value for k based on the given description. Question 43. Question 44. Question 45. Question 46. PROVING A THEOREM Question
47. Question 48. Question 49. Question
50. Question 51. Maintaining Mathematical Proficiency Plot the point in a coordinate plane. Question 52. Question 53. Question 54. Question 55. Copy and complete the table. Question 56. Answer: The given table is: From the above table, The given equation is: y = x + 9 Hence, The completed table is: Question 57. Answer: 3.4 – 3.5 Performance Task: Navajo RugsMathematical Practices Question 1. Question 2. Question 3. Parallel and Perpendicular Lines Chapter Review3.1 Pairs of Lines and AnglesThink of each segment in the figure as part of a line. Which line(s) or plane(s) appear to fit the description? Question 1. line(s) perpendicular to Answer: We know that, The lines that are at 90° are “Perpendicular lines” Hence, From the above figure, The lines perpendicular to \(\overline{Q R}\) are: \(\overline{R M}\) and \(\overline{Q L}\) Question 2. Answer: We know that, The lines that do not have any intersection points are called “Parallel lines” Hence, From the above figure, The line parallel to \(\overline{Q R}\) is: \(\overline {L M}\) Question 3. Answer: We know that, The lines that do not intersect and are not parallel and are not coplanar are “Skew lines” Hence, From the above figure, The lines skew to \(\overline{Q R}\) are: \(\overline{J N}\), \(\overline{J K}\), \(\overline{K L}\), and \(\overline{L M}\) Question 4. 3.2 Parallel Lines and TransversalsFind the values of x and y. Question 5. Answer: The given figure is: From the given figure, We can observe that x and 35° are the corresponding angles We know that, By using the Corresponding Angles Theorem, x = 35° Now, We can observe that 35° and y are the consecutive interior angles So, 35° + y = 180° y = 180° – 35° y = 145° Hence, from the above, We can conclude that x° = 35° and y° = 145° Question 6. Answer: The given figure is: From the given figure, We can observe that 48° and y are the consecutive interior angles and y and (5x – 17)° are the corresponding angles So, By using the Consecutive Interior Angles Theorem, 48° + y° = 180° y° = 180° – 48° y° = 132° Now, By using the corresponding angles theorem, y° = (5x – 17)° 132° = (5x – 17)° 5x = 132° + 17° 5x = 149° x = \(\frac{149}{5}\) x° = 29.8° Hence, from the above, We can conclude that x° = 29.8° and y° = 132° Question 7. Answer: The given figure is: From the above figure, We can observe that 2x° and 2y° are the alternate exterior angles 2y° and 58° are the alternate interior angles So, 2x° = 2y° = 58° So, x° = y° =29° Hence, from the above, We can conclude that x° = y° = 29° Question 8. Answer: The given figure is: From the given figure, We can observe that (5y – 21)° and 116° are the corresponding angles So, (5y – 21)° = 116° 5y° = 116° + 21° 5y° = 137° y° = \(\frac{137}{5}\) y° = 27.4° Now, We can observe that (5y – 21)° ad (6x + 32)° are the alternate interior angles So, (5y – 21)° = (6x + 32)° 5 (28)° – 21° = (6x + 32)° 140 – 21 – 32 = 6x° 6x° = 140° – 53° 6x = 87° x = \(\frac{87}{6}\) x° = 14.5° Hence, from the above, We can conclude that x° = 14.5° and y° = 27.4° 3.3 Proofs with Parallel LinesFind the value of x that makes m || n. Question 9. Answer: The given figure is: We know that, m || n is true only when x and 73° are the consecutive interior angles according to the “Converse of Consecutive Interior angles Theorem” Now, It is given that m || n So, x + 73° = 180° x = 180° – 73° x = 107° Hence, from the above, We can conclude that the value of x is: 107° Question 10. Answer: The given figure is: We know that, m || n is true only when 147° and (x + 14)° are the corresponding angles by using the “Converse of the Alternate Exterior Angles Theorem” Now, It is given that m || n So, (x + 14)°= 147° x° = 147° – 14° x° = 133° Hence, from the above, We can conclude that the value of x is: 133° Question 11. Answer: The given figure is: m || n is true only when 3x° and (2x + 20)° are the corresponding angles by using the “Converse of the Corresponding Angles Theorem” Now, It is given that m || n So, (2x + 20)°= 3x° 3x° – 2x° = 20° x° = 20° Hence, from the above, We can conclude that the value of x is: 20° Question 12. Answer: The given figure is: We know that, m || n is true only when (7x – 11)° and (4x + 58)° are the alternate interior angles by the “Convesre of the Consecutive Interior Angles Theorem” Now, It is given that m || n So, (7x – 11)° = (4x + 58)° 7x° – 4x° = 58° + 11° 3x° = 69° x° = \(\frac{69}{3}\) x° = 23° Hence, from the above, We can conclude that the value of x is: 23° 3.4 Proofs with Perpendicular LinesDetermine which lines, if any, must be parallel. Explain your reasoning. Question 13. Answer: The given figure is: From the given figure, We can observe that x ⊥ z and y ⊥ z We know that, According to the Perpendicular Transversal Theorem, In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also. Hence, from the above, We can conclude that x and y are parallel lines Question 14. Answer: The given figure is: From the given figure, We can observe that w ⊥ y and z ⊥ x We can also observe that w and z is not both ⊥ to x and y We know that, According to the Perpendicular Transversal Theorem, In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also. Hence, from the above, We can conclude that there are not any parallel lines in the given figure Question 15. Answer: The given figure is: From the given figure, We can observe that m ⊥ a, n ⊥ a, l ⊥ b, and n ⊥ b We know that, According to the Perpendicular Transversal Theorem, In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also. Hence, from the above, We can conclude that m and n are parallel lines Question 16. Answer: The given figure is: From the given figure, We can observe that a ⊥ n, b ⊥ n, and c ⊥ m We know that, According to the Perpendicular Transversal Theorem, In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also. Hence, from the above, We can conclude that there are not any parallel lines in the given figure 3.5 Equations of Parallel and Perpendicular LinesWrite an equation of the line passing through the given point that is parallel to the given line. Question 17. Question 18. Question 19. Question 20. Write an equation of the line passing through the given point that is perpendicular to the given line. Question 21. Question 22. Question 23. Question 24. Find the distance front point A to the given line. Question 25. Question 26. Parallel and Perpendicular Lines TestFind the values of x and y. State which theorem(s) you used. Question 1. Answer: The given figure is: From the given figure, We can observe that x and 61° are the vertical angles 61° and y° are the alternate interior angles We know that, According to the “Alternate Interior Angles theorem”, the alternate interior angles are congruent According to the “Vertical Angles Theorem”, the vertical angles are congruent Hence, x° = y° = 61° Question 2. Answer: The given figure is: From the given figure, We can observe that 8x° and 96° are the alternate interior angles (11y + 19)° and 96° are the corresponding angles We know that, According to the “Alternate Interior Angles Theorem”, the alternate interior angles are congruent According to the “Corresponding Angles Theorem”, the corresponding angles are congruent So, 8x° = 96° x° = \(\frac{96}{8}\) x° = 12° Now, (11y + 19)° = 96° 11y° = 96° – 19° 11y° = 77° y° = \(\frac{77}{11}\) y° = 7° Hence, from the above, We can conclude that x° = 12° and y° = 7° Question 3. Answer: The given figure is: From the given figure, We can observe that 42° and 6(2y – 3)° are the consecutive interior angles 42° and (8x + 2)° are the vertical angles We know that, According to the “Consecutive Interior Angles Theorem”, the sum of the consecutive interior angles is 180° According to the “Vertical Angles Theorem”, the vertical angles are congruent So, 42° + 6 (2y – 3)° = 180° 6 (2y°) – 6(3)° = 180° – 42° 12y° – 18° = 138° 12y° = 138° + 18° 12y° = 156° y° = \(\frac{156}{12}\) y° = 13° Now, 42° = (8x + 2)° 8x° = 42° – 2° x° = 40° x° = \(\frac{40}{8}\) x° = 5° Hence, from the above, We can conclude that x° = 5° and y° = 13° Find the distance from point A to the given line. Question 4. Question 5. Find the value of x that makes m || n. Question
6. Answer: The given figure is: From the given figure, We can observe that x° and 97° are the corresponding angles We know that, According to the “Converse of the Corresponding Angles Theorem”, m || n is true only when the corresponding angles are congruent It is given that m || n So, x° = 97° Hence, from the above, We can conclude that x° = 97° Question 7. Answer: The given figure is: From the given figure, We can observe that 8x° and (4x + 24)° are the alternate exterior angles We know that, According to the “Converse of the Alternate Exterior Angles Theorem”, m || n is true only when the alternate exterior angles are congruent It is given that m || n So, 8x° = (4x + 24)° 8x° – 4x° = 24° 4x° = 24° x° = \(\frac{24}{4}\) x° = 6° Hence, from the above, We can conclude that x° = 6° Question 8. Answer: The given figure is: From the given figure, We can observe that (11x + 33)° and (6x – 6)° are the interior angles We know that, According to the “Converse of the Interior Angles Theory”, m || n is true only when the sum of the interior angles are supplementary It is given that m || n So, (11x + 33)°+(6x – 6)° = 180° 17x° + 27° = 180° 17x° = 180° – 27° x° = –\(\frac{153}{17}\) x° = 9° Hence, from the above, We can conclude that x° = 9° Write an equation of the line that passes through the given point and is Question 9. Question 10. Question 11. Answer: The given figure is: It is given that a student claimed that j ⊥ K, j ⊥ l We know that, According to the “Perpendicular Transversal Theorem”, In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also. So, From the above definition, The missing information the student assuming from the diagram is: The line l is also perpendicular to the line j Hence, from the above, We can conclude that the theorem student trying to use is the “Perpendicular Transversal Theorem” Question 12. Answer: You and your family are visiting some attractions while on vacation. You and your mom visit the shopping mall while your dad and your sister visit the aquarium. You decide to meet at the intersection of lines q and p. Each unit in the coordinate plane corresponds to 50 yards. a. Find an equation of line q. b. Find an equation of line p. c. What are the coordinates of the meeting point? d. What is the distance from the meeting point to the subway? Question 13. a. a pair of skew lines Answer: We know that, The “Skew lines” are the lines that are non-intersecting, non-parallel and non-coplanar Hence, From the given figure, We can conclude that the pair of skew lines are: \(\overline{A B}\) and \(\overline{G H}\) b. a pair of perpendicular lines c. a pair of paralIeI lines d. a pair of congruent corresponding angles e. a pair of congruent alternate interior angles Parallel and Perpendicular Lines Cumulative AssessmentQuestion 1. Answer: Step 1: Draw a line segment of any length and name that line segment as AB Step 2: Draw an arc by using a compass with above half of the length of AB by taking the center at A above AB Step 3: Draw another arc by using a compass with above half of the length of AB by taking the center at B above AB Step 4: Repeat steps 3 and 4 below AB Step 5: Draw a line segment CD by joining the arcs above and below AB Step 6: Measure the lengths of the midpoint of AB i.e., AD and DB. By measuring their lengths, we can prove that CD is the perpendicular bisector of AB Question 2. a. Use the numbers and symbols to create the equation of a line in slope-intercept form that passes through the point (4, – 5) and is parallel to the given line. Answer: The given line equation is: x + 2y = 10 The given point is: (4, -5) Now, The given equation in the slope-intercept form is: y = –\(\frac{1}{2}\)x + 5 We know that, The slopes of the parallel lines are the same So, The equation of the line that is parallel to the given line equation is: y = –\(\frac{1}{2}\)x + c To find the value of c, Substitute (4, -5) in the above equation So, -5 = –\(\frac{1}{2}\) (4) + c c = -5 + 2 c = -3 Hence, from the above, We can conclude that the line that is parallel to the given line equation is: y = –\(\frac{1}{2}\)x – 3 b. Use the numbers and symbols to create the equation of a line in slope-intercept form Question 3. Answer: The given figure is: From the given figure, We can conclude that 44° and 136° are the adjacent angles b. Answer: The given figure is: From the given figure, We can conclude that 18° and 23° are the adjacent angles c. Answer: The given figure is: From the given figure, We can conclude that 75° and 75° are alternate interior angles d. Answer: The given figure is: From the given figure, We can conclude that 42° and 48° are the vertical angles Question 4. a. What is the length of the field? Answer: It is given that a coordinate plane has been superimposed on a diagram of the football field where 1 unit is 20 feet. So, From the given figure, The length of the field = | 20 – 340 | = 320 feet Hence, from the above, We can conclude that the length of the field is: 320 feet b. What is the perimeter of the field? c. Turf costs $2.69 per square foot. Your school has a $1,50,000 budget. Does the school have enough money to purchase new turf for the entire field? Question 5. Given ∠1 ≅∠3 Prove ∠2 ≅∠4 Answer: The given table is: Hence, The completed table is: Question 6. Answer: Yes, I support my friend’s claim Explanation: From the given figure, We can observe that 141° and 39° are the consecutive interior angles We know that, According to the consecutive Interior Angles Theorem, 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 Hence, from the above, We can conclude that the claim of your friend can be supported Question 7. (A) are parallel. (B) intersect (C) are perpendicular (D) A, B, and C are noncollinear. Answer: We know that, The “Skew lines” are the lines that are not parallel, non-intersect, and non-coplanar Hene, from the given options, We can conclude that option D) is correct because parallel and perpendicular lines have to be lie in the same plane Question 8. ∠1 ∠2 ∠3 ∠4 ∠5 ∠6 ∠7 ∠8 a. ∠4 ≅ ________ b the Alternate Interior Angles Theorem (Thm. 3.2). Answer: From the given figure, We can conclude that By using the Alternate interior angles Theorem, ∠4 ≅ ∠5 b. ∠2 ≅ ________ by the Corresponding Angles Theorem (Thm. 3. 1) c. ∠1 ≅ ________ by the Alternate Exterior Angles Theorem (Thm. 3.3). d. m∠6 + m ________ = 180° by the Consecutive Interior Angles Theorem (Thm. 3.4). Question 9. Answer: It is given that you and your friend walk to school together every day. You meet at the halfway point between your houses first and then walk to school. Each unit in the coordinate plane corresponds to 50 yards. a. What are the coordinates of the midpoint of the line segment joining the two houses? Answer: From the given figure, We can conclude that the midpoint of the line segment joining the two houses is: M = (150, 250) b. What is the distance that the two of you walk together? |