9. By the Linear Pair Postulate (Post. 2.3):

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1 Chapter Maintaining Mathematical Proficiency. d = ( ) + (9 ) = ( ) + (6) = = d = (8 ( )) + ( 6 7) = (8 + ) + ( ) = () + ( ) = + 69 = d = (0 5) + (8 ( )) = ( 5) + (8 + ) = ( 5) + () = 5 + = 69 =. d = (9 ) + ( ) = ( ) + ( 5) = = What Did You Learn? (p. 5). G A C J B H. For part (a), I started by writing the equation (x + ) = (x + 56), because the angles represented by these two expressions are corresponding angles with respect to lines p and q. So, in order for lines p and q to be parallel by the Corresponding Angles Theorem (Thm..), the expressions must be equal to each other. For part (b), I started by writing the equation, (y + 7) + (y 7) = 80. In order for lines r and s to be parallel, the angles represented by these two expressions must be supplementary because each one forms a linear pair with one of the consecutive interior angles formed by lines r and s and transversal q... Quiz (p. 6). GH is parallel to EF.. GC is skew to EF.. FG is perpendicular to EF.. Plane GCB is parallel to plane ADF. Any three of the four points G, C, B, and F can be used to form the parallel plane. 5. and 5, and and 6 are consecutive interior angles. 6. and 6, and and 5 are alternate interior angles. 9. By the Linear Pair Postulate (Post..): m + 8 = 80 m = By the Alternate Exterior Angles Theorem (Thm..): m = m = 0. m = by the Corresponding Angles Theorem (Thm..). m = by the Vertical Angle Congruence Theorem (Thm..6).. By the Linear Pair Postulate (Post..8): m + 57 = 80 m = m = 57 by the Alternate Interior Angles Theorem (Thm..).. yes; Consecutive Interior Angles Converse (Thm..8) (69 + = 80 ). no. yes; Transitive Property of Parallel Lines (Thm..9) 5. a. All of the bars are parallel to each other by the Transitive Property of Parallel Lines (Thm..9). b. corresponds to by the Corresponding Angles Theorem (Thm..). So, m = m = a. Sample answer: q p and m k b. Sample answer: n m and n k c. Sample answer: Lines k and q are skew, and lines and m are skew. d. Because m k, by the Alternate Exterior Angles Theorem (Thm..).. Explorations (p. 7). a. AB CD ; AB is parallel to the horizontal edge of the paper because points A, O, and B are all the same distance from the edge. Similarly, CD is parallel to the vertical edge of the paper because points C, O, and D are the same distance from the edge. The horizontal and vertical edges form right angles in the corners. So, lines parallel to them will also be perpendicular. b. AO OB ; Point O must be the midpoint of AB because the paper was folded in half. So, AO and OB are congruent by definition of midpoints.. a. Check students work. b. They are all right angles. 7. and 5, and 6, and 7, and and 8 are corresponding angles. 8. and 8, and and 7 are alternate exterior angles. 80 Geometry Copyright Big Ideas Learning, LLC

2 Chapter. slope = 0 7 k = 7 k The slope of the given line is. The slope of the perpendicular line is. = 7 k (7 k) = 7 + k = k = 5 5. Using points A(, ) and B(6, 8), find the coordinates of point P that lies beyond point B along AB so that the ratio of AB to BP is to. In order to keep the ratio, AB BP =, solve this ratio for BP to get BP = AB. Next, find the rise and run from point A to point B. Leave the slope in terms of rise and run and do not simplify. m AB = 8 6 = 6 = rise run. Add of the run to the x-coordinate of B, which is + 6 = 8. Add of the rise to the y-coordinate of B, which is =. So, the coordinates of P are (8, ). 6. The slope of the perpendicular line is. y = x + b Use the y-intercept of y = x + 5, (0, 5). y = x + b 5 = 0 + b 5 = b y = x + 5 Find the intersection of the perpendicular lines y = x + 5 and y = x. x = x + 5 x = x + 0 5x = 0 x = y = = Find the distance between (0, 5) and (, ). distance = ( 0) + ( 5) = + ( ) = + = 5. units 7. If lines x and y are perpendicular to line z, then by the Slopes of Perpendicular Lines Theorem (Thm..), m x m z = and m y m z =. By the Transitive Property of Equality, m x m z = m y m z, and by the Division Property of Equality m x = m y. Therefore, by the Slopes of Parallel Lines Theorem (Thm..), x y. 9. If lines x and y are vertical lines and they are cut by any horizontal transversal, z, then x z and y z by Theorem.. Therefore, x y by the Lines Perpendicular to Transversal Theorem (Thm..). 50. If lines x and y are horizontal, then by definition m x = 0 and m y = 0. So, by the Transitive Property of Equality, m x = m y. Therefore, by the Slopes of Parallel Lines Theorem (Thm..), x y. 5. By definition, the x-axis is perpendicular to the y-axis. Let m be a horizontal line, and let n be a vertical line. Because any two horizontal lines are parallel, m is parallel to the x-axis. Because any two vertical lines are parallel, n is parallel to the y-axis. By the Perpendicular Transversal Theorem, (Thm.), n is perpendicular to the x-axis. Then, by the Perpendicular Transversal Theorem (Thm..), n is perpendicular to m. Maintaining Mathematical Proficiency 5. y 5. y y A(, 6) x C(5, 0) 5 x 5 x B(0, ) 55. y D(, ) x x 0 y = x = = = 9 x y = x = = x 0 y = x 7 0 x y = x = = 5 8. If x y and y z, then by the Slopes of Parallel Lines Theorem (Thm..), m x = m y and m y = m z. Therefore, by the Transitive Property of Equality, m x = m z. So, by the Slopes of Parallel Lines Theorem, (Thm..), x z. Copyright Big Ideas Learning, LLC Geometry 95

3 Chapter..5 What Did You Learn? (p. 6). You can find the distance between two lines only if the two lines are parallel. If you choose a point on one line and find the distance from that point to the other line, the answer will always be the same when the lines are parallel. But if the lines are not parallel, the answer will be different for every point on the line.. After drawing the perpendicular lines going through each endpoint of the given segment, you could pick an arbitrary point on one of the perpendicular lines. Then set the compass to the distance from this point to the corresponding endpoint, and use the same compass setting to mark a point on the other perpendicular line that is the same distance from the other endpoint. Connect these two points to construct the fourth segment of the rectangle. This segment should be congruent and parallel to the original segment and perpendicular to the other two constructed segments.. Because the distance from your house to the school is one-fourth of the distance from the school to the movie theater, you have to use five congruent segments. Four of the segments are between your school and the movie theater and one is between your house and your school. Chapter Review (pp. 6 66). All angles may or may not be right angles, and lines that appear perpendicular to QR are QL, RM, QP, and RN.. The lines that appear parallel to QR are JK, ML, and NP.. The lines that appear skew to QR are KP, KL, JN, and JM.. The plane that appears parallel to LMQ is plane JKPN, which can be defined by any three of these four vertices x = 80 Definition of supplementary angles x = 5 y = 5 Corresponding Angles Theorem (Thm..) The values are x = 5 and y = y = 80 Consecutive Interior Angles Theorem (Thm..) y = (5x 7) = 8 Alternate Interior Angles Theorem (Thm..) 5x = 65 x = The values are x = and y = x = 80 Definition of supplementary angles x = x = 6 y = 58 Corresponding Angles Theorem (Thm..) y = 9 The values are x = 6 and y = (6x + ) = 6 Alternate Exterior Angles Theorem (Thm..) 6x = 8 x = (5y ) + (6x + ) = 80 Definition of supplementary angles 5y = 80 5y = 80 5y + 95 = 80 5y = 85 y = 7 The values are x = and y = By the Consecutive Interior Angles Converse (Thm..8), m n when the marked angles are supplementary. x + 7 = 80 x = 07 The lines are parallel when x = By the Alternate Exterior Angles Converse (Thm..6), m n when the marked angles are congruent. 7 = (x + ) = x The lines are parallel when x =.. Use the Vertical Angles Congruence Theorem (Thm..6) and the Consecutive Interior Angles Converse (Thm..8). x + (x + 0) = 80 5x + 0 = 80 5x = 60 x = The lines are parallel when x =.. Use the Corresponding Angles Converse (Thm..5). (7x ) = (x + 58) x = 69 x = The lines are parallel when x =.. x y; Because x z and y z, lines x and y are parallel by the Lines Perpendicular to a Transversal Theorem (Thm..). 96 Geometry Copyright Big Ideas Learning, LLC

4 Chapter. none; The only thing that can be concluded in this diagram is that x z and w y. In order to say that lines are parallel, you need to know something about both of the intersections between the two lines and a transversal. 5. m n, a b; Because a n and b n, lines a and b are parallel by the Lines Perpendicular to a Transversal Theorem (Thm..). Because m a and n a, lines m and n are parallel by the Lines Perpendicular to a Transversal Theorem (Thm..). Because b and n b, lines and n are parallel by the Lines Perpendicular to a Transversal Theorem (Thm..). Because n and m n, lines and m are parallel by the Transitive Property of Parallel Lines (Thm..9). 6. a b; Because a n and b n, lines a and b are parallel by the Lines Perpendicular to a Transversal Theorem (Thm..). 7. The slope of the parallel line is. y = x + b = + b = b Because m = and b =, an equation of the line is y = x. 8. The slope of the parallel line is. x + b 5 = ( 6) + b 5 = + b 8 = b Because m = and b = 8, an equation of the line is x The slope of the parallel line is. y = x + b 0 = + b 0 = 6 + b 6 = b Because m = and b = 6, an equation of the line is y = x 6.. The slope of the perpendicular line is. x + b = 6 + b = + b = b Because m = and b =, the equation of the line is x.. The slope of the perpendicular line is. y = x + b = 0 + b = 0 + b = b Because m = and b =, an equation of the line is y = x +.. The slope of the perpendicular line is. y = x + b = 8 + b = + b = b Because m = and b =, an equation of the line is y = x +.. The slope of the perpendicular line is 7. y = 7x + b 5 = 7 ( ) + b 5 = 7 + b = b Because m = 7 and b =, an equation of the line is y = 7x. 0. The slope of the parallel line is. x + b = + b = + b = b Because m = and b =, an equation of the line is x. Copyright Big Ideas Learning, LLC Geometry 97

5 Chapter 5. The slope of the perpendicular line is. y = x + b = + b = + b = b The line perpendicular to y = x + is y = x. Find the point of intersection. y = x + Equation y = x Equation x + = x x = 7 x = 7 = 7 7 = 7 6 = So, the perpendicular lines intersect at ( 7, ) Find the distance from ( 7, to (, ). distance = ( 7 ) + ( ) ( ) = ( 7 ) + ( + ) = ( ) + ( ) 9 = + 9 = 8. units 6. The slope of the perpendicular line is. y = x + b = ( ) + b = + b = b The line perpendicular to x + is y = x. Find the point of intersection. y = x Equation x + Equation x = x + x = x + 5x = 5x = 5x 5 = 5 x = 5 = 5 ). 5) Find the distance from ( 5, to (, ). distance = ( 5 ) ( ) + ( 5 ) = ( 5 ) + + ( ) = ( ) + ( 5 ) = ( 6 5 ) 5) + ( 6 = = units Chapter Test (p. 67). x = 6 by the Vertical Angles Congruence Theorem (Thm..6); y = 6 by the Alternate Exterior Angles Theorem (Thm..).. 8x = 96 Corresponding Angles Theorem (Thm..) x = 96 + (y + 7) = 80 Linear Pair Postulate (Post..8) y + 0 = 80 y = 77 y = 7. (8x + ) = Alternate Interior Angles Theorem (Thm..) 8x = 0 x = 5 + [6(y )] = 80 Consecutive Interior Angles Theorem (Post..) + y 8 = 80 y + = 80 y = 56 y = y = ( 5) = = 5 So, the perpendicular lines intersect at ( 5, 5). 98 Geometry Copyright Big Ideas Learning, LLC

6 Chapter. The slope of y = x is, so the line perpendicular to y = x will have a slope of. y = x + b = + b = b The line perpendicular to y = x is y = x +. Find the point of intersection. y = x Equation y = x + Equation x = x + x = x = x = y = ( = ) So, the perpendicular lines intersect at (, Find the distance from (, ) to (, ). distance = ( ( ) ) + ( ) = ( 6 + ) + ( 8 ) = ( ) 7 + ( ) 7 = = units 5. The slope of x is, so the line perpendicular to x will have a slope of. y = x + b 7 = ( ) + b 7 = 9 + b = b ). The line perpendicular to x is y = x. Find the point of intersection. y = x Equation x Equation x = x 9x 6 = x 6 0x 6 = 6 0x = 0 x = 0 y = ( 0 ) = So, the perpendicular lines intersect at (0, ). Find the distance from (, 7) to (0, ). distance = ( 0) + (7 ( )) = ( ) + (9) = = units 6. x = 97 by the Corresponding Angles Converse (Thm..5). 7. 8x = (x + ) Alternate Exterior Angles Converse (Thm..7) x = x = 6 8. Use the Vertical Angles Congruence Theorem (Thm..6) and the Consecutive Interior Angles Converse (Thm..8). (x + ) + (6x 6) = 80 7x + 7 = 80 7x = 5 x = 9 9. a. The slope of the parallel line is. y = x + b = ( 5) + b = 0 + b = b Because m = and b =, an equation of the parallel line is y = x +. b. The slope of the perpendicular line is. y = x + b = ( 5) + b = 5 + b = 5 + b = b = b Because m = and b =, an equation of the perpendicular line is y = x. Copyright Big Ideas Learning, LLC Geometry 99

7 Chapter 0. a. The slope of the parallel line is. y = x + b 9 = ( ) + b 9 = + b 7 = + b 8 = b 8 = b Because m = and b = 8, an equation of the parallel line is y = x 8. b. The slope of the perpendicular line is. y = x + b 9 = ( ) + b 9 = + b 6 = b Because m = and b = 6, an equation of the perpendicular line is y = x 6.. The student assumes k and is trying to use the Perpendicular Transversal Theorem (Thm..).. a. Line q passes through the points (00, 50) and (00, 50) The slope of line q is = =. y = mx + b y = x + b 50 = (00) + b 50 = b 650 = b The equation of line q is y = x b. Line p passes through the points (0, 50) and (50, 00) The slope of line p is 50 0 = =. y = mx + b x + b 50 = (0) + b 50 = b The equation of line p is x +50. c. Find the point of intersection of line q and line p. y = x Equation x + 50 Equation x = x x = x x = 50 0x = 500 x = 50 y = (50) = 00 The coordinates (50, 00) represent the meeting point. d. Find the distance from the meeting point, (50, 00), to the subway, (50, 00). distance = (50 50) + (00 00) = = 00,000 6 The distance from the meeting point and the subway is about 6 yards. AB and LM because the lines are non-intersecting, non-coplanar, and non-parallel.. a. Sample answer: A pair of skew lines is b. Sample answer: A pair of perpendicular lines is IJ because the lines intersect at a right angle. c. A pair of paralel lines is EF and CD and EF because the lines are perpendicular to the same transversal. d. A pair of congruent corresponding angles is and because the angles are corresponding and CD EF. e. A pair of congruent alternate interior angles is and because the angles are alternate interior and CD EF. Chapter Standards Assessment (pp ). Every point on the red arc in Step is the same distance from point A. Because the same compass setting is used, every point on the red arc in Step is the same distance from point B as all of the points in the blue arc are from point A. Also, CD AB because the shortest distance from a point to a line is the perpendicular segment that connects the point to that line. So, points C and D and every point on CD are equidistant from points A and B, which means that M is the midpoint of AB by definition.. x + y = 0 y = x + 0 y = x + 5 a. The slope of the parallel line is. y = x + b 5 = () + b 5 = + b = b Because m = and b =, an equation of the parallel line is y = x. b. The slope of the perpendicular line is. y = x + b = () + b = + b 5 = b Because m = and b = 5, an equation of the perpendicular line is y = x Geometry Copyright Big Ideas Learning, LLC

8 Chapter. a. The angles are supplementary angles because + 6 = 80. b. The angles are adjacent angles because they have a common side and a common vertex. c. The angles are vertical angles because they are non-adjacent and share a common vertex. d. The angles are complementary angles because + 8 = 90.. a. The length of the field is 60 feet. b. The perimeter of the field is = 00 feet. c. The area of the field is = 57,600 square feet. The cost at $.69 per square foot is 57, = $5,9. Because this is greater than $50,000, the school does not have enough money. 9. a. Friend s house: (50, 00), your house: (50, 00) midpoint = (, ) = ( 00, 500 ) = (50, 50) The midpoint of the line segment joining the two houses is (50, 50). b. School: (00, 00) Find the distance from the midpoint to the school. distance = (00 50) + (00 50) = = 65, yd You and your friend walk about 55 yards together. 5. Given Prove STATEMENTS REASONS.. Given.. Vertical Angles Congruence Theorem (Thm..6).. Transitive Property of Congruence.. Vertical Angles Congruence Theorem (Thm..6) Transitive Property of Congruence 6. yes; Because + 9 = 80, the marked angles are supplementary. They are consecutive interior angels, so m n by the Consecutive Angels Converse (Thm..5). 7. D; Skew lines are lines that are non-coplanar, non-intersecting, and non-parallel. 8. a. 5 by the Alternate Interior Angles Theorem (Thm..). b. 6 by the Corresponding Angles Theorem (Thm..). c. 8 by the Alternate Exterior Angles Theorem (Thm..). d. m 6 + m = 80 by the Consecutive Interior Angles Theorem (Thm..). Copyright Big Ideas Learning, LLC Geometry 0

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