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2 ALGEBRA Find x and y so that the quadrilateral is a your answer with the method indicated. 6. A( 2, 4), B(5, 4), C(8, 1), D( 1, 1); Slope Formula 4. Opposite angles of a parallelogram are congruent. So, and. Solve for x. Solve for y. x = 11, y = 14 Since the slope of slope of, ABCD is not a 5. Opposite sides of a parallelogram are congruent. So, and. Solve for x. No; both pairs of opposite sides must be parallel; since the slope of slope of, ABCD is not a Solve for y. x = 4, y = 8 COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a Justify esolutions Manual - Powered by Cognero Page 2

3 7. W( 5, 4), X(3, 4), Y(1, 3), Z( 7, 3); Midpoint Formula Yes; the midpoint of is or. The midpoint of is or. So the midpoint of. By the definition of midpoint, diagonals bisect each other, WXYZ is a. Since the 8. Write a coordinate proof for the statement: If a quadrilateral is a parallelogram, then its diagonals bisect each other. Begin by positioning parallelogram ABCD on the coordinate plane so A is at the origin and the figure is in the first quadrant. Let the length of each base be a units so vertex B will have the coordinates (a, 0). Let the height of the parallelogram be c. Since D is further to the right than A, let its coordinates be (b, c) and C will be at (b + a, c). Once the parallelogram is positioned and labeled, use the midpoint formula to determine whether the diagonals bisect each other. Given: ABCD is a Prove: bisect each other. Yes; the midpoint of. By the definition of midpoint,. Since the diagonals bisect each other, WXYZ is a midpoint of midpoint of so bisect each other. by definition of midpoint Given: ABCD is a Prove: bisect each other. midpoint of midpoint of so bisect each other. by definition of midpoint esolutions Manual - Powered by Cognero Page 3

4 CCSS ARGUMENTS Determine whether each quadrilateral is a Justify your answer. 9. Yes; both pairs of opposite sides are congruent, which meets the conditions stated in Theorem 6.9. No other information is needed. Yes; both pairs of opp. sides are. 12. No; none of the tests for are fulfilled. We know that one pair of opposite sides are congruent and one diagonal bisected the second diagonal of the quadrilateral. These do not meet the qualifications to be a No; none of the tests for are fulfilled Yes; one pair of opposite sides are parallel and congruent. From the figure, one pair of opposite sides has the same measure and are parallel. By the definition of congruence, these segments are congruent. By Theorem 6.12 this quadrilateral is a Yes; one pair of opp. sides is. No; none of the tests for are fulfilled. Only one pair of opposite sides have the same measure. We don't know if they are parallel Yes; the diagonals bisect each other. By Theorem 6.11 this quadrilateral is a Yes; the diagonals bisect each other. No; none of the tests for are fulfilled. Consecutive angles are supplementary but no other information is given. Based on the information given, this is not a No; none of the tests for are fulfilled. No; none of the tests for are fulfilled. esolutions Manual - Powered by Cognero Page 4

5 15. PROOF If ACDH is a parallelogram, B is the midpoint of, and F is the midpoint of, write a flow proof to prove that ABFH is a 16. PROOF If WXYZ is a parallelogram,, and M is the midpoint of, write a paragraph proof to prove that ZMY is an isosceles triangle. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ACDH is a parallelogram, B is the midpoint of and F is the midpoint of. You need to prove that ABFH is a Use the properties that you have learned about parallelograms and midpoints to walk through the proof. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given WXYZ is a parallelogram,, and M is the midpoint of. You need to prove that ZMY is an isosceles triangle. Use the properties that you have learned about parallelograms, triangle congruence, and midpoints to walk through the proof. Given: WXYZ is a parallelogram, is the midpoint of. Prove: ZMY is an isosceles triangle. Since WXYZ is a parallelogram,, and M is the midpoint of, so. It is given that, so by SAS. By CPCTC,. So, ZMY is an isosceles triangle, by the definition of an isosceles triangle.. M Opp. sides are. Opp. sides are. Given: WXYZ is a parallelogram, is the midpoint of. Prove: ZMY is an isosceles triangle. Since WXYZ is a parallelogram,, and M is the midpoint of, so. It is given that, so by SAS. By CPCTC,. So, ZMY is an isosceles triangle, by the definition of an isosceles triangle. 17. REPAIR Parallelogram lifts are used to elevate large vehicles for maintenance. In the diagram, ABEF and BCDE are parallelograms. Write a twocolumn proof to show that ACDF is also a. M esolutions Manual - Powered by Cognero Page 5

6 You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABEF and BCDE are parallelograms. You need to prove that ACDH is a Use the properties that you have learned about parallelograms to walk through the proof. Given: ABEF is a parallelogram; BCDE is a Prove: ACDF is a 18. Statements (Reasons) 1. ABEF is a parallelogram; BCDE is a (Given) 2. (Def. of ) 3. (Trans. Prop.) 4. ACDF is a (If one pair of opp. sides is, then the quad. is a.) ALGEBRA Find x and y so that the quadrilateral is a Opposite sides of a parallelogram are congruent. Solve for x. Solve for y. Statements (Reasons) 1. ABEF is a parallelogram; BCDE is a (Given) 2. (Def. of ) 3. (Trans. Prop.) 4. ACDF is a (If one pair of opp. sides is, then the quad. is a.) x = 2, y = 29 Given: ABEF is a parallelogram; BCDE is a Prove: ACDF is a esolutions Manual - Powered by Cognero Page 6

7 Opposite sides of a parallelogram are congruent. Solve for x. Opposite angles of a parallelogram are congruent. Alternate interior angles are congruent. Solve for y. Use substitution. x = 8, y = 9 Substitute in. x = 30, y = 15.5 esolutions Manual - Powered by Cognero Page 7

8 ALGEBRA Find x and y so that the quadrilateral is a 21. Diagonals of a parallelogram bisect each other. So, and. Solve for y. 22. Opposite angles of a parallelogram are congruent. So,. We know that consecutive angles in a parallelogram are supplementary. So, Solve for x. Substitute in. Substitute in So, or 40. x = 11, y = 7 x = 40, y = 20 esolutions Manual - Powered by Cognero Page 8

9 23. Opposite sides of a parallelogram are congruent. So, and. Solve for x in terms y. Substitute in. Since both pairs of opposite sides are parallel, ABCD is a Substitute in to solve for x. So, x = 4 and y = 3. x = 4, y = 3 COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a Justify your answer with the method indicated. 24. A( 3, 4), B(4, 5), C(5, 1), D( 2, 2); Slope Formula Yes; slope of slope of. So,. Slope of Slope of. So,. Since both pairs of opposite sides are parallel, ABCD is a esolutions Manual - Powered by Cognero Page 9

10 25. J( 4, 4), K( 3, 1), L(4, 3), M(3, 3); Distance Formula Since the pairs of opposite sides are not congruent, JKLM is not a Since the slope of slope of and the slope of slope of, VWXY is not a No; both pairs of opposite sides must be congruent. The distance between K and L is between L and M is and J is. The distance. The distance between M. The distance between J and K is. Since, both pairs of opposite sides are not congruent, JKLM is not a No; a pair of opposite sides must be parallel and congruent. Slope of, slope of, slope of, and slope of. Since the slope of 26. V(3, 5), W(1, 2), X( 6, 2), Y( 4, 7); Slope Formula slope of and the slope of slope of, VWXY is not a esolutions Manual - Powered by Cognero Page 10

11 Yes; a pair of opposite sides must be parallel and congruent. Slope of slope of, so. so,. So, QRST is a 27. Q(2, 4), R(4, 3), S( 3, 6), T( 5, 1); Distance and Slope Formulas Slope of slope of, so. Since QR = ST,. So, QRST is a 28. Write a coordinate proof for the statement: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given and you need to prove that ABCD is a Use the properties that you have learned about parallelograms to walk through the proof. Given: Prove: ABCD is a parallelogram esolutions Manual - Powered by Cognero Page 11

12 slope of. The slope of is 0. slope of.the slope of is 0. Therefore,. So by definition of a parallelogram, ABCD is a Given: Prove: ABCD is a parallelogram slope of. The slope of is undefined. slope of. The slope of is undefined. Therefore,. So, are right angles. slope of. The slope of is 0. slope of.the slope of is 0. Given: ABCD is a is a right angle. Prove: are right angles. Therefore,. So by definition of a parallelogram, ABCD is a 29. Write a coordinate proof for the statement: If a parallelogram has one right angle, it has four right angles. Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are B(0, b), C(a, b), and D(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given that ABCD is a parallelogram and is a right angle. You need to prove that rest of the angles in ABCD are right angles. Use the properties that you have learned about parallelograms to walk through the proof. Given: ABCD is a is a right angle. Prove: are right angles. slope of. The slope of is undefined. slope of. The slope of is undefined. Therefore,. So, are right angles. 30. PROOF Write a paragraph proof of Theorem You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given. You need to prove that ABCD is a Use the properties that you have learned about esolutions Manual - Powered by Cognero Page 12

13 parallelograms and angles and parallel lines to walk through the proof. Given: Prove: ABCD is a Opposite sides are parallel, so ABCD is a 31. PANTOGRAPH A pantograph is a device that can be used to copy an object and either enlarge or reduce it based on the dimensions of the pantograph. Draw to form two triangles. The sum of the angles of one triangle is 180, so the sum of the angles for two triangles is 360. So,. Since. By substitution,. So,. Dividing each side by 2 yields angles are supplementary and. So, the consecutive. Likewise,. So, these consecutive angles are supplementary and. Opposite sides are parallel, so ABCD is a Given: Prove: ABCD is a Draw to form two triangles. The sum of the angles of one triangle is 180, so the sum of the angles for two triangles is 360. So,. Since. By substitution,. So,. Dividing each side by 2 yields angles are supplementary and. So, the consecutive. Likewise,. So, these consecutive angles are supplementary and. a. If, write a paragraph proof to show that. b. The scale of the copied object is the ratio of CF to BE. If AB is 12 inches, DF is 8 inches, and the width of the original object is 5.5 inches, what is the width of the copy? a. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given. You need to prove that BCDE is a Use the properties that you have learned about parallelograms to walk through the proof. Given: Prove: BCDE is a We are given that. AC = CF by the definition of congruence. AC = AB + BC and CF = CD + DF by the Segment Addition Postulate and AB + BC = CD + DF by substitution. Using substitution again, AB + BC = AB + DF, and BC = DF by the Subtraction Property. by the definition of congruence, and by the Transitive Property. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram, so BCDE is a By the definition of a parallelogram,. b. The scale of the copied object is. BE = CD. So, BE = 12. CF = CD + DF. = esolutions Manual - Powered by Cognero Page 13

14 = 20 Therefore,. Write a proportion. Let x be the width of the copy.. Solve for x. 12x = 110 x 9.2 The width of the copy is about 9.2 in. a. Given: Prove: BCDE is a We are given that. AC = CF by the definition of congruence. AC = AB + BC and CF = CD + DF by the Segment Addition Postulate and AB + BC = CD + DF by substitution. Using substitution again, AB + BC = AB + DF, and BC = DF by the Subtraction Property. by the definition of congruence, and by the Transitive Property. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram, so BCDE is a By the definition of a parallelogram,. b. about 9.2 in. PROOF Write a two-column proof. 32. Theorem 6.11 You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given. You need to prove that ABCD is a Use the properties that you have learned about parallelograms and triangle congruence to walk through the proof Given: Prove: ABCD is a Statements (Reasons) 1. (Given) 2. (Vertical.) 3. (SAS) 4. (CPCTC) 5. ABCD is a (If both pairs of opp. sides are, then quad is a.) Given: Prove: ABCD is a Statements (Reasons) 1. (Given) 2. (Vertical.) 3. (SAS) 4. (CPCTC) 5. ABCD is a (If both pairs of opp. sides are, then quad is a.) esolutions Manual - Powered by Cognero Page 14

16 ACBE. CCSS REASONING Name the missing coordinates for each 36. From the x-coordinates of W and X, has a length of a units. Since X is on the x-axis it has coordinates (a, 0). Since horizontal segments are parallel, the endpoints of have the same y- coordinate, c. The distance from Z to Y is the same as WX, a units. Since Z is at (-b, c), what should be added to -b to get a units? The x-coordinate of Y is a b. Thus the missing coordinates are Y(a b, c) and X (a, 0). Y(a b, c), X(a, 0) 35. Since AB is on the x-axis and horizontal segments are parallel, position the endpoints of so that they have the same y-coordinate, c. The distance from D to C is the same as AB, also a + b units, let the x- coordinate of D be b and of C be a. Thus, the missing coordinates are C(a, c) and D( b, c). C(a, c), D( b, c) esolutions Manual - Powered by Cognero Page 16

18 The slopes of are the same so the segments are parallel. Use the Distance Formula to find AB and DC. The slopes of are the same so the segments are parallel. Use the Distance Formula to find AB and DC. Thus, AB = DC and. Therefore, ABCD is a parallelogram because if one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a Given: RSTV is a quadrilateral. A, B, C, and D are midpoints of sides respectively. Prove: ABCD is a Thus, AB = DC and. Therefore, ABCD is a parallelogram because if one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a 39. MULTIPLE REPRESENTATIONS In this problem, you will explore the properties of rectangles. A rectangle is a quadrilateral with four right angles. a. GEOMETRIC Draw three rectangles with varying lengths and widths. Label one rectangle ABCD, one MNOP, and one WXYZ. Draw the two diagonals for each rectangle. b. TABULAR Measure the diagonals of each rectangle and complete the table at the right. Place quadrilateral RSTV on the coordinate plane and label coordinates as shown. (Using coordinates that are multiples of 2 will make the computation easier.) By the Midpoint Formula, the coordinates of A, B, C, and D are Find the slopes of. slope of slope of c. VERBAL Write a conjecture about the diagonals of a rectangle. a. Rectangles have 4 right angles and have opposite sides congruent. Draw 4 different rectangles with diagonals. esolutions Manual - Powered by Cognero Page 18

19 b. b. Use a ruler to measure the length of each diagonal. c. Sample answer: The diagonals of a rectangle are congruent. c. Sample answer: The measures of the diagonals for each rectangle are the same. The diagonals of a rectangle are congruent. a. 40. CHALLENGE The diagonals of a parallelogram meet at the point (0, 1). One vertex of the parallelogram is located at (2, 4), and a second vertex is located at (3, 1). Find the locations of the remaining vertices. First graph the given points. The midpoint of each diagonal is (0, 1). Let and be the coordinates of the remaining vertices. Here, diagonals of a parallelogram meet at the point (0, 1). So, and. esolutions Manual - Powered by Cognero Page 19

23 49. SAT/ACT A parallelogram has vertices at (0, 0), (3, 5), and (0, 5). What are the coordinates of the fourth vertex? A (0, 3) B (5, 3) C (5, 0) D (0, 3) E (3, 0) First graph the given points. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of ABCD with the given vertices. 50. A( 3, 5), B(6, 5), C(5, 4), D( 4, 4) Since the diagonals of a parallelogram bisect each other, their intersection point is the midpoint of and Find the midpoint of with endpoints ( 3, 5) and (5, 4). Use the Midpoint Formula. Substitute. The coordinates of the intersection of the diagonals of parallelogram ABCD are (1, 0.5). The vertices (3, 5) and (0, 5) lie on the same horizontal line. The distance between them is 3. So, the fourth vertex must also lie in the same horizontal line as (0, 0) and should be 3 units away. The only point which lie on the same horizontal line as (0, 0) is (3, 0) and it is also 3 units away from (0, 0). So, the correct choice is E. E (1, 0.5) 51. A(2, 5), B(10, 7), C(7, 2), D( 1, 4) Since the diagonals of a parallelogram bisect each other, their intersection point is the midpoint of and Find the midpoint of with endpoints (2, 5) and (7, 2). Use the Midpoint Formula. Substitute. The coordinates of the intersection of the diagonals of parallelogram ABCD are (4.5, 1.5). (4.5, 1.5) esolutions Manual - Powered by Cognero Page 23

24 Find the value of x. 52. Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x. 54. Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x. 35 esolutions Manual - Powered by Cognero Page 24

25 55. FITNESS Toshiro was at the gym for just over two hours. He swam laps in the pool and lifted weights. Prove that he did one of these activities for more than an hour. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given that Toshiro swam laps in the pool and lifted weights for a total of just over 2 hours. You need to prove that he did one of these activities for more than 1 hour. Use a proof by contradiction method. Assume the opposite of what you need to prove and then find a contradiction.. You need to prove. Use the properties that you have learned about triangles to walk through the proof. Given: P + W > 2 (P is time spent in the pool; W is time spent lifting weights.) Prove: P > 1 or W > 1 Step 1: Assume P 1 and W 1. Step 2: P + W 2 Step 3: This contradicts the given statement. Therefore he did at least one of these activities for more than an hour. Given: P + W > 2 (P is time spent in the pool; W is time spent lifting weights.) Prove: P > 1 or W > 1 Step 1: Assume P 1 and W 1. Step 2: P + W 2 Step 3: This contradicts the given statement. Therefore he did at least one of these activities for more than an hour. PROOF Write a flow proof. 56. Given: Prove: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given esolutions Manual - Powered by Cognero Page 25

26 57. Given: Prove: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given. You need to prove. Use the properties that you have learned about triangles to walk through the proof. esolutions Manual - Powered by Cognero Page 26

27 Use slope to determine whether XY and YZ are perpendicular or not perpendicular. 58. X( 2, 2), Y(0, 1), Z(4, 1) Substitute the coordinates of the points in slope formula to find the slopes of the lines. The product of the slopes of the lines is not 1. Therefore, the lines are not perpendicular. not perpendicular 59. X(4, 1), Y(5, 3), Z(6, 2) Substitute the coordinates of the points in slope formula to find the slopes of the lines. The product of the slopes of the lines is not 1. Therefore, the lines are not perpendicular. not perpendicular esolutions Manual - Powered by Cognero Page 27

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