# 6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

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1 ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 1. If, find. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So, Therefore,. 2. If AB = 2x + 3 and BC = x + 7, find CD. A rhombus is a parallelogram with all four sides congruent. So, So, AB = 2(4) + 3 = 11. CD is congruent to AB, so CD = 11. esolutions Manual - Powered by Cognero Page 1

2 3. PROOF Write a two-column proof to prove that if ABCD is a rhombus with diagonal. 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 ABCD is a rhombus with diagonal. You need to prove. Use the properties that you have learned about rhombi to walk through the proof. Given: ABCD is a rhombus with diagonal. Prove: Proof: Statements(Reasons) 1. ABCD is a rhombus with diagonal. (Given) 2. (Diag. of rhombus bisects ) 3. (Refl. Prop.) 4. (Def. of rhombus) 5. (SAS) 6. (CPCTC) 4. GAMES The checkerboard below is made up of 64 congruent black and red squares. Use this information to prove that the board itself is a square. Sample answer: 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 the checkerboard is made up of 64 congruent squares. You need to prove that the board is a square. Use the properties that you have learned about squares to walk through the proof. Since each side of the board is 8 squares in length and each of the squares is congruent, the lengths of all four sides of the board are equal. Since we know that each of the four quadrilaterals that form the corners of the board are squares, we know that the measure of the angle of each vertex of the board is 90. Therefore, the board is a square. esolutions Manual - Powered by Cognero Page 2

3 COORDINATE GEOMETRY Given each set of vertices, determine whether rhombus, a rectangle, or a square. List all that apply. Explain. 5. Q(1, 2), R( 2, 1), S(1, 4), T(4, 1) First graph the quadrilateral. QRST is a If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals. So, the parallelogram is a rectangle. Check whether the two diagonals are perpendicular. QS has a slope of. RT has a slope of. These slopes are opposite reciprocals. The diagonals are perpendicular. So, it is a rhombus. Since the diagonals are both congruent and perpendicular to each other the parallelogram is a rectangle, rhombus and square. esolutions Manual - Powered by Cognero Page 3

4 6. Q( 2, 1), R( 1, 2), S(4, 1), T(3, 2) First graph the quadrilateral. If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals. The diagonals are not congruent. So, the parallelogram is not a rectangle. Check whether the two diagonals are perpendicular. The diagonals are not perpendicular. So, it is not a rhombus either. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 7. If AB = 14, find BC. A rhombus is a parallelogram with all four sides congruent. So, Therefore, BC = 14. esolutions Manual - Powered by Cognero Page 4

5 8. If, find. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So, Therefore, 9. If AP = 3x 1 and PC = x + 9, find AC. The diagonals of a rhombus bisect each other. 3x 1 = x + 9 2x = 10 x = 5 Therefore, AC = 2(5 + 9) = If DB = 2x 4 and PB = 2x 9, find PD. The diagonals of a rhombus bisect each other. So, Therefore, PD = PB = 2(7) 9 = If, find. In a rhombus, consecutive interior angles are supplementary. Each pair of opposite angles of a rhombus is congruent. So, 12. If, find x. The diagonals of a rhombus are perpendicular to each other. esolutions Manual - Powered by Cognero Page 5

6 PROOF Write a two-column proof. 13. Given: Prove: WXYZ is a rhombus. 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 WXYZ is a rhombus. Use the properties that you have learned about parallelograms and rhombi to walk through the proof. Given: Prove: WXYZ is a rhombus. Proof: Statements(Reasons) 1. (Given) 2. WXYZ is a. (Both pairs of opp. sides are.) 3. WXYZ is a rhombus. (If one pair of consecutive sides of a are, the is a rhombus.) esolutions Manual - Powered by Cognero Page 6

7 14. Given: QRST is a parallelogram. Prove: QRST is a square. 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 QRST is a parallelogram;. You need to prove that QRST is a square. Use the properties that you have learned about parallelograms and squares to walk through the proof. Given: QRST is a parallelogram; Prove: QRST is a square. Proof: Statements(Reasons) 1. QRST is a parallelogram;. (Given) 2. QRST is a rectangle. (If the diagonals of a are, the is a rectangle.) 3. is a right angle. (Def of rt. ) 4. (Def. of perpendicular) 5. QRST is a rhombus. (If the diagonals of a are, is a rhombus.) 6. QRST is a square. (Thm.6.2, if a quadrilateral is a rectangle and a rhombus, then it is a square.) esolutions Manual - Powered by Cognero Page 7

8 15. Given: JKQP is a square.. Prove: JKLM is a parallelogram. 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 JKQP is a square and. You need to prove that JKLM is a parallelogram. Use the properties that you have learned about parallelograms to walk through the proof. Given: JKQP is a square.. Prove: JKLM is a parallelogram. Proof: Statements (Reasons) 1. JKQP is a square.. (Given) 2. JKQP is a parallelogram. (All squares are parallelograms.) 3. (Def. of ) 4. (Opp. Sides of are.) 5. JP = KQ (Def of segs.) 6. JM = MP, KL = LQ (Def. of bisects) 7. JP = JM + MP, KQ = KL + LQ (Seg. Add Post.) 8. JP = 2JM, KQ = 2KL (Subst.) 9. 2JM = 2KL (Subst.) 10. JM = KL (Division Prop.) 11. (Def. of segs.) 12. JKLM is a parallelogram. (If one pair of opp. sides is, then the quad. is a.) esolutions Manual - Powered by Cognero Page 8

9 16. Given: ACDH and BCDF are parallelograms;. Prove: ABFH is a rhombus. 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 and BCDF are parallelograms;.you need to prove that ABHF is a rhombus. Use the properties that you have learned about parallelograms and rhombi to walk through the proof. Given: ACDH and BCDF are parallelograms;. Prove: ABHF is a rhombus. Proof: Statements (Reasons) 1. ACDH and BCDF are parallelograms;. (Given) 2. (Def. of ) 3. (Trans. Prop) 4. (Def. of ) 5. AC = HD (Def of segs.) 6. AC = AB + BC, HD = HF + FD (Seg. Add. Post.) 7. AC HD = AB + BC HF FD (Subt. Prop.) 8. AB = HF (Subst.) 9. (Def. of segs.) 10. (Subst.) 11. ABFH is a rhombus. (Def. of rhombus) esolutions Manual - Powered by Cognero Page 9

11 COORDINATE GEOMETRY Given each set of vertices, determine whether rectangle, or a square. List all that apply. Explain. 19. J( 4, 1), K(1, 1), L(4, 3), M( 1, 3) First graph the quadrilateral. JKLM is a rhombus, a If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals. The diagonals are not congruent. So, the parallelogram is not a rectangle. Check whether the two diagonals are perpendicular. The diagonals are perpendicular. So, it is a rhombus. esolutions Manual - Powered by Cognero Page 11

12 20. J( 3, 2), K(2, 2), L(5, 2), M(0, 2) First, graph the quadrilateral. If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals. The diagonals are not congruent. So, the parallelogram is not a rectangle. Check whether the two diagonals are perpendicular. The diagonals are perpendicular. So, it is a rhombus. esolutions Manual - Powered by Cognero Page 12

13 21. J( 2, 1), K( 4, 3), L(1, 5), M(3, 1) First graph the quadrilateral. If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals. The diagonals are not congruent. So, the parallelogram is not a rectangle. Check whether the two diagonals are perpendicular. The diagonals are not perpendicular. So, it is not a rhombus either. esolutions Manual - Powered by Cognero Page 13

14 22. J( 1, 1), K(4, 1), L(4, 6), M( 1, 6) First graph the quadrilateral. If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals. The diagonals are congruent. So, the parallelogram is a rectangle. Check whether the two diagonals are perpendicular. The diagonals are perpendicular. So, it is a rhombus. Since the diagonals are both congruent and perpendicular to each other the parallelogram is a rectangle, rhombus and square. ABCD is a rhombus. If PB = 12, AB = 15, and, find each measure. 23. AP The diagonals of a rhombus are perpendicular to each other. So, by the Pythagorean Theorem, AP 2 = AB 2 PB 2. esolutions Manual - Powered by Cognero Page 14

15 24. CP All the four sides of a rhombus are congruent and the diagonals are perpendicular to each other. So, by the Pythagorean Theorem, CP 2 = BC 2 PB 2. BC = AB. Substitute AB for BC. 25. All the four sides of a rhombus are congruent. So, is an isosceles triangle. Then, 26. The diagonals are perpendicular to each other. So, in the right triangle PAB, All the four sides of a rhombus are congruent. So, is an isosceles triangle. Then, WXYZ is a square. If WT = 3, find each measure. 27. ZX The diagonals of a square are congruent and bisect each other. So, ZX = WY = 2(WT) = XY The diagonals of a square are congruent and bisect each other at right angles. So, YT = XT = WT = 3. By the Pythagorean Theorem, XY 2 = YT 2 + XT 2. esolutions Manual - Powered by Cognero Page 15

16 The diagonals of a square are perpendicular to each other. So, In a square, each diagonal bisects a pair of opposite angles. So, Classify each quadrilateral. 31. Refer to the photo on p The diagonals are congruent and bisect each other at right angle. Therefore, the quadrilateral is a square. 32. Refer to the photo on p The two pairs of opposite angles are congruent and the adjacent sides are congruent. Therefore, the quadrilateral is a rhombus. esolutions Manual - Powered by Cognero Page 16

17 33. Refer to the photo on p The two pairs of opposite sides are congruent and one of the angles is a right angle. Therefore, the quadrilateral is a rectangle. PROOF Write a paragraph proof. 34. Theorem 6.16 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 ABCD is a rhombus. You need to prove that each diagonal bisects a pair of opposite angles.. Use the properties that you have learned about rhombi to walk through the proof. Given: ABCD is a rhombus. Prove: Each diagonal bisects a pair of opposite angles. Proof: We are given that ABCD is a rhombus. By definition of rhombus, ABCD is a parallelogram. Opposite angles of a parallelogram are congruent, so. because all sides of a rhombus are congruent. by SAS. by CPCTC. by SAS. by CPCTC. By definition of angle bisector, each diagonal bisects a pair of opposite angles. esolutions Manual - Powered by Cognero Page 17

18 35. Theorem 6.17 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 ABCD is a parallelogram;. You need to prove that ABCD is a rhombus. Use the properties that you have learned about rhombi to walk through the proof. Given: ABCD is a parallelogram;. Prove: ABCD is a rhombus. Proof: We are given that ABCD is a parallelogram. The diagonals of a parallelogram bisect each other, so because congruence of segments is reflexive. We are also given that are right angles by the definition of perpendicular lines. Then. Thus, because all right angles are congruent. Therefore, by SAS. by CPCTC. Opposite sides of parallelograms are congruent, so. Then since congruence of segments is transitive,. All four sides of ABCD are congruent, so ABCD is a rhombus by definition. 36. Theorem 6.18 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 a diagonal of a parallelogram bisects an angle of a parallelogram. You need to prove that the parallelogram is a rhombus. Use the properties that you have learned about parallelograms and rhombi to walk through the proof. Given: ABCD is a parallelogram; diagonal bisects. Prove: ABCD is a rhombus. Proof: It is given that ABCD is a parallelogram. Since opposite sides of a parallelogram are parallel, definition, are alternate interior angles of parallel sides. Since alternate interior angles are congruent,. Congruence of angles is symmetric, therefore. it is given that bisects, so by definition. By the Transitive Property,. The sides opposite congruent angles in a triangle are congruent, therefore, of consecutive sides of the parallelogram is congruent, ABCD is a rhombus.. By. So, since a pair esolutions Manual - Powered by Cognero Page 18

19 37. Theorem 6.19 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 ABCD is a parallelogram;. You need to prove that ABCD is a rhombus. Use the properties that you have learned about rhombi to walk through the proof. Given: ABCD is a parallelogram;. Prove: ABCD is a rhombus. Proof: Opposite sides of a parallelogram are congruent, so. We are given that. So, by the Transitive Property,. So,. Thus, ABCD is a rhombus by definition. 38. Theorem 6.20 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 ABCD is a rectangle and a rhombus. You need to prove that ABCD is a square. Use the properties that you have learned about squares to walk through the proof. Given: ABCD is a rectangle and a rhombus. Prove: ABCD is a square. Proof: We know that ABCD is a rectangle and a rhombus. ABCD is a parallelogram, since all rectangles and rhombi are parallelograms. By the definition of a rectangle, are right angles. By the definition of a rhombus, all of the sides are congruent. Therefore, ABCD is a square since ABCD is a parallelogram with all four sides congruent and all the angles are right. esolutions Manual - Powered by Cognero Page 19

21 PROOF Write a coordinate proof of each statement. 41. The diagonals of a square are perpendicular. Begin by positioning square ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units. Then the rest of the vertices are B(a, 0), C(a, a), and D(0, 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 ABCD is a square and you need to prove that. Use the properties that you have learned about squares to walk through the proof. Given: ABCD is a square. Prove: Proof: slope of slope of The slope of is the negative reciprocal of the slope of, so they are perpendicular. esolutions Manual - Powered by Cognero Page 21

22 42. The segments joining the midpoints of the sides of a rectangle form a rhombus. Begin by positioning rectangle 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(a, 0), C(a, b), and D(0, b). 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 ABCD is a rectangle and Q, R, S, and T are midpoints of their respective sides.and you need to prove that QRST is a rhombus. Use the properties that you have learned about rhombi to walk through the proof. Given: ABCD is a rectangle. Q, R, S, and T are midpoints of their respective sides. Prove: QRST is a rhombus. Proof: Midpoint Q is. Midpoint R is. Midpoint S is. Midpoint T is. QR = RS = ST = QT QRST is a rhombus. esolutions Manual - Powered by Cognero Page 22

24 44. REPAIR The window pane shown needs to be replaced. What are the dimensions of the replacement pane? The window pane is in the shape of a square. The diagonal of a square is the hypotenuse of a right triangle with two consecutive sides of the square as its legs. Let x be the length of each side of the square. So, by the Pythagorean Theorem, Therefore, the length of each side of the square is about 15 inches. esolutions Manual - Powered by Cognero Page 24

25 45. MULTIPLE REPRESENTATIONS In this problem, you will explore the properties of kites, which are quadrilaterals with exactly two distinct pairs of adjacent congruent sides. a. GEOMETRIC Draw three kites with varying side lengths. Label one kite ABCD, one PQRS, and one WXYZ. Then draw the diagonals of each kite, labeling the point of intersection N for each kite. b. TABULAR Measure the distance from N to each vertex. Record your results in a table like the one shown. c. VERBAL Make a conjecture about the diagonals of a kite. a. Sample answer: Draw three different kites, each with the intersection of the diagonals labeled N. b. Use a ruler to measure each length in the table. c. Sample answer: From the measurements recorded in the table, N is the midpoint of each of the short diagonals. The shorter diagonal of a kite is bisected by the longer diagonal. esolutions Manual - Powered by Cognero Page 25

27 48. CHALLENGE The area of square ABCD is 36 square units and the area of is 20 square units. If, find the length of. Since the area of the square is 36 square units, the length of each side of the square is 6 units. All the four angles of a square are right angles. So, by the Pythagorean Theorem, The area of is 20 square units. So, Also, we have So, by HL postulate, AE = CF by CPCTC Therefore, CF = 2. esolutions Manual - Powered by Cognero Page 27

28 49. OPEN ENDED Find the vertices of a square with diagonals that are contained in the lines y = x and y = x + 6. Justify your reasoning. Sample answer: First graph the lines y = x and y = -x + 6. There are 6 units from the origin to the y-intercept of y = -x + 6 and 6 units from the origin to the x-intercept of y = -x + 6. So, three of the vertices of a square will be at (0, 0), ((0, 6), and (6, 0). The point 6 units above (6, 0) is (6, 6). (0, 0), (6, 0), (0, 6), (6, 6); the diagonals are perpendicular, and any four points on the lines equidistant from the intersection of the lines will be the vertices of a square. esolutions Manual - Powered by Cognero Page 28

29 50. WRITING IN MATH Compare all of the properties of the following quadrilaterals: parallelograms, rectangles, rhombi, and squares. Sample answer: Parallelogram: Opposite sides of a parallelogram are parallel and congruent. Opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other and each diagonal separates a parallelogram into two congruent triangles. Rectangle: A rectangle has all the properties of a parallelogram. A rectangle has four right angles. The diagonals of a rectangle are congruent. Rhombus: A rhombus has all of the properties of a parallelogram. All sides of a rhombus are congruent. The diagonals of a rhombus are perpendicular and bisect the angles of the rhombus. Square: A square has all of the properties of a parallelogram. A square has all of the properties of a rectangle. A square has all of the properties of a rhombus. esolutions Manual - Powered by Cognero Page 29

30 51. JKLM is a rhombus. If CK = 8 and JK = 10, find JC. A 4 C 8 B 6 D 10 The diagonals of a rhombus are perpendicular to each other. So, by the Pythagorean Theorem, JC 2 = JK 2 CK 2. Therefore, the correct choice is B. esolutions Manual - Powered by Cognero Page 30

31 52. EXTENDED RESPONSE The sides of square ABCD are extended by sides of equal length to form square WXYZ. a. If CY = 3 cm and the area of ABCD is 81 cm 2, find the area of WXYZ. b. If the areas of ABCD and WXYZ are respectively, find DZ. c. If AB = 2CY and the area of ABCD = g square meters, find the area of WXYZ in square meters. a. Since the area of the square is 81 square cm, the length of each side of the square is 9 cm. Since the sides of the square ABCD are extended to form square WXYZ, DZ = AW = BX = CY = 3 cm. Area of each of the triangle is The area of the square WXYZ is the sum of the 4 congruent triangles and the area of the square. b. In the right triangle WZD, WZ = 13 and WD = AW + AD = DZ + 7 = x + 7 where x is the length of the segment By the Pythagorean Theorem, Solve the equation for x. Since x is a length, it cannot be negative. Therefore, the length DZ is about 5 cm. c. If AB = 2CY, the lengths of the two legs of the right triangle WAX are WA = CY and AX = 2CY + CY = 3CY. Then by the Pythagorean theorem, Since AB = 2CY, But AB 2 = the area of the square ABCD = g. Therefore, the area of the square WXYZ = WX 2 = 2.5g square meters. esolutions Manual - Powered by Cognero Page 31

32 53. ALGEBRA What values of x and y make quadrilateral ABCD a parallelogram? F x = 3, y = 2 G, y = 1 H x = 2, y = 3 J x = 3, y = 1 Each pair of opposite sides of a parallelogram is congruent. So, 13x 3y = 17 and 5x + y = 13. Solve the system of two equations to find the values of x and y. Multiply the second equation by 3 and then add to the 1st equation to eliminate the y-term. 3(5x + y = 13) = 15x + 3y = 39 13x 3y = x + 3y = 39 28x + 0y = 56 28x = 56 x = 2 Use the value of x to find the value of y. Therefore, the correct choice is H. 54. SAT/ACT What is 6 more than the product of 3 and a certain number x? A 3x 6 B 3x C x D 3x + 6 E 6 + 3x The product of 3 and x is 3x. Six more than 3x is 3x + 6. Therefore, the correct choice is D. esolutions Manual - Powered by Cognero Page 32

34 60. One pairs of opposite sides is parallel and congruent. Theorem 6.12 states that if one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. So no other information is needed to determine if it is a parallelogram. Therefore, it is a parallelogram. 61. MEASUREMENT Monifa says that her backyard is shaped like a triangle and that the lengths of its sides are 22 feet, 23 feet, and 45 feet. Do you think these measurements are correct? Explain your reasoning. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Since = 45, the sides of Monifa s backyard cannot be 22 ft, 23 ft and 45 ft. 62. COORDINATE GEOMETRY Identify the transformation and verify that it is a congruence transformation. 63. is a reflection of. Find the lengths of the sides of the two triangles. AB = 5, BC = 4, AC =, XY = 5, YZ = 4, XZ =.Since all pairs of corresponding sides are congruent, by SSS. Solve each equation. esolutions Manual - Powered by Cognero Page 34

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