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Study Guide and Review State whether each sentence is true or false. If false, replace the underlined word or phrase to make a true sentence. 1. No angles in an isosceles trapezoid are congruent. SOLUTION: By definition, an isosceles trapezoid is a trapezoid in which the legs are congruent, both pairs of base angles are congruent, and the diagonals are congruent. false, both pairs of base angles ANSWER: false, both pairs of base angles 2. If a parallelogram is a rectangle, then the diagonals are congruent. SOLUTION: A rectangle is a parallelogram with four right angles, opposite sides parallel, opposite sides congruent, opposite angles congruent, consecutive angles are supplementary, and the diagonals bisect each other. The statement is true. ANSWER: true 3. A midsegment of a trapezoid is a segment that connects any two nonconsecutive vertices. SOLUTION: The midsegment of a trapezoid is the segment that connects the midpoints of the legs of the trapezoid. A diagonal is a segment that connects any two nonconsecutive vertices. false, diagonal ANSWER: false, diagonal 4. The base of a trapezoid is one of the parallel sides. SOLUTION: One of the parallel sides of a trapezoid is its base. The statement is true. ANSWER: true 5. The diagonals of a rhombus are perpendicular. SOLUTION: A rhombus has perpendicular diagonals. The statement is true. ANSWER: true 6. The diagonal of a trapezoid is the segment that connects the midpoints of the legs. SOLUTION: A diagonal is a segment that connects any two nonconsecutive vertices. The midsegment of a trapezoid is the segment that connects the midpoint of the legs of the trapezoid. false, midsegment ANSWER: false, midsegment eSolutions Manual - Powered by Cognero 7. A rectangle is not always a parallelogram. Page 1 SOLUTION: A rhombus has perpendicular diagonals. The statement is true. ANSWER: Study Guide and Review true 6. The diagonal of a trapezoid is the segment that connects the midpoints of the legs. SOLUTION: A diagonal is a segment that connects any two nonconsecutive vertices. The midsegment of a trapezoid is the segment that connects the midpoint of the legs of the trapezoid. false, midsegment ANSWER: false, midsegment 7. A rectangle is not always a parallelogram. SOLUTION: By definition a rectangle is a parallelogram with four right angles. false, is always ANSWER: false, is always 8. A quadrilateral with only one set of parallel sides is a parallelogram. SOLUTION: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A trapezoid is a quadrilateral with exactly one pair of parallel sides. false, trapezoid ANSWER: false, trapezoid 9. A rectangle that is also a rhombus is a square. SOLUTION: By definition, a square is a parallelogram with four congruent sides and four right angles. The statement is true. ANSWER: true 10. The leg of a trapezoid is one of the parallel sides. SOLUTION: By definition, a trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. The nonparallel sides are called legs. false, nonparallel ANSWER: false, nonparallel Find the sum of the measures of the interior angles of each convex polygon. 11. decagon SOLUTION: A decagon has ten sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures. Substitute n = 10 in . eSolutions Manual - Powered by Cognero Page 2 The nonparallel sides are called legs. false, nonparallel ANSWER: Study Guide and Review false, nonparallel Find the sum of the measures of the interior angles of each convex polygon. 11. decagon SOLUTION: A decagon has ten sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures. Substitute n = 10 in . ANSWER: 1,440 12. 15-gon SOLUTION: A 15-gon has fifteen sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures. Substitute n = 15 in . ANSWER: 2,340 13. SNOWFLAKES The snowflake decoration at the right is a regular hexagon. Find the sum of the measures of the interior angles of the hexagon. SOLUTION: A hexagon has six sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures. Substitute n = 6 in . ANSWER: 720 The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon. 14. 135 eSolutions Manual - Powered by Cognero SOLUTION: Let n = the number of sides in the polygon. Since all angles of a regular polygon are congruent, the sum of the Page 3 ANSWER: Study Guide and Review 720 The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon. 14. 135 SOLUTION: Let n = the number of sides in the polygon. Since all angles of a regular polygon are congruent, the sum of the interior angle measures is 135n. By the Polygon Interior Angles Sum Theorem, the sum of the interior angle . measures can also be expressed as ANSWER: 8 15. ≈ 166.15 SOLUTION: Let n = the number of sides in the polygon. Since all angles of a regular polygon are congruent, the sum of the interior angle measures is about 166.15n. By the Polygon Interior Angles Sum Theorem, the sum of the interior angle . measures can also be expressed as ANSWER: 26 Use ABCD to find each measure. 16. SOLUTION: We know that consecutive angles in a parallelogram are supplementary. So, Substitute. ANSWER: 65° 17. AD SOLUTION: eSolutions Manual - Powered by Cognero We know that opposite sides of a parallelogram are congruent. So, Page 4 ANSWER: Study 65°Guide and Review 17. AD SOLUTION: We know that opposite sides of a parallelogram are congruent. So, ANSWER: 18 18. AB SOLUTION: We know that opposite sides of a parallelogram are congruent. So, ANSWER: 12 19. SOLUTION: We know that opposite angles of a parallelogram are congruent. So, ANSWER: 115° ALGEBRA Find the value of each variable in each parallelogram. 20. SOLUTION: Since the opposite sides of a parallelogram are congruent, 3x – 6 = x + 4. Solve for x. 3x – 6 = x + 4 Opp. sides of a parallelogram are . 2x – 6 = 4 Subtract x from each side. 2x = 10 Add 6 to each side. x = 5 Divide each side by 2. Since alternate interior angles are congruent, . 5y = 60 y = 12 So, x = 5 and y = 12. ANSWER: x = 5, y = 12 eSolutions Manual - Powered by Cognero Page 5 So, x = 5 and y = 12. ANSWER: Study x =Guide 5, y =and 12 Review 21. SOLUTION: Since the opposite sides are congruent, 3y + 13 = 2y + 19. Solve for y. 3y + 13 = 2y + 19 y =6 Since the opposite angles are congruent, 2x + 41 = 115. Solve for x. 2x + 41 = 115 2x = 74 x = 37 ANSWER: x = 37, y = 6 22. DESIGN What type of information is needed to determine whether the shapes that make up the stained glass window below are parallelograms? SOLUTION: Sample answer: Review the definition of and theorems about parallelograms. A quadrilateral is a parallelogram if both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and parallel or if both pairs of opposite sides are parallel..The shapes can also be parallelograms if both pairs of opposite angles are congruent or if the diagonals bisect each other. ANSWER: Sample answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and parallel, then the shapes are parallelograms. The shapes can also be parallelograms if both pairs of opposite angles are congruent or if the diagonals bisect each other. Determine whether each quadrilateral is a parallelogram. Justify your answer. 23. SOLUTION: The diagonals of the figure bisect each other. By Theorem 6.11 if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. No other information is needed to determine that the figure is a parallelogram. ANSWER: eSolutions - Powered yes, Manual Theorem 6.11 by Cognero Page 6 ANSWER: Sample answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and parallel, shapes are parallelograms. The shapes can also be parallelograms if both pairs of opposite angles Study Guidethen andthe Review are congruent or if the diagonals bisect each other. Determine whether each quadrilateral is a parallelogram. Justify your answer. 23. SOLUTION: The diagonals of the figure bisect each other. By Theorem 6.11 if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. No other information is needed to determine that the figure is a parallelogram. ANSWER: yes, Theorem 6.11 24. SOLUTION: One pair of opposite sides are parallel and congruent. By Theorem 6.12 if one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. No other information is needed to determine that the figure is a parallelogram. ANSWER: yes, Theorem 6.12 25. PROOF Write a two-column proof. Given: Prove: Quadrilateral EBFD is a parallelogram. SOLUTION: 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 EBFD is a parallelogram. Use the properties that you have learned about parallelograms to walk through the proof. Given: Prove: Quadrilateral EBFD is a parallelogram. eSolutions Manual - Powered by Cognero 1. ABCD, (Given) 2. AE = CF (Def. of segs) Page 7 you are given . You need to prove that EBFD is a parallelogram. Use the properties that you have learned about parallelograms to walk through the proof. Study Guide and Review Given: Prove: Quadrilateral EBFD is a parallelogram. 1. ABCD, (Given) 2. AE = CF (Def. of segs) 3. (Opp. sides of a ) 4. BC = AD (Def. of segs) 5. BC = BF + CF, AD = AE +ED (Seg. Add. Post.) 6. BF + CF = AE + ED (Subst.) 7. BF + AE = AE + ED (Subst.) 8. BF = ED (Subt. Prop.) 9. (Def. of segs) 10. (Def. of ) 11. Quadrilateral EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it is a parallelogram.) ANSWER: Given: Prove: Quadrilateral EBFD is a parallelogram. 1. ABCD, (Given) 2. AE = CF (Def. of segs) 3. (Opp. sides of a ) 4. BC = AD (Def. of segs) 5. BC = BF + CF, AD = AE +ED (Seg. Add. Post.) 6. BF + CF = AE + ED (Subst.) 7. BF + AE = AE + ED (Subst.) 8. BF = ED (Subt. Prop.) 9. (Def. of segs) 10. (Def. of ) 11. Quadrilateral EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it is a parallelogram.) ALGEBRA Find x and y so that the quadrilateral is a parallelogram. 26. eSolutions Manual - Powered by Cognero SOLUTION: We know that opposite angles of a parallelogram are congruent. Page 8 9. (Def. of segs) 10. (Def. of ) 11.Guide Quadrilateral EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it is a Study and Review parallelogram.) ALGEBRA Find x and y so that the quadrilateral is a parallelogram. 26. SOLUTION: We know that opposite angles of a parallelogram are congruent. So, 12x + 72 = 25x + 20 and 3y + 36 = 9y - 12. Solve for x. 12x + 72 = 25x + 20 72 = 13x + 20 52 = 13x 4 = x Solve for y. 3y + 36 = 9y - 12 36 = 6y - 12 48 = 6y 8 = y When x = 4 and y = 8 the quadrilateral is a parallelogram. ANSWER: x = 4, y = 8 27. SOLUTION: We know that diagonals of a parallelogram bisect each other. So, . Solve for x. Alternate interior angles in a parallelogram are congruent. Solve for y. 5y = 60 So, y = 12. When x = 5 and y = 12 the quadrilateral is a parallelogram. ANSWER: x = 5, y = 12 eSolutions Manual - Powered by Cognero 28. PARKING The lines of the parking space shown below are parallel. How wide is the space (in inches)? Page 9 When x = 4 and y = 8 the quadrilateral is a parallelogram. ANSWER: Study x =Guide 4, y =and 8 Review 27. SOLUTION: We know that diagonals of a parallelogram bisect each other. So, . Solve for x. Alternate interior angles in a parallelogram are congruent. Solve for y. 5y = 60 So, y = 12. When x = 5 and y = 12 the quadrilateral is a parallelogram. ANSWER: x = 5, y = 12 28. PARKING The lines of the parking space shown below are parallel. How wide is the space (in inches)? SOLUTION: Since the distance between two parallel lines is the same, we can write the equation 6x + 12 = 5x + 20 and then solve for x. 6x + 12 = 5x + 20 x + 12 = 20 x = 8 Substitute x = 8 in 5x + 20. 5x + 20 = 5(8) + 20 = 60 So, the length of the space is 60 inches. ANSWER: 60 in. ALGEBRA Quadrilateral EFGH is a rectangle. eSolutions Manual - Powered by Cognero Page 10 So, the length of the space is 60 inches. ANSWER: Study Guide and Review 60 in. ALGEBRA Quadrilateral EFGH is a rectangle. 29. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Substitute. ANSWER: 33 30. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Substitute. ANSWER: 77 31. If FK = 32 feet, find EG. SOLUTION: We know that diagonals of a rectangle are congruent and bisect each other. So, EG = FH, FK = KH, and EK = KG. FH = FK + KH Diagonals of a rectangle bisect each other. = FK + FK FK = KH, substitution = 32 + 32 Substitute. = 64 Add. EG is the same length as FH so EG = 64 feet. ANSWER: 64 32. Find SOLUTION: All four angles of a rectangle are right angles. So, ANSWER: 180 33. If EF = 4x – 6 and HG = x + 3, find EF. eSolutions Manual - Powered by Cognero SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, EF = HG. Page 11 All four angles of a rectangle are right angles. So, ANSWER: Study Guide and Review 180 33. If EF = 4x – 6 and HG = x + 3, find EF. SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, EF = HG. EF = HG Opp. sides of rectangle are congruent. 4x – 6 = x + 3 Substitution. 3x – 6 = 3 Subtract x from each side. 3x = 9 Add 6 to each side. x = 3 Divide each side by 3. Substitute x = 3 into 4x - 6 to find EF. EF = 4x – 6 Original equation. = 4(3) – 6 x = 3 = 12 – 6 Multiply. = 6 Subtract. So, EF = 6. ANSWER: 6 ALGEBRA ABCD is a rhombus. If EB = 9, AB = 12 and , find each measure. 34. AE SOLUTION: The diagonals of a rhombus are perpendicular. So, use the Pythagorean Theorem. Since the length must be positive, AE = 7.9. ANSWER: 7.9 35. SOLUTION: All the four sides of a rhombus are congruent. So, is an isosceles triangle. Therefore, ANSWER: eSolutions 55 Manual - Powered by Cognero 36. CE Page 12 Since the length must be positive, AE = 7.9. ANSWER: Study 7.9Guide and Review 35. SOLUTION: All the four sides of a rhombus are congruent. So, is an isosceles triangle. Therefore, ANSWER: 55 36. CE SOLUTION: The diagonals of a rhombus are perpendicular. Use AE to find CE. Use the Pythagorean Theorem. Since the length must be positive, AE = 7.9. CE = AE = 7.9 ANSWER: 7.9 37. SOLUTION: The diagonals are perpendicular to each other. So, in the right triangle EAB, All the four sides of a rhombus are congruent. So, is an isosceles triangle. Therefore, ANSWER: 35 38. LOGOS A car company uses the symbol shown at the right for their logo. If the inside space of the logo is a rhombus, what is the length of FJ? SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, FG = FJ = 2.5 cm. ANSWER: 2.5 cm eSolutions Manual - Powered by Cognero COORDINATE GEOMETRY Given each set of vertices, determine whether QRST is a rhombus, a rectangle, or a square. List all that apply. Explain. Page 13 All the four sides of a rhombus are congruent. So, is an isosceles triangle. Therefore, ANSWER: Study Guide and Review 35 38. LOGOS A car company uses the symbol shown at the right for their logo. If the inside space of the logo is a rhombus, what is the length of FJ? SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, FG = FJ = 2.5 cm. ANSWER: 2.5 cm COORDINATE GEOMETRY Given each set of vertices, determine whether QRST is a rhombus, a rectangle, or a square. List all that apply. Explain. 39. Q(12, 0), R(6, -6), S(0, 0), T(6, 6) SOLUTION: First graph the quadrilateral. Use the distance formula to find the length of each side of QRST. eSolutions Manual - Powered by Cognero Page 14 Study Guide and Review So, all sides are congruent. The quadrilateral is a rhombus. Check to see whether we can say more: are consecutive sides perpendicular? Since the products of the slopes of consecutive sides are -1, the sides are perpendicular. So, the quadrilateral is also a rectangle and a square. ANSWER: Rectangle, rhombus, square; all sides are , consecutive are . 40. Q(–2, 4), R(5, 6), S(12, 4), T(5, 2) SOLUTION: First graph the quadrilateral. Use the distance formula to find the length of each side of QRST. eSolutions Manual - Powered by Cognero Page 15 Study Guide and Review All the sides are congruent. 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, QRST is not a rectangle. Since the figure is not a rectangle, it also Since cannot be a square. Check whether the two diagonals are perpendicular. Undefined slope and 0 slope are perpendicular, so the diagonals are perpendicular. It is a rhombus. ANSWER: Rhombus; all sides are , diagonals are . Find each measure. 41. GH SOLUTION: Use the Pythagorean Theorem. Since the length must be positive, GH = 19.2. ANSWER: 19.2Manual - Powered by Cognero eSolutions 42. Page 16 Undefined slope and 0 slope are perpendicular, so the diagonals are perpendicular. It is a rhombus. ANSWER: Study Guide and Review Rhombus; all sides are , diagonals are . Find each measure. 41. GH SOLUTION: Use the Pythagorean Theorem. Since the length must be positive, GH = 19.2. ANSWER: 19.2 42. SOLUTION: The trapezoid WZXY is an isosceles trapezoid. So, each pair of base angles is congruent. So, The sum of the measures of the angles of a quadrilateral is 360. Let . So, ANSWER: 68 43. DESIGN Renee designed the square tile as an art project. a. Describe a way to determine if the trapezoids in the design are isosceles. b. If the perimeter of the tile is 48 inches and the perimeter of the red square is 16 inches, what is the perimeter of one of the trapezoids? eSolutions Manual - Powered by Cognero Page 17 So, ANSWER: Study Guide and Review 68 43. DESIGN Renee designed the square tile as an art project. a. Describe a way to determine if the trapezoids in the design are isosceles. b. If the perimeter of the tile is 48 inches and the perimeter of the red square is 16 inches, what is the perimeter of one of the trapezoids? SOLUTION: a. Sample answer: A trapezoid is isosceles if the legs are congruent. The legs of the trapezoids are part of the diagonals of the square tile. The diagonals of a square bisect opposite angles, so each base angle of each trapezoid measures 45°. One pair of sides is parallel and the base angles are congruent. b. The perimeter of a square is given by 4s, where s is the side length. Solving 48 = 4s1 and 16 = 4s2, we find that the tile is 12 in. long on a side and the red square is 4 in. long on a side. Now all that remains is to find the two other sides of the trapezoid. A diagonal of the tile is . So the length of each nonparallel side of in, and a diagonal of the red square is a trapezoid is in. Add to find the perimeter of the trapezoid. ANSWER: a. Sample answer: The legs of the trapezoids are part of the diagonals of the square. The diagonals of a square bisect opposite angles, so each base angle of a trapezoid measures 45°. One pair of sides is parallel and the base angles are congruent. b. eSolutions Manual - Powered by Cognero Page 18 Practice Test - Chapter 6 Find the sum of the measures of the interior angles of each convex polygon. 1. hexagon SOLUTION: A hexagon has six sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures. Substitute n = 6 in . ANSWER: 720 2. 16-gon SOLUTION: A 16-gon has sixteen sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures. Substitute n = 16 in . ANSWER: 2520 3. ART Jen is making a frame to stretch a canvas over for a painting. She nailed four pieces of wood together at what she believes will be the four vertices of a square. a. How can she be sure that the canvas will be a square? b. If the canvas has the dimensions shown below, what are the missing measures? SOLUTION: a. Sample answer: A quadrilateral is a square if it has diagonals that are congruent and perpendicular or all sides are congruent with 4 right angles. She should measure the angles at the vertices to see if they are 90 or she can check to see if the diagonals are congruent and perpendicular. b. Each side of a square have the same measure so x = 2 ft. Each angle of a square is a right angle so y = 90. ANSWER: a. Sample answer: She should measure the angles at the vertices to see if they are 90 or she can check to see if the diagonals are congruent and perpendicular. b. x = 2 ft, y = 90 Quadrilateral ABCD is an isosceles trapezoid. eSolutions Manual - Powered by Cognero Page 1 ANSWER: a. Sample answer: She should measure the angles at the vertices to see if they are 90 or she can check to see if the diagonals congruent Practice Testare - Chapter 6 and perpendicular. b. x = 2 ft, y = 90 Quadrilateral ABCD is an isosceles trapezoid. 4. Which angle is congruent to ? SOLUTION: The base angles of an isosceles triangle are congruent so is congruent to . ANSWER: 5. Which side is parallel to ? SOLUTION: The bases of a trapezoid are parallel so is parallel to . ANSWER: 6. Which segment is congruent to ? SOLUTION: The diagonals of an isosceles trapezoid are congruent so is congruent to . ANSWER: The measure of the interior angles of a regular polygon is given. Find the number of sides in the polygon. 7. 900 SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior . angle measures can also be expressed as ANSWER: 7 8. 1980 SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior . angle measures can also be expressed as eSolutions Manual - Powered by Cognero Page 2 ANSWER: Practice Test - Chapter 6 7 8. 1980 SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior . angle measures can also be expressed as ANSWER: 13 9. 2880 SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior . angle measures can also be expressed as ANSWER: 18 10. 5400 SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior . angle measures can also be expressed as ANSWER: 32 11. MULTIPLE CHOICE If QRST is a parallelogram, what is the value of x? A 11 C 13 B 12 D 14 SOLUTION: Diagonals of a parallelogram bisect each other. So, 14x – 34 = 12x – 8. eSolutions Manual Solve for x.- Powered by Cognero 14x – 34 = 12x – 8 Diag. bisect each other. 2x – 34 = – 8 Subtract 12x from each side. Page 3 ANSWER: Practice Test - Chapter 6 32 11. MULTIPLE CHOICE If QRST is a parallelogram, what is the value of x? A 11 C 13 B 12 D 14 SOLUTION: Diagonals of a parallelogram bisect each other. So, 14x – 34 = 12x – 8. Solve for x. 14x – 34 = 12x – 8 Diag. bisect each other. 2x – 34 = – 8 Subtract 12x from each side. 2x = 26 Add 34 to each side. x = 13 Divide each side by 2. So, the correct option is C. ANSWER: C If CDFG is a kite, find each measure. 12. GF SOLUTION: Use the Pythagorean Theorem. Since the length must be positive, GF = 5. ANSWER: 5 13. SOLUTION: eSolutions Manual - Powered by Cognero Since a kite can only have one pair of opposite congruent angles and Let x be the measure of angle D. Page 4 Since the length must be positive, GF = 5. ANSWER: Practice Test - Chapter 6 5 13. SOLUTION: Since a kite can only have one pair of opposite congruent angles and Let x be the measure of angle D. The sum of the measures of the angles of a quadrilateral is 360. So, ANSWER: 122 ALGEBRA Quadrilateral MNOP is a rhombus. Find each value or measure. 14. SOLUTION: Since the diagonals of a rhombus are perpendicular, by the definition of perpendicular lines. ANSWER: 90 15. If PR = 12, find RN. SOLUTION: In a rhombus, diagonals bisect each other. So, RN = PN = 12. ANSWER: 12 16. If , find . SOLUTION: Since MNOP is a rhombus, diagonal eSolutions Manual - Powered by Cognero bisects . Therefore, . Page 5 SOLUTION: In a rhombus, diagonals bisect each other. So, RN = PN = 12. ANSWER: Practice Test - Chapter 6 12 16. If , find . SOLUTION: Since MNOP is a rhombus, diagonal bisects . Therefore, . ANSWER: 62 17. CONSTRUCTION The Smiths are building an addition to their house. Mrs. Smith is cutting an opening for a new window. If she measures to see that the opposite sides are congruent and that the diagonal measures are congruent, can Mrs. Smith be sure that the window opening is rectangular? Explain. SOLUTION: Sample answer: Yes, that is enough to show that the opening is a rectangle. Since each pair of opposite sides are the same length, the opening is a parallelogram. By Theorem 6.14, if the diagonals of a parallelogram are congruent then it is a rectangle. ANSWER: Sample answer: Yes. If it is a rectangle, the diagonals are congruent. Use JKLM to find each measure. 18. SOLUTION: Opposite angles of a parallelogram are congruent. So, ANSWER: 109 19. JK SOLUTION: Opposite sides of a parallelogram are congruent. So, JK = 6. ANSWER: 6 20. eSolutions Manual - Powered by Cognero SOLUTION: Consecutive angles in a parallelogram are supplementary. Page 6 Opposite sides of a parallelogram are congruent. So, JK = 6. ANSWER: Practice Test - Chapter 6 6 20. SOLUTION: Consecutive angles in a parallelogram are supplementary. So, ANSWER: 71 ALGEBRA Quadrilateral DEFG is a rectangle. 21. If DF = 2(x + 5) – 7 and EG = 3(x – 2), find EG. SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG. Use the value of x to find EG. EG = 3(9 – 2) = 21 ANSWER: 21 22. If , find SOLUTION: . . Substitute in . ANSWER: 22 eSolutions Manual Cognero = 14- +Powered 2x andbyGF = 4(x 23. If DE SOLUTION: – 3) + 6, find GF. Page 7 ANSWER: Practice Test - Chapter 6 22 23. If DE = 14 + 2x and GF = 4(x – 3) + 6, find GF. SOLUTION: Opposite sides of a rectangle are congruent. DE = GF 14 + 2x = 4( x – 3) + 6 Substitute. 14 + 2x = 4x – 12 + 6 Distributive Property 14 + 2x = 4x – 6 Simplify. 14 – 2x = –6 Subtract 4x from each side. –2x = –20 Subtract 14 from each side. 2x = 20 Divide each side by -1. x =10 Divide each side by 2. Use the value of x to find GF. GF = 4(x – 3) + 6 Original equation = 4(10 - 3) + 6 Substitute. = 4(7) + 6 Subtract. = 34 Simplify. ANSWER: 34 Determine whether each quadrilateral is a parallelogram. Justify your answer. 24. SOLUTION: Each pair of opposite angles are congruent. By Theorem 6.10 if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. No other information is needed to determine that the figure is a parallelogram. ANSWER: Yes, opposite angles are congruent. 25. SOLUTION: There are 2 pairs of consecutive angles that are congruent. Since opposite sides are not congruent, this fails Theorem 6.9. If both pairs of opposite sides are congruent, the quadrilateral is a parallelogram. This is not a parallelogram. ANSWER: No, opposite sides are not congruent. eSolutions Manual - Powered by Cognero Page 8