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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. Thus, the statement "No angles in an isosceles trapezoid are congruent." is false. It should be "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. Thus, the statement "A midsegment of a trapezoid is a segment that connects any two nonconsecutive vertices." is false. It should be " 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. Thus, the statement "The diagonal of a trapezoid is the segment that connects the midpoints of the legs." is false. It eSolutions Manual - Powered by Cognero Page 1 should be the " midsegment". ANSWER: 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. Thus, the statement "The diagonal of a trapezoid is the segment that connects the midpoints of the legs." is false. It should be the " midsegment". ANSWER: false, midsegment 7. A rectangle is not always a parallelogram. SOLUTION: By definition a rectangle is a parallelogram with four right angles. Thus, the statement "A rectangle is not always a parallelogram." is false. It should be "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. Thus, the statement "A quadrilateral with only one set of parallel sides is a parallelogram." is false. It should be "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. Thus, the statement "The leg of a trapezoid is one of the parallel sides." is false. It should be "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 = 10 in by Cognero . eSolutions Manual n - Powered Page 2 The nonparallel sides are called legs. Thus, the statement "The leg of a trapezoid is one of the parallel sides." is false. It should be "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: We know that opposite sides of a parallelogram are congruent. eSolutions So, Manual - Powered by Cognero ANSWER: Page 4 ANSWER: Study 65Guide 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: 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 29. If , find 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. 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, eSolutions Manual - Powered by Cognero ANSWER: 68 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. 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 eSolutions Manual - Powered by Cognero Page 17