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Study Guide and Review 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 . 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 . 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 . 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 eSolutions Manual - Powered by Cognero 15. ≈ 166.15 Page 1 Substitute n = 6 in . Study Guide and Review 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 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 Use ABCD to find each measure. 16. SOLUTION: We know that consecutive angles in a parallelogram are supplementary. So, Substitute. 17. AD SOLUTION: We know that opposite sides of a parallelogram are congruent. So, 18. AB SOLUTION: We know that opposite sides of a parallelogram are congruent. So, eSolutions Manual - Powered by Cognero 19. SOLUTION: Page 2 17. AD SOLUTION: WeGuide knowand that Review opposite sides of a parallelogram are congruent. Study So, 18. AB SOLUTION: We know that opposite sides of a parallelogram are congruent. So, 19. SOLUTION: We know that opposite angles of a parallelogram are congruent. So, 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. 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 eSolutions Manual - Powered by Cognero Determine whether each quadrilateral is a parallelogram. Justify your answer. Page 3 5y = 60 y = 12 Study So,Guide x = 5and andReview y = 12. 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 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. 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. 25. PROOF Write a two-column proof. Given: Prove: Quadrilateral EBFD is a parallelogram. SOLUTION: eSolutions - Powered by Cognero YouManual need to walk through the Page 4 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. SOLUTION: One pair of opposite sides are parallel and congruent. By Theorem 6.12 if one pair of opposite sides of a quadrilateral both parallel and congruent, then the quadrilateral is a parallelogram. No other information is needed Study Guide andisReview to determine that the figure is a parallelogram. 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. 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. 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 + 20by Cognero 72 = 13x eSolutions Manual - Powered 52 = 13x 4 = x Page 5 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. ALGEBRA Quadrilateral EFGH is a rectangle. 29. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Substitute. 30. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Substitute. 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. eSolutions Manual by Cognero FH = FK +- Powered KH Diagonals of a rectangle bisect each other. = FK + FK FK = KH, substitution = 32 + 32 Substitute. Page 6 All four angles of a rectangle are right angles. So, Substitute. Study Guide and Review 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. 32. Find SOLUTION: All four angles of a rectangle are right angles. So, 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. 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. eSolutions Manual - Powered by Cognero Since the length must be positive, AE = 7.9. Page 7 = 4(3) – 6 x = 3 = 12 – 6 Multiply. = 6 Subtract. Guide and Review Study So, EF = 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. 35. SOLUTION: All the four sides of a rhombus are congruent. So, is an isosceles triangle. Therefore, 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 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, 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) eSolutions Manual - Powered by Cognero SOLUTION: First graph the quadrilateral. Page 8 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, Study Guide and Review 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. 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. eSolutions Manual - Powered by Cognero Find each measure. 41. GH Page 9 Since the products of the slopes of consecutive sides are -1, the sides are perpendicular. So,Guide the quadrilateral is also a rectangle and a square. Study and Review Find each measure. 41. GH SOLUTION: Use the Pythagorean Theorem. Since the length must be positive, GH = 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, eSolutions Manual - Powered by Cognero Page 10