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Substitute n = 15 in . Study Guide and Review Find the sum of the measures of the interior angles of each convex polygon. 11. decagon 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 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 = 6 in by Cognero . eSolutions Manual n - Powered 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 Use ABCD to find each measure. Page 1 19. Study Guide and Review Use ABCD to find each measure. SOLUTION: We know that opposite angles of a parallelogram are congruent. So, ALGEBRA Find the value of each variable in each parallelogram. 20. 16. SOLUTION: We know that consecutive angles in a parallelogram are supplementary. So, Substitute. SOLUTION: Since the opposite sides of a parallelogram are congruent, 2x + 9 = 4x – 5. Solve for x. 2x + 9 = 4x – 5 Opp. sides of a parallelogram are . 9 = 2x – 5 Subtract 2x from each side. 14 = 2x Add 5 to each side. 7 = x Divide each side by 2. 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, 19. Since the sides of a parallelogram are parallel, the alternate interior angles are congruent. Thus, the alternate interior angles at top and bottom must both have a measure of 4y as shown . Since the opposite sides of the parallelogram are parallel, the consecutive interior angles must be supplementary. So, set the sum of 42, 23, 4y, and 83 equal to 180 and solve for y. SOLUTION: We know that opposite angles of a parallelogram are congruent. So, ALGEBRA Find the value of each variable in each parallelogram. 20. eSolutions Manual - Powered by Cognero SOLUTION: So, x = 7 and y = 8. Page 2 Since the opposite angles are congruent, 2x + 41 = 115. Solve for x. 2x + 41 = 115 2x = 74 x = 37 Study So,Guide x = 7and andReview y = 8. ALGEBRA Find x and y so that the quadrilateral is a parallelogram. 21. SOLUTION: 26. Since the opposite sides are congruent, 3y + 13 = 2y + 19. Solve for y. 3y + 13 = 2y + 19 y =6 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 Since the opposite angles are congruent, 2x + 41 = 115. Solve for x. 2x + 41 = 115 2x = 74 x = 37 Solve for y. 3y + 36 = 9y - 12 36 = 6y - 12 48 = 6y 8 = y ALGEBRA Find x and y so that the quadrilateral is a parallelogram. When x = 4 and y = 8 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 27. SOLUTION: We know that diagonals of a parallelogram bisect each other. So, . Solve for x. Solve for y. 3y +Manual 36 = 9y - 12 by Cognero eSolutions - Powered 36 = 6y - 12 48 = 6y Alternate interior angles in a parallelogram are congruent. Page 3 48 = 6y 8 = y Study Guide Review When x =and 4 and y = 8 the quadrilateral is a parallelogram. Solve for y. 5y = 60 So, y = 12. When x = 5 and y = 12 the quadrilateral is a parallelogram. 28. PARKING The lines of the parking space shown below are parallel. How wide is the space (in inches)? 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. 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. ALGEBRA Quadrilateral EFGH is a rectangle. 29. If 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 eSolutions Manual x- Powered by + Cognero Substitute = 8 in 5x 20. 5x + 20 = 5(8) + 20 , find . SOLUTION: All four angles of a rectangle are right angles. So, Substitute. 30. If SOLUTION: , find . Page 4 5x + 20 = 5(8) + 20 = 60 Study Guide and Review So, the length of the space is 60 inches. ALGEBRA Quadrilateral EFGH is a rectangle. = 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, 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. 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 eSolutions Manual - Powered by Cognero 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 Page 5 Theorem. = 4(3) – 6 x = 3 = 12 – 6 Multiply. = 6 Subtract. Guide and Review Study 35. SOLUTION: All the four sides of a rhombus are congruent. So, triangle. Therefore, So, EF = 6. ALGEBRA ABCD is a rhombus. If EB = 9, AB = 12 and , find each measure. is an isosceles 36. CE SOLUTION: The diagonals of a rhombus are perpendicular. Use AE to find CE. Use the Pythagorean Theorem. 34. AE SOLUTION: The diagonals of a rhombus are perpendicular. So, 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, Since the length must be positive, AE = 7.9. 35. SOLUTION: All the four sides of a rhombus are congruent. So, triangle. Therefore, All the four sides of a rhombus are congruent. So, triangle. Therefore, is an isosceles 36. CE is an isosceles 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: The diagonals of a rhombus are perpendicular. Use AE to find CE. Use the Pythagorean Theorem. SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, FG = FJ = 2.5 cm. eSolutions Manual - Powered by Cognero Since the length must be positive, AE = 7.9. Page 6 Find each measure. 41. GH The diagonals are perpendicular to each other. So, in the right triangle EAB, AllGuide the four sides of a rhombus are congruent. So, Study and Review triangle. Therefore, Since the length must be positive, GH = 19.2. is an isosceles 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. 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 . Find each measure. 41. GH So, SOLUTION: Use the Pythagorean Theorem. Since the length must be positive, GH = 19.2. 42. eSolutions Manual - Powered by Cognero Page 7