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Find the Sum of the Measures of the Interior Angles of Each Convex Polygon

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Find the Sum of the Measures of the Interior Angles of Each Convex Polygon 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 Page 1 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, 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, 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. 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. So, x = 7 and y = 8. 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 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. 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: 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 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. 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 , 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 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. 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 Page 2 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, 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, 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. 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 .
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