This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems you've sent. 5) If the exterior angles of a regular polygon are in the ratio of 2:3, find the size of the exterior angle. The phrasing of this question is unusual, as all exterior angles of a regular polygon are equal. We will assume it means the ratio of the exterior angle to the interior angle is 2:3. a) Find the size of the exterior angle. Step 1: Let the exterior angle be E and the interior angle be I. We assume the ratio of the exterior angle to the interior angle is 2:3. (E)/(I) = (2)/(3) E = (2)/(3)I Step 2: Use the property that the interior angle and exterior angle of a polygon sum to 180^. I + E = 180^ Step 3: Substitute E = (2)/(3)I into the sum equation and solve for I. I + (2)/(3)I = 180^ (5)/(3)I = 180^ I = 180^ × (3)/(5) = 36^ × 3 = 108^ Step 4: Calculate the exterior angle E. E = 180^ - I = 180^ - 108^ = 72^ Alternatively, E = (2)/(3)I = (2)/(3)(108^) = 72^. The exterior angle is 72^. (This corresponds to a regular pentagon, as 360^ / 72^ = 5 sides). b) What is the interior angle sum of a heptagon? Step 1: Identify the number of sides for a heptagon. A heptagon has n=7 sides. Step 2: Use the formula for the sum of interior angles of a polygon. The sum of interior angles of an n-sided polygon is given by: Sum = (n-2) × 180^ Step 3: Substitute n=7 into the formula and calculate the sum. Sum = (7-2) × 180^ = 5 × 180^ = 900^ The interior angle sum of a heptagon is 900^. 7) Find the value of x in the figures below. a) The figure is a quadrilateral. The sum of the interior angles of a quadrilateral is 360^. The angles are (x-5)^, (x-1)^, (x+11)^, and (x+19)^. Step 1: Set up the equation by summing all interior angles. (x-5) + (x-1) + (x+11) + (x+19) = 360 Step 2: Combine like terms. 4x + (-5 - 1 + 11 + 19) = 360 4x + 24 = 360 Step 3: Solve for x. 4x = 360 - 24 4x = 336 x = (336)/(4) x = 84 The value of x is 84. b) The figure is a 5-pointed star (pentagram). The sum of the angles at the points of a pentagram is 180^. All five angles are given as x. Step 1: Set up the equation by summing the angles at the points. x + x + x + x + x = 180 5x = 180 Step 2: Solve for x. x = (180)/(5) x = 36 The value of x is 36. c) The figure is a rectangle. In a rectangle, opposite sides are equal in length. The left side is 2x and the right side is x+1. Step 1: Set the expressions for the opposite sides equal to each other. 2x = x+1 Step 2: Solve for x. 2x - x = 1 x = 1 The value of x is 1. d) The figure shows two angles on a straight line. Angles on a straight line sum to 180^. The angles are x and 2x. Step 1: Set up the equation by summing the angles. x + 2x = 180 Step 2: Combine like terms. 3x = 180 Step 3: Solve for x. x = (180)/(3) x = 60 The value of x is 60. e) The figure is a hexagon. The sum of the interior angles of an n-sided polygon is (n-2) × 180^. For a hexagon, n=6. The angles are (x+9)^, (x-1)^, (x+1)^, (x+3)^, (x+5)^, and (x+7)^. Step 1: Calculate the sum of interior angles for a hexagon. Sum = (6-2) × 180^ = 4 × 180^ = 720^ Step 2: Set up the equation by summing all given angles. (x+9) + (x-1) + (x+1) + (x+3) + (x+5) + (x+7) = 720 Step 3: Combine like terms. 6x + (9 - 1 + 1 + 3 + 5 + 7) = 720 6x + 24 = 720 Step 4: Solve for x. 6x = 720 - 24 6x = 696 x = (696)/(6) x = 116 The value of x is 116. f) The figure shows three angles around a point. Angles around a point sum to 360^. The angles are x, 110^, and 150^. Step 1: Set up the equation by summing the angles. x + 110 + 150 = 360 Step 2: Combine the constant terms. x + 260 = 360 Step 3: Solve for x. x = 360 - 260 x = 100 The value of x is 100. That's 2 down. 3 left today — send the next one.