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 your questions: 4. Fourth angle of a quadrilateral Step 1: Recall the sum of angles in a quadrilateral. The sum of the interior angles of any quadrilateral is 360^. Step 2: Calculate the fourth angle. Let the fourth angle be x. x + 100^ + 60^ + 80^ = 360^ x + 240^ = 360^ x = 360^ - 240^ x = 120^ The fourth angle of the quadrilateral is 120^. 5. Size of each angle of a regular decagon Step 1: Determine the number of sides of a decagon. A decagon is a polygon with 10 sides, so n=10. Step 2: Calculate the sum of interior angles. The sum of the interior angles of a polygon with n sides is given by the formula (n-2) × 180^. Sum of angles = (10-2) × 180^ Sum of angles = 8 × 180^ Sum of angles = 1440^ Step 3: Calculate the size of each angle. For a regular polygon, all interior angles are equal. Divide the sum of angles by the number of sides. Each angle = (1440^)/(10) Each angle = 144^ The size of each angle of a regular decagon is 144^. 6. What is a polygon? A polygon is a closed two-dimensional shape made up of straight line segments connected end-to-end. The line segments do not cross each other, and the shape encloses a single region. 7. Angle between directions a) W and E West and East are opposite directions. The angle between opposite directions is 180^. The angle between W and E is 180^. b) NE and SW Northeast and Southwest are opposite directions. The angle between opposite directions is 180^. The angle between NE and SW is 180^. 8. 3-figure bearing of B from A Step 1: Understand 3-figure bearings. A 3-figure bearing is an angle measured clockwise from the North direction, expressed with three digits. Step 2: Identify the direction of B from A. From the diagram, point B is in the Southeast quadrant relative to point A. The line AB appears to be exactly halfway between South and East. Step 3: Calculate the bearing. North is 000^. East is 090^. South is 180^. The direction Southeast is exactly halfway between South and East. Therefore, the bearing is 180^ - 45^ = 135^. Alternatively, it is 90^ (to East) + 45^ (from East to Southeast) = 135^. The 3-figure bearing of B from A is 135^. That's 2 down. 3 left today — send the next one.