The detailed answer covers definitions of polygons, their properties, angle characteristics, and explores the concepts of regular polygons, exterior angles, and rectangular shapes.
Polygon: a plane figure with multiple angles and sides.
Regular Polygon: a polygon with equal sides and angles like an equilateral triangle or a square.
Concave Polygon: a polygon with at least one interior angle greater than 180 degrees.
Convex Polygon: a polygon with all interior angles less than 180 degrees.
Interior angle properties: Sum of interior angles in a polygon = (n-2) * 180 degrees where n is the number of sides.
Exterior angle properties: Sum of exterior angles in any polygon is always 360 degrees.
Ratio of exterior to interior angle: In a regular polygon, each exterior angle = 360/n degrees where n is the number of sides. Given a ratio of 1:4, the polygon must have 5 sides (Pentagon).
Measure of each exterior angle of a regular pentagon: Each exterior angle of a regular pentagon = 360/5 = 72 degrees.
Making a rectangular slab in different ways: The mason can ensure the slab is rectangular by measuring the diagonals; if they are equal, it's a rectangle. Alternatively, by ensuring all angles are right angles or opposite sides are parallel, the slab can be rectangular.
Square defined as a rhombus: A square can be defined as a rhombus since it has equal angles but differs in that all sides must be equal for a square, not just opposite sides.
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