Curves and Polygons in the Plane MTH 232 Curves and Polygons in the Plane

Curves and Regions A curve is the set of points that a pencil (crayon, marker) can trace without lifting until all the points are covered. If each point is touched only once, the curve is simple. If the initial point (where you start) is the same as the final point (where you finish), the curve is closed. If the initial (final) point is the only point touched more than once, the curve is simple and closed. A simple, closed curve divides the plane into three regions: The curve itself; The interior of the curve; The exterior of the curve. The interior and exterior are called the regions defined by the curve.

Concave and Convex Figures; Polygons A figure is convex if for each pair of points P and Q in the interior of the figure, the line segment PQ lies entirely in the interior. A figure is concave if is not convex. A polygon is a simple closed curve made up of finitely many line segments. The endpoints of the line segments are called vertices and the segments themselves are called sides.

More About Polygons Polygons are sometimes classified by the number of sides (or vertices) they have (e.g., a pentagon has five sides). Interior angles are formed by two sides with a common vertex. Exterior angles are formed by extending a side beyond one of the vertices. An interior angle and its adjacent exterior angle are supplementary.

Theorem Sums of the Angle Measures in a Complex Polygon: The sum of the measures of the exterior angles of a convex polygon is 360 degrees. The sum of the measures of the interior angles of a polygon with n sides is 180(n – 2) degrees.

Classification of Triangles By Angle Measure. A triangle is: acute if all three angles are acute; right if one angle is a right triangle; obtuse if one interior angle is obtuse. By Side Length. A triangle is: scalene if no two sides have the same length; isosceles if (exactly) two sides have the same length; equilateral if all three sides have the same length.

Classification of Quadrilaterals A kite has two distinct pairs of congruent adjacent sides. A trapezoid has (at least) one pair of parallel sides. An isosceles trapezoid has a pair of congruent angles along one of the parallel sides. A parallelogram has two pair of parallel sides (opposite sides and angles are congruent, and consecutive angles are supplementary). A rhombus is a parallelogram with all sides the same length. A rectangle is a parallelogram with all right angles. A square is (1) a rhombus with four equal angles, or (2) a rectangle with all equal sides.

Regular Polygons Regular polygons are both equilateral (all sides congruent) and equiangular(all angles congruent). For a regular polygon with n sides: Each interior angles measures 180(n – 2)/n. Each exterior angle measures 360/n. Each central angle measures 360/n.