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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:
1. The curve itself;
2. The interior of the curve;
3. 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:
a) The sum of the measures of the exterior
angles of a convex polygon is 360 degrees.
b) 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:
1. acute if all three angles are acute;
2. right if one angle is a right triangle;
3. obtuse if one interior angle is obtuse.
• By Side Length. A triangle is:
1. scalene if no two sides have the same length;
2. isosceles if (exactly) two sides have the same
length;
3. 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:
1. Each interior angles measures 180(n – 2)/n.
2. Each exterior angle measures 360/n.
3. Each central angle measures 360/n.