Polygons and Convexity Lesson 2.5 Polygons and Convexity pp. 61-67

Objectives: 1. To define a polygon and its parts. 2. To identify convex and concave polygons. 3. To classify polygons according to the number of sides.

Concave sets always have some points that cannot be connected by a segment without leaving the set.

EXAMPLE 1 Show that an angle is a concave set. Label two points on the angle so that the connection segment is not contained in the angle. Notice that most of AB is in the interior of the angle instead of being part of the angle itself. A B

Definition A convex set has the property that any two of its points determine a segment contained in the set. A concave set is a set that is not convex.

Definition A polygon is a simple closed curve that consists only of segments. A side of a polygon is one of the segments that define the polygon.

Definition A vertex of a polygon is an endpoint of a side of the polygon.

Definition An angle of a polygon is an angle with two properties: its vertex is a vertex of the polygon and each side of the angle contains a side of the polygon.

Definition A polygonal curve is a curve that is either not closed or not simple, made only of line segments.

Definition A polygonal region is a polygon together with its interior.

EXAMPLE 2 Classify these polygonal regions as convex or concave. b) c) d)

Definition Triangle A 3-sided polygon. Quadrilateral A 4-sided polygon. Pentagon A 5-sided polygon. Hexagon A 6-sided polygon. Heptagon A 7-sided polygon.

Definition Octagon A 8-sided polygon. Nonagon A 9-sided polygon. Decagon A 10-sided polygon. Hendecagon A 11-sided polygon. Dodecagon A 12-sided polygon.

Definition An equilateral polygon is a polygon in which all sides have the same length. An equiangular polygon is a convex polygon in which all angles have the same degree measure.

Definition A regular polygon is a polygon that is both equilateral and equiangular. A diagonal of a polygon is any segment that connects two vertices but is not a side of the polygon.

Definition The interior of a convex polygon is the intersection of the interiors of its angles. The exterior of a convex polygon is the union of the exteriors of its angles.

Homework pp. 65-67

►A. Exercises 1. Classify as convex or concave.

►A. Exercises 1. Classify by number of sides. If it is not a polygon, explain why.

►A. Exercises 3. Classify as convex or concave.

►A. Exercises 3. Classify by number of sides. If it is not a polygon, explain why.

►A. Exercises 7. Classify as convex or concave.

►A. Exercises 7. Classify by number of sides. If it is not a polygon, explain why.

►B. Exercises 20-23. No. of Name of No. of No. of Total sides polygon vertices diag./vert. diag. V P D 5 7 n

1 2 2

►B. Exercises 20-23. No. of Name of No. of No. of Total sides polygon vertices diag./vert. diag. V P D 5 7 n pentagon 5 2 5

1 2 3 4 4

►B. Exercises 20-23. No. of Name of No. of No. of Total sides polygon vertices diag./vert. diag. V P D 5 7 n pentagon 5 2 5 heptagon 7 4 14 n-gon n n-3 ½n(n-3)

■ Cumulative Review True/False 28. If two lines do not intersect, then they are parallel.

■ Cumulative Review True/False 29. Skew lines may be coplanar.

■ Cumulative Review True/False 30. A diameter is a chord of a circle.

■ Cumulative Review True/False 31. If l and m are skew lines and m and n are skew lines, then l and n are skew lines.

■ Cumulative Review True/False 32. Is the union of an angle and its interior a region? Explain.