1. Math 310
9.2
Polygons

2. Curve & Connected
Idea
The idea of a curve is something you could draw on paper without lifting your pencil.
The idea of connected is that a set can’t be split into two disjoint sets.

3. Simple Curve
Def
A simple curve is a curve that does not cross itself, with the possible exception of having the same beginning and ending points.

4. Closed Curve
Def
A closed curve is a curve that has the same beginning and ending point.

5. Polygon
Def
A polygon is a planar figure, that is a simple closed curve consisting entirely of straight segments. Where two segments meet is called a vertex of the polygon. Each segment is called a side of the polygon.

6. Convex and Concave Polygons
Def
A convex polygon is one in which every point in the interior of the polygon can be joined by a line segment that stays entirely inside the polygon. A concave polygon would be one in which there exist at least two points, such that when joined by a line segment, the segment passes outside of the polygon.

7. Polygonal Region
Def
A polygon divides the plane its in, into three regions, the interior, exterior, and the curve itself. The interior and the curve itself comprise the polygonal region we are concerned with.

8. Types of Polygons
Polygons are classified, primarily, according to the number of sides that they have. This is summarized in the following table.

9. Types of Polygons (cont)
Number of Sides (or vertices)
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
n-gon n

10. Parts of a Polygon
Side
Vertex
Interior angle
Exterior angle
diagonal

11. Interior Angle
Def
An interior angle of a polygon is created by three consecutive vertices.

12. Exterior Angle
Def
An exterior angle of a polygon is created by extending a side out from the polygon and reading the angle between the contiguous side.

13. Diagonal
Def
A diagonal of a polygon is a segment connecting any two non-consecutive vertices.

14. Ex exterior interior interior angle exterior angle side vertex
diagonal
exterior angle

15. Congruency
Idea
Two segments are congruent if the tracing of one can be fitted exactly onto the other. Two angles are congruent if their measures are the same. Two polygons are congruent if all their parts are congruent.
Notation:

16. Ex.

17. Regular Polygon
Def
A polygon is called regular if all its angles and sides are congruent.

18. Ex

19. Further Classification of Polygons
Triangles
Quadrilaterals

20. Triangles Right triangle Acute triangle Obtuse triangle
Scalene triangle
Isosceles triangle
Equilateral triangle

21. Right Triangle
Def
A right triangle contains exactly one right angle.

22. Acute & Obtuse Triangle
Def
An acute triangle contains all acute angles while an obtuse triangle contains exactly one obtuse angle.

23. Scalene Triangle
Def
A scalene triangle has three non-congruent sides. (ie all the sides are of different length.)

24. Isosceles Triangle Def
An isosceles triangle has at least two congruent sides.

25. Equilateral Triangle Def
An equilateral triangle is a triangle with all sides congruent.

26. Quadrilaterals Trapezoid Kite Isosceles trapezoid Parallelogram
Rectangle
Rhombus
Square

27. Trapezoid
Def
A trapezoid is a quadrilateral with at least one pair of parallel sides.

28. Isosceles Trapezoid Def
An isosceles trapezoid is a trapezoid with the non-parallel sides being congruent

29. Kite
Def
A quadrilateral with two non-overlapping pairs of congruent adjacent sides.
NON-VERTEX ANGLES
VERTEX ANGLES

30. Parallelogram
Def
A parallelogram is a quadrilateral with opposite sides parallel.

31. Rectangle
Def
A rectangle is a parallelogram with a right angle.

32. Rhombus
A rhombus is a quadrilateral with four congruent sides.

33. Square
A square is a rhombus with a right angle.

34. Quadrilateral Hierarchy
By looking at the definitions, we see that some of the quadrilateral definitions fulfill one or more of the other definitions automatically. For example, the definition of a square also satisfies the definition of a rhombus. This allows us to set up a hierarchy among the quadrilaterals which will be very useful when discussing their various properties.

35. Quadrilateral Hierarchy
Trapezoid
Kite
Parallelogram
Isosceles Trapezoid
Rhombus
Rectangle
Square