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Math 310

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  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)

  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. exterior interior interior angle side vertex diagonal Ex exterior angle 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 Quadrilateral Hierarchy Trapezoid Kite Parallelogram Isosceles Trapezoid Rhombus Rectangle Square