2. Section 3.5 Polygons

3. A polygon is:  A closed plane figure made up of several line segments they are joined together.  The sides to not cross each other.  Exactly two lines meet at every vertex. Not Polygons: Polygons:

4. Side - one of the line segments that make up the polygon. Vertex - point where two sides meet. Two or more of these points are called vertices. Diagonal - a line connecting two vertices that isn't a side. Interior Angle - Angle formed by two adjacent sides inside the polygon. Exterior Angle - Angle formed by two adjacent sides outside the polygon.

5. Convex Polygons: A polygon such that: o Every interior angle is less than 180° o Every line segment between two vertices does not go on the exterior of the polygon. (It remains inside or on the boundaries of the polygon) Nonconvex Polygons: A polygon such that: o At least one interior angle has measure greater than 180° o There exists a line segment between two vertices is on the exterior of the polygon.

6. Polygons are classified by the number of sides they have. Number of SidesName 3 Triangle3 Triangle 4 Quadrilateral4 Quadrilateral 5 Pentagon5 Pentagon 6 Hexagon6 Hexagon 7 Heptagon7 Heptagon 8 Octagon8 Octagon 10 Decagon10 Decagon

7. Diagonals of a Polygon  To find the number of diagonals in each polygon we need to know n, the number of sides  Use the formula:

8. Suppose you start with a pentagon. If you pick any vertex of that figure, and connect it to all the other vertices, how many triangles can you form? three triangles If you start with vertex A and connect it to all other vertices (it's already connected to B and E by sides) you form three triangles. Each triangle contains 180 0. So the total number of degrees in the interior angles of a pentagon is: 3 x 180° = 540°

9. We can apply this to any convex polygon. The sum of the measure of the angles of a convex polygon with n sides is: (n-2)180 We can also use this formula to find the number of sides in a polygon.

10. Example 1: Find the number of degrees in the sum of the interior angles of an octagon. An octagon has 8 sides. So n = 8. Using the formula, that gives us (8-2)180= (6)180 = 1080° Example 2: Find the number of degrees in the sum of the interior angles of a quadrilateral. A quadrilateral has 4 sides. So n = 4. Using the formula, that gives us (4-2)180= (2)180 = 360°

11. Example 3: How many sides does a polygon have if the sum of its interior angles is 720 0 ? Since, this time, we know the number of degrees, we set the formula equal to 720 °, and solve for n. (n-2) 180 = 720 n-2 = 4 n= 6

12. A polygon that is both equiangular (all angles congruent) and equilateral (all sides congruent). Find the measure of each interior angle and exterior angle of a regular hexagon: The sum of interior angles is (6-2)180=720 Since all six angles are congruent, each interior angle has a measure of 720/6=120 Since each polygon has exterior angles that add to 360, each exterior angle has measure 360/6=60 120 °

13. The sum of the measures of the exterior angles of any convex polygon, one at each vertex, is 360° exterior angle An exterior angle of a polygon is formed by extending one side of the polygon. See Geometer’s Sketchpad for Demonstration.

14. Find the sum of the exterior angles of : A Pentagon: A Decagon: A 15-Sided Polygon: A 7-Sided Polygon:360° 360° 360° 360° The sum of the measures of the exterior angles of any convex polygon, one at each vertex, is always 360°! Remember...

15. Example Using an Octagon  # of Diagonals:  Sum of the Interior Angles:  Each Interior Angle:  Sum of the Exterior Angles:  Each Exterior Angle:

16. Example Using an Heptagon  # of Diagonals:  Sum of the Interior Angles:  Each Interior Angle:  Sum of the Exterior Angles:  Each Exterior Angle:

17. Example Using a Quadrilateral  # of Diagonals:  Sum of the Interior Angles:  Each Interior Angle:  Sum of the Exterior Angles:  Each Exterior Angle: