Convex vs. Concave Polygons Interior Angles of Polygons Exterior Angles of Polygons Polygons

To be or not to be…  Polygons consist of entirely segments  Consecutive sides can only intersect at endpoints. Nonconsecutive sides do not intersect.  Vertices must only belong to one angle  Consecutive sides must be noncollinear.

A rose by any other name…  To name a polygon, start at a vertex and either go clockwise or counterclockwise. ab c d e f

Diagonals  A diagonal of a polygon is any segment that connects two nonconsecutive (nonadjacent) vertices of the polygon.

Convex polygons A polygon in which each interior angle has a measure less than 180.

Polygons can be CONCAVE or CONVEX CONVEX CONCAVE

Classify each polygon as convex or concave.

Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon 15 sides Pentadecagon

Important Terms EQUILATERAL - All sides are congruent EQUIANGULAR - All angles are congruent REGULAR - All sides and angles are congruent

# of sides # of triangles Sum of measures of interior angles 31 1(180) = (180) = (180) = (180) = 720 n n-2 (n-2)  180

Regular Polygons No. of sidesNameAngle SumInterior Angle 3triangle quadrilateral 180°60° 360°90° pentagon540°108° hexagon720°120° heptagon 900° 128 7/9° octagon1080°135° nonagon1260° 140° decagon1440°144°

If a convex polygon has n sides, then the sum of the measure of the interior angles is (n – 2)(180°)

Use the regular pentagon to answer the questions. A)Find the sum of the measures of the interior angles. B)Find the measure of ONE interior angle 540° 108°

Exterior angles of a triangle The exterior angle of a triangle is equal to the sum of the interior opposite angles. interior opposite angles exterior angle A B C D i.e.  ACD =  ABC +  BAC

20° C A B D E Find  CED = 40° 40°  CDE = 40° 40°  EAB 60° = 120° 120° 55°  CAE = 85° 85°  ACE 35° = 35°  ABE = 20° 20°  AEB = 120° 120° Example

Two more important terms Exterior Angles Interior Angles

b Exterior angles of a polygon Exterior angles of a polygon add to 360°. At each vertex:interior angle + exterior angle = 180° a c e a + b + c + d + e = 360° d

In any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°

1 3 2

Find the measure of ONE exterior angle of a regular hexagon. 60°

Find the measure of ONE exterior angle of a regular heptagon. 51.4°

Each exterior angle of a polygon is 18 . How many sides does it have? n = 20

The sum of the measures of five interior angles of a hexagon is 535 o. What is the measure of the sixth angle? 185°

x + 3x + 5x + 3x = 360 o 12x = 360 o x = 30 o Use substitution to solve for each angle measure. The measure of the exterior angle of a quadrilateral are x, 3x, 5x, and 3x. Find the measure of each angle. 30°, 90°, 150°, and 90°

If each interior angle of a regular polygon is 150 , then how many sides does the polygon have? n = 12

Find  ABC = 120° 120° Example  ADC = 60° 60°  BAC = 30° 30°  CAD = 30° 30° ABCDE is a regular hexagon with centre O. C A B D E F O  ACD  ODE  EOD = 90° = 60° 60° = 60° 60°