G4.1 Introducing Polygons Objectives: Define “polygon” Classify polygons by their sides Classify polygons by their angles. Discover the sum of the interior angles of any polygon Use the Polygon-Sum conjecture to solve for missing angles

Concept: Define Polygon POLYGON: A 2-dimensional object made of line segments connected at their endpoints to enclose an area. POLYGONS NOT POLYGONS

Concept: Define Concave & Convex CONVEX: Any polygon that has all of it’s verticies pointing away from the interior of the polygon ON A PEGBOARD, A RUBBER BAND WILL COPY THE CONVEX SHAPE CONCAVE: Any polygon that has a vertex that points to the interior of the polygon.

Concept: NAMES OF POLYGONS SIDES Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon N n-gon

Concept: Classify by sides and angles Equilateral: All sides are equal Equiangular: All angles are equal Regular: All angles are equal and all sides are equal. This is just like a triangle classification.

Concept: How Many Diagonals? FIND THE NUMBER DIAGONALS CAN BE FORMED FROM A SINGLE VERTEX IN ANY POLYGON? 6 SIDES = 3 DIAGONALS

Concept: How Many Diagonals? TRY AGAIN. CAN YOU MAKE A CONJECTURE ABOUT THE RELATIONSHIP BETWEEN THE NUMBER OF SIDES IN THE POLYGON AND THE NUMBER OF DIAGONALS FROM ONE VERTEX? 8 SIDES = 5 DIAGONALS DIAGONALS = N – 3

Concept: How Many TRIANGLES? FIND THE NUMBER OF TRIANGLES FORMED BY DIAGONALS FROM ONE VERTEX 8 SIDES = 6 TRIANGLES

Concept: How Many TRIANGLES? IS THERE A RELATIONSHIP BETWEEN THE NUMBER OF SIDES IN A POLYGON AND THE NUMBER OF TRIANGLES CREATED? 8 SIDES = 6 TRIANGLES TRIANGLES = N – 2

Concept: How Many DEGREES? HOW MANY DEGREES ARE IN EVERY TRIANGLE? HOW MANY TRIANGLES IN THE POLYGON? CAN YOU MAKE A CONJECTURE or FORMUL ABOUT THE SUM OF THE DEGREES IN A POLYGON? 4 SIDES = 2 TRIANGLES

CONCEPT: POLYGON SUM CONJECTURE EACH TRIANGLE HAS 180° IF N IS THE NUMBER OF SIDES OF A POLYGON THEN: INTERIOR ANGLE SUM = (N–2)180º

2 3 1 4 5 INT ANGLE SUM = ( 5 – 2 ) 180° ( 3 ) 180° = 540°

CONCEPT: REGULAR POLYGONS REGULAR POLYGONS HAVE EQUAL SIDES AND EQUAL ANGLES SO WE CAN FIND THE MEASURE OF EACH INTERIOR ANGLE

N EACH INTERIOR ANGLE OF A REGULAR POLYGON = (N – 2 ) 180 REMEMBER N = NUMBER OF SIDES

REGULAR HEXAGON INT ANGLE SUM = (6 – 2 ) 180 = 720° EACH INT ANGLE = 720 = 120° 6

CONCEPT: Using Polygon-Sum Find the measures of g and h. Quadrilateral Sum = 360º Pentagon Sum = 540º

CONCEPT: Using Polygon-Sum –225 –225 3g = 315 ___ ___ 3 3 g = 105º

CONCEPT: Using Polygon-Sum A circle has 360º 108 + 130 + x = 360º 238 + x = 360º –238 –238 x = 122º

CONCEPT: Using Polygon-Sum 90 + 122 + 66 + h = 360º 278 + h = 360º 122º –278 –278 h = 82º

Concept: Polygon Diagonal Sum Conj. Remember worksheet 5.1 where you had to find the number of diagonals of a polygon. Counting proved to be difficult. WE DIDN’T DO THAT PART!! It would prove to be time consuming and difficult

d = n(n–3) 2 Concept: Polygon Diagonal Sum Conj. GOOD NEWS!!! There’s an App for that!!! Okay, not realy. But there is an equation. d = n(n–3) 2 Check it out: D = 6(6-3) = 6(3) = 18 = 9 2 2 2

Homework: PM G4.1