3.4: the polygon angle-sum theorem

3.4: THE POLYGON ANGLE-SUM THEOREM

OBJECTIVE: STUDENTS WILL BE ABLE TO…• TO CLASSIFY POLYGONS, AND TO FIND THE SUMS OF

INTERIOR AND EXTERIOR ANGLES OF POLYGONS

HELLO!!! It’s going to be a good day

POLYGON:• CLOSED PLANE FIGURE• AT LEAST 3 SIDES THAT ARE SEGMENTS• SIDES INTERSECT ONLY AT ENDPOINTS• ADJACENT SIDES ARE NOT COLLINEAR POLYGONS NOT POLYGONS

TO NAME A POLYGON:• START AT ANY VERTEX• LIST VERTICES CONSECUTIVELYNAME THE POLYGON. NAME THE SIDES AND

ANGLES.

E D

C

B

A

COMMON POLYGONS:SIDES NAME

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n n- gon

DIAGONAL OF A POLYGON

• A SEGMENT IN A POLYGON DRAWN FROM A VERTEX TO A NONADJACENT VERTEX

CONVEX VS. CONCAVE

CONVEX POLYGON• NO DIAGONAL WITH

POINTS OUTSIDE POLYGON

CONCAVE POLYGON• AT LEAST ONE DIAGONAL

OUTSIDE POLYGON

POLYGON-ANGLE-SUM THEOREM

THE SUM OF THE MEASURES OF ANGLES IN AN N-GON :

(N-2)180EXAMPLE:FIND THE SUM OF THE MEASURES OF A 15-GON.

FINDING MISSING ANGLE MEASURES

• FIGURE OUT WHAT THE SUM OF THE INTERIOR ANGLES SHOULD BE

• ADD UP EXPRESSIONS AND SET THEM EQUAL TO THE SUM FROM PART ONE.

EXAMPLE: FIND Y.

120°

85°

y°

53°

Example:1. The sum of the measures of a given

polygon is 720. How many sides are in the polygon?

2.Pentagon ABCDE has 5 congruent angles. Find the measure of each angle.

POLYGON-EXTERIOR-ANGLE SUM THEOREM

THE SUM OF THE MEASURES OF EXTERIOR ANGLES OF A POLYGON IS ALWAYS 360⁰.

EXAMPLE:

WHAT IS THE SUM OF EXTERIOR ANGLES FOR A 500-GON?

EQUILATERAL POLYGON: All sides

EQUIANGULAR POLYGON: All angles

REGULAR POLYGON: equilateral and equiangular

If you have a REGULAR POLYGON, to find: ONE Interior Measure: ONE Exterior measure:

nn 180)2(

n360

Examples:1. The sum of the measures of angles of a

polygon with n sides is 1980. Find n, the number of sides.

2. Find the measures of one interior angle and one exterior angle of a REGULAR hexagon.

3. The measure of an exterior angle of a regular polygon is 30⁰. What is the measure of an interior angle? How many sides are there?

TYPE 2: WRITE AND ANSWER THE FOLLOWING ON A SEPARATE PIECE OF PAPER. (2 POINTS

EACH QUESTION)

1. FIND THE SUM OF THE INTERIOR AND EXTERIOR ANGLES OF AN OCTAGON.

2. FIND THE MEASURE OF ONE INTERIOR ANGLE AND ONE EXTERIOR ANGLE OF A REGULAR 15-GON.

3. THE SUM OF THE INTERIOR ANGLES OF A POLYGON IS 4680°. FIND THE NUMBER OF SIDES.