Pre-AP Bellwork ) Solve for x  (4x + 2)° (8 + 6x)

Pre-AP Bellwork ) Find the values of the variables and then the measures of the angles. 2 z° w° x° y° 30° (2y – 6)°

3 3-4 Polygon Angle-Sum Theorem Geometry

Definitions:  Polygon—a plane figure that meets the following conditions: It is formed by 3 or more segments called sides, such that no two sides with a common endpoint are collinear. Each side intersects exactly two other sides, one at each endpoint.  Vertex – each endpoint of a side. Plural is vertices. You can name a polygon by listing its vertices consecutively. For instance, PQRST and QPTSR are two correct names for the polygon above. SIDE

Example 1: Identifying Polygons  State whether the figure is a polygon. If it is not, explain why.  Not D – has a side that isn’t a segment – it’s an arc.  Not E– because two of the sides intersect only one other side.  Not F because some of its sides intersect more than two sides/ Figures A, B, and C are polygons.

Polygons are named by the number of sides they have – MEMORIZE Number of sidesType of Polygon 3Triangle 4Quadrilateral 5Pentagon 6Hexagon 7Heptagon 8Octagon 9Nonagon 10Decagon 12Dodecagon nn-gon

Convex or Concave??? 7 A convex polygon has no diagonal with points outside the polygon. A concave polygon has at least one diagonal with points outside the polygon

8 Measures of Interior and Exterior Angles  You have already learned the name of a polygon depends on the number of sides in the polygon: triangle, quadrilateral, pentagon, hexagon, and so forth. The sum of the measures of the interior angles of a polygon also depends on the number of sides.

9 Measures of Interior and Exterior Angles  For instance... Complete this table Polygon# of sides # of triangles Sum of measures of interior ’s Triangle 31 1●180=180 Quadrilateral 2●180=360 Pentagon Hexagon Nonagon (9) n-gon n

Pre-AP Bellwork ) Find the sum of the interior angles of a dodecagon. 10

11 Measures of Interior and Exterior Angles  What is the pattern? (n – 2) ● 180.  This relationship can be used to find the measure of each interior angle in a regular n-gon because the angles are all congruent.

12 Ex. 1: Finding measures of Interior Angles of Polygons  Find the value of x in the diagram shown: 88 136 142 105 xx

13 SOLUTION:  S(hexagon)= (6 – 2) ● 180 = 4 ● 180 = 720.  Add the measure of each of the interior angles of the hexagon. 88 136 142 105 xx

14 SOLUTION: 136 + 136 + 88 + 142 + 105 +x = 720 x = 720 X = 113 The measure of the sixth interior angle of the hexagon is 113.

15 Polygon Interior Angles Theorem  The sum of the measures of the interior angles of a convex n-gon is (n – 2) ● 180  COROLLARY: The measure of each interior angle of a regular n-gon is: ● (n-2) ● 180 or

EX.2 Find the measure of an interior angle of a decagon…. n=10 16

17 Ex. 2: Finding the Number of Sides of a Polygon  The measure of each interior angle is 140. How many sides does the polygon have?  USE THE COROLLARY

18 Solution: = 140 (n – 2) ●180= 140n 180n – 360 = 140n 40n = 360 n = 90 Corollary to Thm Multiply each side by n. Distributive Property Addition/subtraction props. Divide each side by 40.

19 Copy the item below.

20 EXTERIOR ANGLE THEOREMS 3-10

21 Ex. 3: Finding the Measure of an Exterior Angle

22 Ex. 3: Finding the Measure of an Exterior Angle 3-10 Simplify.

23 Ex. 3: Finding the Measure of an Exterior Angle 3-10

24 Using Angle Measures in Real Life Ex. 4: Finding Angle measures of a polygon

25 Using Angle Measures in Real Life Ex. 5: Using Angle Measures of a Regular Polygon

26 Using Angle Measures in Real Life Ex. 5: Using Angle Measures of a Regular Polygon

27 Using Angle Measures in Real Life Ex. 5: Using Angle Measures of a Regular Polygon Sports Equipment: If you were designing the home plate marker for some new type of ball game, would it be possible to make a home plate marker that is a regular polygon with each interior angle having a measure of: a. 135°? b. 145°?

28 Using Angle Measures in Real Life Ex. : Finding Angle measures of a polygon