Pre-AP Bellwork 10-19

1
Pre-AP Bellwork 10-19

• 3) Solve for x..

30?
(8 6x)?
(4x 2)
2
Pre-AP Bellwork 10-24

• 5) Find the values of the variables and then the
  measures of the angles.

z
x
w
30
y
(2y 6)
3
3-4 Polygon Angle-Sum Theorem

• Geometry

4
Definitions
SIDE

• Polygona 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.

5
Example 1 Identifying Polygons

• State whether the figure is a polygon. If it is
  not, explain why.
• Not D has a side that isnt a segment its 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.
6
Polygons are named by the number of sides they
have MEMORIZE
Number of sides Type of Polygon
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n n-gon

7
Convex or Concave???

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 3 1 1?180?180?
Quadrilateral 2?180?360?
Pentagon
Hexagon
Nonagon (9)
n-gon n
10
Pre-AP Bellwork 10 - 24

• 6) Find the sum of the interior angles of a
  dodecagon.

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

142?
88?
136?
105?
136?
x?
13
SOLUTION

• S(hexagon) (6 2) ? 180? 4 ? 180?
  720?.
• Add the measure of each of the interior angles of
  the hexagon.

14
SOLUTION

• 136? 136? 88? 142? 105? x? 720?.
• 607 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
16
EX.2 Find the measure of an interior angle of a
decagon.

• n10

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?
Corollary to Thm. 11.1
(n 2) ?180? 140?n
Multiply each side by n.
180n 360 140?n
Distributive Property
Addition/subtraction props.
40n 360
n 90
Divide each side by 40.
19
Copy the item below.
20
EXTERIOR ANGLE THEOREMS
3-10
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 LifeEx. 4 Finding
Angle measures of a polygon
25
Using Angle Measures in Real LifeEx. 5 Using
Angle Measures of a Regular Polygon
26
Using Angle Measures in Real LifeEx. 5 Using
Angle Measures of a Regular Polygon
27
Using Angle Measures in Real LifeEx. 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
• 135?
• 145?

28
Using Angle Measures in Real LifeEx. Finding
Angle measures of a polygon