Transcript Quadrilaterals - DavisEric.com

Quadrilaterals
Chapter 6
Angles of Polygons
Chapter 6.1
Types of Polygons
# of Sides
Type of
Polygon
# of
Sides
Type of
Polygon
3
Triangle
8
Octagon
4
Quadrilateral
9
Nonagon
5
Pentagon
10
Decagon
6
Hexagon
12
Dodecagon
7
Heptagon
n
n-gon
Regular Polygons
• Equilateral  all sides are congruent
• Equiangular  all angles are congruent
• Regular Polygon  Equilateral and Equiangular
Identify
Polygons
Name
Quadrilateral
Octagon
Pentagon
Equilateral?
Yes
Yes
Yes
Equiangular?
No
No
Yes
Regular?
No
No
Yes
Diagonals
• A segment that connects two nonconsecutive
vertices in a polygon.
Diagonals
Diagonals
Interior Angles of a Polygon
Angle sum of a Pentagon
= 3 (180) = 5400
Angle sum of a Hexagon
= 4 (180) = 7200
Interior Angle Sum Theorem
• The sum of the measures of the interior
angles of any convex n-gon is
= (n – 2) • 180o
# of s
180o per 
• The measure of each interior angle of a
regular n-gon is
(n  2) 180

n
Interior Angles of Regular Polygons
ARCHITECTURE A mall is designed so that five
walkways meet at a food court that is in the shape
of a regular pentagon. Find the sum of the
measures of the interior angles of the pentagon.
Since the pentagon is a convex
polygon, we can use the Interior
Angle Sum Theorem.
Interior Angle Sum Thm.
Simplify.
Answer: The sum of the measures of the angles is 540.
A decorative window is designed to
have the shape of a regular octagon.
Find the sum of the measures of the
interior angles of the octagon.
A. 1440
B. 1260
C. 1080
D. 900
Find the value of X
Interior angle sum of a 7-gon is
o
(7-2)•180
95o
5 • 180o
155o
900o
128+102+143+95+ 155+137 + X = 900
760 + X = 900
137o
X = 140
X
143o
102o
128o
The measure of an interior angle of
a regular polygon is 135. Find the
number of sides in the polygon.
(n  2)(180 )
135 
n
135n  (n  2)(180)
135 n  180 n  360
 45n  360
 360
n
8
 45
Answer: The polygon has 8 sides.
The measure of an interior angle of a
regular polygon is 144. Find the number
of sides in the polygon.
A. 12
B. 9
C. 11
D. 10
(n  2)(180 )
144 
n
Interior Angles of a Quadrilateral
(theorem)
• The sum of the measures of the interior
angles of a quadrilateral is 360o.
1
2
4
3
m1 + m2 + m3 + m4 = 360o
Sample Problem
85o
2xo
xo
65o
85 + 65 + x + 2x = 360o
150 + 3x = 360o
3x = 210o
x = 70o
Interior Angles of Nonregular Polygons
Find the measure of each interior angle.
Since
angles is
the sum of the measures of the interior
Interior Angles of Nonregular Polygons
Sum of measures of int. 
Substitution
Combine like terms.
Subtract 8 from each side.
Divide each side by 32.
Use the value of x to find the measure of each angle.
Answer: mR = 55, mS = 125, mT = 55, mU = 125
Find the value of x.
A. x = 7.8
B. x = 22.2
C. x = 15
D. x = 10
Exterior Angles of a Polygon
• The sum of the exterior angles
of any convex polygon is 360o
Sketchpad
• The measure of each exterior
360
angle of a regular polygon is
n
Exterior Angles
Find the measures of an exterior angle and an interior
angle of convex regular nonagon ABCDEFGHJ.
At each vertex, extend a side to form one exterior angle.
Exterior Angles
The sum of the measures of the exterior angles is 360. A
convex regular nonagon has 9 congruent exterior angles.
Divide each side by 9.
Answer: The measure of each exterior angle is 40.
Since each exterior angle and its corresponding
interior angle form a linear pair, the measure of
the interior angle is 180 – 40 or 140.
Find the measures of an exterior angle
and an interior angle of convex regular
hexagon ABCDEF.
A. 30; 150
B. 45; 135
C. 60; 120
D. 20; 160
Homework
Chapter 6.1
• Pg 321
1-10, 15-20, 23-29