Download Interior Angles of Quadrilaterals

Sect. 6.1
Polygons
Goal 1
Describing Polygons
Goal 2
Interior Angles of
Quadrilaterals
Describing Polygons
Polygon: a plane figure with following properties.
From the Greek poly = many and gon = angle
1. A closed figure formed by 3 or more line
segments called Sides
2. The sides intersect at points called the vertices.
3. The angle between two sides is called an interior
angle or vertex angle.
4. You name a polygon by listing its vertices
consecutively.
Describing Polygons
These are Polygons
These are Not
Polygons
Describing Polygons
Polygons are named by the number of Sides
# of Sides
Polygon Name
# of Sides
Polygon Name
3
Triangle
10
Decagon
4
Quadrilateral
11
Hendecagon
5
Pentagon
12
Dodecagon
6
Hexagon
13
13-gon
7
Heptagon
14
14-gon
8
Octagon
15
15-gon
9
Nonagon
n
n-gon
Describing Polygons
• Polygon
is Convex if for any two points inside
polygon, the line segment joining these two points is
also inside. A figure not convex is Concave.
Convex
Concave
Rubber Band Test: If you can wrap a rubber
band around the polygon, and it touches all parts
of every side, then it is convex.
Describing Polygons
The following figures are convex.
Describing Polygons
The following figures are concave. Note the
red line segment drawn between two points
inside the figure that also passes outside of
the figure.
Interior Angles of Quadrilaterals
Regular Polygons
• If all sides of a polygon have equal lengths and if all
angles have the same measure then the polygon is called
a Regular Polygon.
• A regular triangle is called an Equilateral Triangle.
• A regular quadrilateral is called a Square.
Some common regular polygons.
Equilateral Triangle
Regular Pentagon
Regular Octagon
Square
Regular Hexagon
Some different types of REGULAR
POLYGONS. These shapes are defined as
• a convex polygon
• with all sides congruent and
• all angles congruent
Describing Polygons
Quadrilateral: A four-sided polygon.
Interior Angles of Quadrilaterals
Diagonal
The line segment connecting two
nonadjacent vertices in a polygon.
B
BD, BF , and DE
A
E
are all diagonals
D
F
Interior Angles of Quadrilaterals
A quadrilateral has 4 sides. If we draw a line segment
connecting 2 of its opposite vertices, we'll have 2
triangles.
So, what could you
conclude about the total
degree measure inside the
quadrilateral?
Could you say this about
any quadrilateral?
Interior Angles of Quadrilaterals
Theorem 6.1 Interior Angles of a Quadrilateral
The sum of the measures of the interior angles of a Quadrilateral
is 360°.
B
mA + mB + mC + mD = 360°
A
C
D
Describing Polygons
Number of sides
Type of Polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Hetagon
8
Octagon
9
Nonagon
10
Decagon
11
Hendecagon
12
Dodecagon
100
Hectagon
Sum of interior angles
Describing Polygons
Convex Polygon
# of Sides
# of Triangles
Sum of Angles
Triangle
3
1
180 x 1 = 180
Quadrilateral
4
2
Pentagon
5
Hexagon
6
Heptagon
7
Octagon
8
Nonagon
9
n-gon
n
The Angle Sum of a Convex Polygon:
Interior Angles of Quadrilaterals
Example 1
Find mB, mC, and mD
B
55°
A
C
x°
x°
D
Interior Angles of Quadrilaterals
Example 2
Find mB, mC, and mD. Is quadrilateral
ABCD regular?
B
A
x°
80°
(x - 20)°
x°
D
C
Describing Polygons
Homework:
pp. 325 – 328 12 – 20 even 24 – 38 even, 42 – 52 even