Transcript Angles of a Polygon

Quadrilaterals
Chapter 6
Objectives
1. Recognize and name convex polygons and
regular polygons.
2. Find the measures of interior angles and
exterior angles of convex polygons.
Definition: Polygon
A closed geometric figure in a plane formed by
connecting line segments endpoint to endpoint
with each segment intersecting exactly two
others.
Definition: Convex Polygon
A convex polygon is a simple polygon whose interior is
a convex set. The following properties of a simple
polygon are all equivalent to convexity:
Every internal angle is less than 180 degrees or equal
to 180 degrees.
Every line segment between two vertices of the
polygon does not go exterior to the polygon (i.e., it
remains inside or on the boundary of the polygon).
Classifying Polygons
Classifying Polygons
Polygons are also classified according to the
number of sides.
A triangle is a three-sided polygon.
A quadrilateral is a four-sided polygon.
A pentagon is a five-sided polygon.
A hexagon is a six-sided polygon.
A heptagon is a seven-sided polygon.
An octagon is an eight-sided polygon.
A nonagon is a nine-sided polygon.
A decagon is a ten-sided polygon.
Classifying Polygons
Regular Polygons
When a polygon is both equilateral and equiangular, it is
referred to as a regular polygon. For a polygon to be
regular, it must also be convex.
Definition: Diagonal
A diagonal of a polygon is any segment that joins two
nonconsecutive vertices.
Segments QS , SU , UR , RT and QT are the diagonals in this polygon.
Angle Measures in a Polygon
Let n = the number of sides of the polygon
then,
the polygon has n vertices
the polygon has n pairs of supplementary angles
the sum of the measures of all the angles = 180n
Angle Measures in a Polygon
If n = the number of sides of the polygon, then
180n = (interior angle sum) + (exterior angle sum);
180n = 180(n-2) + (exterior angle sum);
180n = 180n – 360 + (exterior angle sum);
360 = exterior angle sum.
Theorem 3-13
The sum of the measures of the angles of a
convex polygon with n sides is (n - 2)180º.
Theorem 3-14
The sum of the measures of the exterior angles
of any convex polygon, one angle at each vertex,
is 360º.
Review Quiz
Find the measure of an interior angle:
1. For a regular quadrilateral.
2. For a regular octagon.
3. For a regular polygon with 15 sides.
4. For a regular polygon with 20 sides.
Review Quiz - Solutions
Find the measure of an interior angle:
1.For a regular quadrilateral.
90º
2.For a regular octagon.
135º
3.For a regular polygon with 15 sides.
156º
4.For a regular polygon with 20 sides.
162º