Geometry Notes
Section 6.1 Angles of Polygons
Polygon: a closed plane figure formed from 3 or more segments
Each segment is called a side of the polygon.
A common endpoint of 2 sides is called a vertex of the polygon.
A segment that connects any two nonconsecutive vertices is a diagonal.
Do-Now: Tell whether the figure is a polygon. If it is a polygon,
name it by the number of sides.
1.
2.
3.
4.
5.
6.
7.
Equilateral Polygon: all the sides are congruent
Equiangular Polygon: all the angles are congruent
# of sides
3
4
5
6
7
8
9
10
11
12
13
n
Polygon
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
hendecagon
dodecagon
13-gon
n-gon
Regular Polygon: both equilateral and equiangular
If a polygon is not regular, it is called irregular.
Center of a Regular Polygon: the point that is equidistant from all the vertices
Central Angle of a Regular Polygon: an angle whose vertex is the center of the polygon
& whose sides pass through 2 consecutive vertices
Convex polygon – all interior angles are less than 180
– all diagonals are contained in the interior of the polygon
A regular polygon is always convex.
Concave polygon – at least one interior angle is more than 180
– any part of a diagonal is in the exterior of the polygon
Examples:
1. Name a quadrilateral that is equiangular but not equilateral. ______________________
2. Name a quadrilateral that is equilateral but not equiangular. ______________________
3. Classify the polygon as regular or irregular and as convex or concave.
a.
b.
c.
d.
e.
Recall an interior angle is inside a polygon and is formed by two sides of the polygon.
An exterior angle is formed by one side of the polygon and the extension of an
adjacent side, so that an exterior angle is outside of the polygon.
 An interior angle and its exterior angle are __________________________.
interior 
exterior 
Interior Angles To find the sum of the interior angle measures of a convex polygon, draw all possible
diagonals from one vertex of the polygon, to create a set of triangles. The sum of the angle measures of
all the triangles equals the sum of the angle measures of the polygon.
In each convex polygon,
the number of triangles
formed is two less than
the number of sides n.
Interior Angles Sum Theorem: The sum of the measures of the interior angles of a
convex polygon with n sides is ________________.
To find the measure of each interior angle of a regular polygon with n sides: _________________
Examples:
1. Find the sum of the measures of the interior angles of an octagon.
2. Find the sum of the measures of the interior angles of a 20-gon.
3. Find the measure of one interior angle a regular decagon.
4. Determine how many sides a regular polygon has if the measure of one interior angle is 160.
5. The measures of the interior angles of polygon ABCDEF are (x+2),(x‒8),(x+7),(x‒3),(x+6),(x‒4).
Find the measures of the angles.
Exterior Angles In the polygons shown, an exterior angle
has been measured at each vertex. Notice that in each case,
the sum of the exterior angle measures is 360°.
Exterior Angles Sum Theorem: The sum of the exterior angles of a convex polygon,
one at each vertex, is ______________.
Examples:
1. Find the sum of the exterior angles of a 20-gon.
2. Find the measure of one exterior angle of a regular octagon.
3. Determine how many sides a regular polygon has if the measure of one exterior angle is 20.
4. Find x.
Assignment: Sec. 6.1 p.398 # 12 – 36 even, 40, 46, 49 (16 probs)