Download Sum of Interior and Exterior Angles in Polygons

Essential Question –
How can I find angle measures in polygons without
using a protractor?
Key Standard – MM1G3a
Polygons
 A polygon is a closed figure formed by a finite number of
segments such that:
1. the sides that have a common endpoint
are noncollinear, and
2. each side intersects exactly two other
sides, but only at their endpoints.
Nonexamples
Polygons
 Can be concave or convex.
Concave
Convex
Polygons are named by number of sides
Number of Sides
3
4
5
Polygon
Triangle
Quadrilateral
Pentagon
6
7
8
9
10
Hexagon
Heptagon
12
n
Dodecagon
n-gon
Octagon
Nonagon
Decagon
Regular Polygon
 A convex polygon in which all the sides are congruent
and all the angles are congruent is called a regular
polygon.
 Draw a:
Quadrilateral
Hexagon
Pentagon
Heptagon
Octogon
 Then draw diagonals to create triangles.
 A diagonal is a segment connecting two
nonadjacent vertices (don’t let segments cross)
 Add up the angles in all of the triangles in
the figure to determine the sum of the
angles in the polygon.
 Complete this table
Polygon
# of sides
# of triangles
Sum of
interior angles
Polygon
# of sides
# of triangles
Sum of interior
angles
Triangle
3
1
180°
Quadrilateral
4
2
2 · 180 = 360°
Pentagon
5
3
3 · 180 = 540°
Hexagon
6
4
4 · 180 = 720°
Heptagon
7
5
5 · 180 = 900°
Octagon
8
6
n-gon
n
n-2
6 · 180 = 1080°
(n – 2) · 180°
Polygon Interior Angles Theorem
The sum of the measures of the interior angles of a
convex n-gon is (n – 2) • 180.
Examples –
1. Find the sum of the measures of the interior angles of a 16–
(16 – 2)*180 = 2520°
gon.
2. If the sum of the measures of the interior angles of a convex
polygon is 3600°, how many sides does the polygon have.
(n – 2)*180 = 3600
180n – 360 = 3600
+ 360 + 360
3. Solve for x.
4x - 2
82
180n = 3960
180
180
n = 22 sides
108
(4 – 2)*180 = 360
2x + 10 108 + 82 + 4x – 2 + 2x + 10 = 360
6x + 198 = 360
6x = 162
6
6
x = 27
Draw a quadrilateral and extend the sides.
There are two sets of angles formed when the sides
of a polygon are extended.
• The original angles are called interior angles.
• The angles that are adjacent to the
interior angles are called exterior angles.
These exterior angles can be formed when any side
is extended.
What do you notice about the interior angle and
the exterior angle? They form a line.
What is the measure of a line?
180°
What is the sum of an interior angle with the
exterior angle? 180°
If you started at Point A, and
followed along the sides of
the quadrilateral making the
exterior turns that are
marked, what would happen?
You end up back where you
started or you would make a
circle.
A
D
B
C
What is the measure of the
degrees in a circle? 360°
Polygon Exterior Angles Theorem
 The sum of the measures of the exterior angles of a
convex polygon, one at each vertex, is 360°.
 Each exterior angle of a regular polygon is 360
n
where n is the number of sides in the polygon
Example
Find the value for x.
Sum of exterior angles is 360°
(4x – 12)⁰
68⁰
60⁰
54⁰
(3x + 13)⁰
65⁰
(4x – 12) + 60+ (3x + 13) + 65 + 54+ 68 = 360
7x + 248 = 360
– 248 – 248
7x = 112
7
7
x = 12
What is the sum of the exterior angles in an octagon? 360°
What is the measure of each exterior angle in a regular
octagon?
360°/ = 45°