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Polygons
Lesson 3-4: Polygons
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Polygons
Definition: A closed figure formed by a finite number of coplanar
segments so that each segment intersects exactly two
others, but only at their endpoints.
These figures are not polygons
These figures are polygons
Lesson 3-4: Polygons
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Classifications of a Polygon
Convex: No line containing a side of the polygon contains a point
in its interior
Concave:
A polygon for which there is a line
containing a side of the polygon and
a point in the interior of the polygon.
Lesson 3-4: Polygons
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Classifications of a Polygon
Regular: A convex polygon in which all interior angles have the
same measure and all sides are the same length
Irregular: Two sides (or two interior angles) are not congruent.
Lesson 3-4: Polygons
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Polygon Names
3 sides
Triangle
4 sides
Quadrilateral
5 sides
Pentagon
6 sides
Hexagon
7 sides
Heptagon
8 sides
Octagon
9 sides
Nonagon
10 sides
Decagon
12 sides
n sides
Dodecagon
n-gon
Lesson 3-4: Polygons
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Convex Polygon Formulas…..
Diagonals of a Polygon: A segment connecting nonconsecutive
vertices of a polygon
For a convex polygon with n sides:
The sum of the interior angles is
 n  2 180
The measure of one interior angle is
 n  2  180
n
The sum of the exterior angles is 360
The measure of one exterior angle is
360
n
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Examples:
1. Sum of the measures of the interior angles of a 11-gon is
(n – 2)180°  (11 – 2)180 °  1620
2. The measure of an exterior angle of a regular octagon is
360 360

 45
n
8
3. The number of sides of regular polygon with exterior angle 72 ° is
n
360
360
n
5
exterior angle
72
4. The measure of an interior angle of a regular polygon with 30 sides
 n2  180  (302) 180  28 180  168
n
30
Lesson 3-4: Polygons
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TRIANGLE FUNDAMENTALS
Triangle Sum Theorem:
The sum of the interior angles in a triangle is
180˚
Example:
5. In Δ ABC, m<A = 45°, m<B = 90°, find
m<C.
m<C= 180°- (45° + 90°)
m<C= 45°
Example:
6. If 2x, x and 3x are the measures of the
angles of a triangle, find the angles.
2x+x+3x=180°
6x= 180°
x=30°
Angles are 90°, 30° and 60°
Exterior Angle Theorem
The measure of the exterior angle of a triangle is equal to
the sum of the measures of the remote interior angles.
mACD  mA  mB
Remote Interior Angles
Exterior Angle
Example 7:
B
80
x
A
(3x-22)
D
C
3x - 22 = x + 80
3x – x = 80 + 22
2x = 102
X=51 m<A=51°
Find the mA.
Corollaries:
1.If two angles of one triangle are congruent to two
angles of a second triangle, then the third angles of
the triangles are congruent.
2. Each angle in an equiangular triangle is 60˚
3. Acute angles in a right triangle are complementary.
4. There can be at most one right or obtuse angle in a
triangle.