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### Polygons

Polygons
Closed shapes or figures in a plane with three or more sides are called polygons.
Vertices
The endpoints of the sides of polygon are called vertices. When name a polygon, its vertices are named consecutive either clockwise or counterclockwise.
Consecutive sides
Consecutive sides are refer to two sides that have an endpoint in common. In the figure above, side DA and AB are consecutive sides.
The polygon can be named as ABCD or ADCB.
Opposite sides of a quadrilateral
Opposite sides of a quadrilateral do not intersect. AD and BC are a pair of opposite sides.
AB and DC are a pair of opposite sides.
Opposite angles of a quadrilateral
Opposite angles are located at alternate vertices of a quadrilateral. In the quadrilateral ABCD,
angle A and angle C are a pair of opposite angles.
angle B and angle D are a pair of opposite angles.
Diagonal
A diagonal is a line segment that joins opposite (alternate) vertices. In the quadrilateral ABCD, line segment BD is a diagonal.
Number of sides of a polygons
The name of a polygons based on the sides of the polygons. The polygon with 3 sides is a triangle. The polygon with 4 sides is a quadrilateral. The polygon width 5 sides is a pentagon. The polygon with 6 sides is a hexagon. The polygon with 8 sides is a octagon.
Regular polygon
If each sides of a polygon are equal and each interior angles are equal, then the polygon is a regular polygon.
Sum of interior angles
If a convex polygon has n sides, then the sum of its interior angles is the products of n-2 and 180 degrees.
Sum of interior angles of a polygon = (n - 2) × 180o In the figure above, the polygon ABCDE has 5 sides.
The polygon has (5 - 2) = 3 triangles
The sum of the interior angle of the polygon ABCDE = (5 - 2) × 180o = 540o
Exterior angle
If a polygon is convex, then the sum of its exterior angles, one by each vertex, is 360 degrees. The figure above is a polygon,
sum of exterior angles of the polygon = angle 1 + angle 2 + angle 3 + angle 4 + angle 5 = 360o
Example 1: In the figure above, name all pairs of
a. opposite sides
b. consecutive sides
c. opposite angles
d. consecutive angles
Solution:
a. opposite sides are AB and CD; AC and BD.
b. consecutive sides are CA and AB; AB and BD; BD and DC; DC and CA.
c. opposite angles are angle A and angle D; angle B and angle C.
d. consecutive angles are angle A and angle B; angle B and angle D; angle D and angle C; angle C and angle A.
Example 2:
Find the sum of the interior angles in the figure below. Solution:
The figure above is an octagon.
An octagon has 8 sides. n = 8
sum of interior angles of the octagon = (n - 2) × 180o = (8 - 2) × 180o = 1080o
Example 3:
In the figure below, what is the sum of interior angles of this polygon? Solution:
The polygon ABCDEF has 6 sides, n = 6
It has (n - 2) = 4 non-overlapping triangles, then
sum of interior angles of the polygon = (n - 2) × 180o
= (6 - 2) & timeo
= 4 × 180o
= 720o
Therefore, the sum of the interior angles of the polygon is 720o.