5.1 Sum of Interior Angles of Convex Polygons

Описание: 5.1 Sum of Interior Angles of Convex Polygons
Concave and convex polygons
What is polygons?
A polygon is a simple closed plane figure formed by three or more lines segments joined end to end, no two of which in succession are collinear.
The line segments forming the polygon are called sides.
The common end point of any two sides is called vertex of the polygon.
The angles formed inside the polygon are called interior angles.
For instance:- Triangle is the simplest polygon. Other polygons are quadrilaterals (four sides), and pentagons (five sides), and so on.
Common naming of polygon
If a polygon has 𝑛 sides, it is named as “𝑛-gon”.
Example:
Polygon with 16 sides is called 16-gon.
Question 1
How many sides and vertices does a hexagon have?
Solution:
A hexagon is a polygon with 6 sides and 6 vertices

Concave and convex polygons
Convex polygon measure of each interior angle is less than 180 degrees.
Vertices are always outwards.
Concave polygon when there is at least one interior angle whose measure is more than 180 degrees.
Vertices present inwards and also outwards.
Note: If none of the side extensions intersect the polygon, then the polygon is convex; otherwise it is concave.
Is the quadrilateral in figure 5.1 convex or concave?
Solution:
convex since the measure of each interior angle is less than 180°.
We check by measuring each angle using protractor.
Sum of interior angles of a convex polygon
Sum of the measure of interior angles of any triangle is 180°.
For any triangle 𝐴𝐵𝐶, note that
𝑎 + 𝑏 + 𝑐 = 180° .
Example
Determine the measure of interior angles of a quadrilateral.
Solution
a Quadrilateral 𝐴𝐵𝐶𝐷.
Draw a line segment from one vertex, say 𝐴 ,to its opposite vertex 𝐶.
Line segment 𝐴𝐶 is one of the diagonals of the given quadrilateral.
the quadrilateral is divided into two triangles so that the sum of interior angles of a quadrilateral is two times the sum of interior angles of a triangle.
The sum of the measure of interior angles of a quadrilateral is 360°.
Note that a quadrilateral can be divided into two triangles.
This is 𝑛 × 180° = 2 × 180°
where 𝑛 = 2 the number of triangles forming the quadrilateral.
Into how many triangles can we divide the pentagon?
Pentagon is a five sided polygon.
Draw a line segment (diagonal) from 𝐴 to another vertex in a polygon (You do not have to draw lines to the adjacent vertices, since they are already connected by a side).
pentagon is divided into three triangles. So that the sum of interior angles of a pentagon is three times the sum of interior angles of a triangle.
Note also that a pentagon can be divided into three triangles.
b. What is the sum of the measures of interior angles of a pentagon?
Sum of the measure of interior angles of a pentagon
= 𝑛 × 180°
= 3 × 180°
= 540°
where 𝑛 = 3 is the number of triangles forming the pentagon.
Deriving sum of the interior angles of a polygon
As the number of sides of a polygon increases; the number of triangles also increases. Furthermore, the sum of measure of interior angles of the polygon increases.
If 𝑛 is the number of sides of a polygon, then 𝑛 − 2 non overlapping triangles are formed.
Sum of the measure of interior angles of 𝑛 sided polygon is equal to 𝑛 − 2 × 180°
Example:
Find the sum of the measure of interior angles of hexagon.
Solution:
Hexagon is a 6-sided polygon .
𝑛 = 6.
the sum of measure of interior angles of a hexagon
=180° × 𝑛 − 2
= 180° × 4
= 720°.