Introduction to angles of a pentagon

Introduction to angles of a pentagon:

              A pentagon is a closed two dimensional figure that is the union of line segments in a plane. A pentagon has the five sides and five angles.

              Pentagon can be classified as

•  Convex pentagon
• Concave pentagon

             For  a given pentagon, we can measure the following angles of Pentagon.

•  Interior angle
• Exterior angle

Interior Angles of a Pentagon:

             An angle located within a closed figure or the angle formed inside a polygon by two adjacent sides is referred to as an Interior angle.   

             For finding the interior angles of a pentagon, start with one vertex and join it to all other vertices to form three triangles inside the pentagon.

            A pentagon has 5 sides and 5 vertices and three triangles are formed by connecting the vertices, i.e., number of triangles inside a pentagon is two less than the sides of the pentagon. So, the sum of all interior angles in a polygon is given by,

                                      Sum of all interior angles = 180 ( n – 2 )

                     For a pentagon, number of sides n = 5 and the sum of all interior angles are

                                                                                    = 180 ( n – 2 )

                                                                                    = 180 ( 5 – 2 )

                                                                                    = 180 * 3

                                                                                    = 540 degrees.

                     Each interior angle of a pentagon  = 540 / 5

                                                                                   = 108 degrees.

Ex 1: Find the number of sides of pentagon if the sum of interior angles of pentagon is 540 degrees.

Sol: Step 1: Sum of interior angles of pentagon = ( n - 2) x 180°, n is number of sides.

                        540°   =  ( n - 2 ) x 180°

         Step 2:  Divide by 180° to each side.

                        540°/180° = ( n - 2 ) x 180° / 180°

                             3           =  n - 2

         Step 3: Add 2 to each side.

                            3 + 2      = n - 2 + 2

                                5         =  n

Therefore, nmber of sides for pentagon is five.

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Exterior Angles of a Pentagon:

            An exterior angle of a pentagon is an angle formed by two sides of the pentagon which normally shares a common vertex. In a vertex, two exterior angles can be formed.

           In each vertex the angle formed by extending a side is equal to 180 degree

                                       Supplementary angle  = interior angle + exterior angle

           For a n sided polygon,

                                    Sum of all exterior angles, = 180 n – 180 ( n - 2 )

                                                                                     = 180 n – 180 n + ( - 180  x  – 2 )

                                                                                     = 0 + 360

                                                                                     = 360 degrees. 

                      So,The sum of exterior angles of a pentagon = 360 degrees

                      In a pentagon, for 5 vertices 5 exterior angles can be formed.      

                       Each exterior angle of a pentagon = 360 / 5 = 72 degrees.