Question

Please solve the following question



Sides of the Pentagon are produced.

Find the sum of angle 1,2,3,4 and 5​

Answer 1

Answer:

180°

Step-by-step explanation:

Let a, b, c, d, e be the interior angles of the pentagon as in the diagram.

Also, let x and y be the two other angles in the triangle with ∠1, as in the diagram.

∠1 = 180° - x - y

   = ( 180° - x ) + ( 180° - y ) - 180°

   = a + b - 180°

Similarly:

∠2 = b + c - 180°,          ∠3 = c + d - 180°,

∠4 = d + e - 180°,          ∠5 = e + a - 180°

So

∠1 + ∠2 + ∠3 + ∠4 + ∠5

= 2 ( a + b + c + d + e ) - 5×180°

The sum of the interior angles of a polygon with n sides is (n-2)×180°, so for a pentagon with n=5 sides, we have

a + b + c + d + e = 3×180°

Therefore

∠1 + ∠2 + ∠3 + ∠4 + ∠5

= 2 ( a + b + c + d + e ) - 5×180°

= 6×180° - 5×180°

= 180°