State true or false:

Triangle is a polygon whose sum of exterior angles is double the sum of interior angles.

Question

State true or false: Triangle is a polygon whose sum of exterior angles is double the sum of interior angles.

Accepted Answer

Given a statement, triangle is a polygon whose sum of exterior angles is double the sum of interior angles.We need to find whether the given statement is true or false.The angle sum property of a polygon says that the sum of all the interior angles of a triangle is always

180
    ∘

. Now, we know that with every interior angle, there exists an exterior angle which is formed by a side of a triangle and an extension of the adjacent side. So, let us consider a triangle with its interior and exterior angles as follows,Now, applying the angle sum property in the above triangle, we get,

∠x+∠y+∠z=
    
     180
    °
   
   .........(i)
 Now, we know that a pair of an exterior and interior angle at a location is always supplementary. Using this fact, we can express the exterior angles of the triangle in terms of its interior angles as follows,

p+x=
      180
      ∘

p=
      180
      ∘
     
     −x........(ii)

q+y=
      180
      ∘

q=
      180
      ∘
     
     −y........(iii)

r+z=
      180
      ∘

r=
      180
      ∘
     
     −z.........(iv)

Adding the equations (ii), (iii) and (iv), we get,

p+q+r=
      180
      ∘
     
     −x+
      180
      ∘
     
     −y+
      180
      ∘
     
     −z

p+q+r=
      540
      ∘
     
     −x−y−z

p+q+r=
      540
      ∘
     
     −(x+y+z).........(v)

Substituting the value of

∠x+∠y+∠z
  from equation (i) in (iv), we get,

p+q+r=
      540
      ∘
     
     −
      180
      ∘

p+q+r=
      360
      ∘

So, we can observe that the sum of the exterior angles of the triangle is

360
    ∘

, which is just double of the sum of its interior angles, i.e.,

180
    ∘

.Hence, the statement is true.Hence, the correct option is (1).

State true or false:

Triangle is a polygon whose sum of exterior angles is double the sum of interior angles.

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

download the app

Download the App