State whether the statements are true or false: The interior angles of a triangle are in the ratio 1:2:3, then the ratio of its exterior angles is 3:2:1.

Question

Accepted Answer

Here, we are given the ratio of the interior angle measures of a triangle as 1:2:3 and we have, to check if the ratio of its exterior angle measures is 3:2:1, and accordingly state if the given statement is true or not. So, we know that according to the angle sum property of a triangle, the sum of all the interior angle measures of a triangle is

180
    °

.Now, based on the ratio of interior angles, let us assume that the interior angles of the triangle are

x,2x
  and

3x
 . Using the angle sum property of a triangle, we get,

x+2x+3x=
      180
      °

6x=
      180
      °

x=
      30
      °

Now, we have, the interior angles as follows,

x=
      30
      ∘

2x=2×
      30
      ∘
     
     =
      60
      ∘

3x=3×
      30
      ∘
     
     =
      90
      ∘

Also, we know that a pair of interior and exterior angle at the same vertex of a triangle always forms a linear pair, so, using this fact, the relation between the exterior and interior angles of a triangle is,Exterior angle =

180
    ∘
   
   −
  interior angleFrom the above relation, we have, the exterior angles of the given triangle corresponding to each interior angle measure as found above, as follows,

180
      ∘
     
     −
      30
      ∘
     
     =
      150
      ∘

180
      ∘
     
     −
      60
      ∘
     
     =
      120
      ∘

180
      ∘
     
     −
      90
      ∘
     
     =
      90
      ∘

So, the exterior angles are found to be

150
    ∘
   
   ,
    
     120
    ∘

and

90
    ∘

. Now, taking their ratio, we get,

150
    °
   
   :
    
     120
    °
   
   :
    
     90
    °
   
   =5:4:3
 But, as per the statement, the ratio of the exterior angles of the given triangle is 3:2:1, which is wrong.Hence, the given statement is false.Hence, the correct option is (2).

State whether the statements are true or false: The interior angles of a triangle are in the ratio 1:2:3, then the ratio of its exterior angles is 3:2:1.

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