Tell whether the following statement is true or false:

The bisectors of base angles of a triangle cannot enclose a right angle in any case.

Question

Tell whether the following statement is true or false:The bisectors of base angles of a triangle cannot enclose a right angle in any case.

Accepted Answer

Given a statement, “the bisectors of base angles of a triangle cannot enclose a right angle in any case.”Using angle sum property in ABC,

∠A+∠B+∠C=
    
     180
    °
   
   ………(1)
  Dividing equation by 2,

⇒
    
     ∠A
    2
   
   +
    
     ∠B
    2
   
   +
    
     ∠C
    2
   
   =

180
      °

2

⇒
    
     ∠A
    2
   
   +
    
     ∠B
    2
   
   +
    
     ∠C
    2
   
   =
    
     90
    °
   
   ……..(2)
  Therefore,

⇒
    
     ∠A
    2
   
   +∠OBC+∠OCB=
    
     90
    °

As, OC is angle bisector of B and OB is angle bisector of A, therefore,

⇒∠OBC=
    1
    2
   
   ∠B and ∠OCB=
    1
    2
   
   ∠C
   Now,

⇒∠OBC+∠OCB=
    
     90
    °
   
   −
    
     ∠A
    2
   
   ………(3)
  Using angle sum property in triangle BOC,

⇒∠BOC+
    
     90
    °
   
   −
    
     ∠A
    2
   
   =
    
     180
    °

⇒∠BOC=
    
     180
    °
   
   −
    
     90
    °
   
   +
    
     ∠A
    2

⇒∠BOC=
    
     90
    °
   
   +
    
     ∠A
    2

Therefore, bisectors of a base angle are more than right angle.Given statement is true.Hence, option 1 is correct.

Detailed Solution