In

ΔABC
if

∠C=3∠B
and

∠B=
2
3

[∠A+∠B]
, find the values of

∠A,∠B,∠C
are ____ respectively.

Question

In

ΔABC
   if

∠C=3∠B
   and

∠B=
    2
    3
   
   [∠A+∠B]
  , find the values of

∠A,∠B,∠C
  are ____ respectively.

Accepted Answer

The ΔABC is represented in below pictureLet the angles in the above triangle are

∠CAB=α

∠ABC=β

∠ACB=γ
  From the given data,

∠ABC=β=
      2
      3
     
     [∠CAB+∠ABC]

β=
      2
      3
     
     [α+β]

β−
      2
      3
     
     β=
      2
      3
     
     α

1
      3
     
     β=
      2
      3
     
     α

Multiplying the above equation with 3, then

β=2α

α=
      1
      2
     
     β.....(i)

Now given that

∠ACB=3∠ABC

γ=3β........(ii)

Finding the sum of the angles in the triangle,

∠CAB+∠ABC+∠ACB=α+β+γ
  From equations (i) and (ii) substituting the values of

α,γ
   in the above equation, then

∠CAB+∠ABC+∠ACB=α+β+γ

∠CAB+∠ABC+∠ACB=
      1
      2
     
     β+β+3β

∠CAB+∠ABC+∠ACB=
      9
      2
     
     β.......(iii)

But we know that the sum of angles in a triangle is equal to

180
    °

, so

∠CAB+∠ABC+∠ACB=
    
     180
    °

From equation (iii),

9
        2
       
       β=
        
         180
        ∘

9β=
        
         360
        ∘

β=

360
          ∘

9

=
        
         40
        ∘

From equation (i) the value of

α
   is,

α=
      1
      2
     
     β

=
      1
      2
     
     (
      40
      °
     
     )

=
      20
      °

From equation (ii), the value of

γ
  is

γ=3β

γ=3(
      40
      °
     
     )

γ=
      120
      °

So, the angles of the triangle are

∠CAB=∠A=α

∠A=
      20
      °

∠ABC=∠B=β

∠B=
      40
      °

In if  and , find the values of are 20°, 40° and 120° respectively.

In $Δ A B C$ if $∠ C = 3 ∠ B$ and $∠ B = 23 [∠ A + ∠ B]$ , find the values of $∠ A, ∠ B, ∠ C$ are ____ respectively.

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In ΔABC if ∠C=3∠B and ∠B= 2 3 [∠A+∠B] , find the values of ∠A,∠B,∠C are ____ respectively.