Question

Q.6. In triangle ABC, AO and BO are the angle bisectors of ZA and B respectively.
Find the value of x
45

Answer 1

Step-by-step explanation:

As BO and CO are the angle bisectors of external angles of△ABC, Then

∠1=∠2∠4=∠3

We know, ∠A+∠ABC+∠ACB=180∘…eqn(1)

And ∠ABC=180−2∠1∠ACB=180−2∠4

Putting it in the eqn (1), we get

∠A+180−2∠1+180−2∠4=180⇒∠1+∠4=90+21∠A…eqn(2)

Also we know from the figure, ∠BOC+∠1+∠4=180∘

∠BOC=180−∠1−∠4

From eqn (2)

∠BOC=180−90−21∠A⇒∠BOC=90∘−21∠A

Answer 2

The value of ∠x is 247.5°

GIVEN

∆ABC is a triangle.

AO is the angle bisector of ∠A

BO is the angle bisector of ∠B

TO FIND

The value of ∠x

SOLUTION

We can simply solve the above problem as follows;

In ΔABC

∠A + ∠B + ∠C = 180° (Sum of interior angles of a triangle is 180°)

∠C = 45°

Therefore,

∠A + ∠B + 45 = 180

∠A + ∠B = 180-45 = 135

Dividing the whole equation by 2

∠A/2 + ∠B/2 = 67.5 (Equation 1)

In Δ AOB

∠BAO + ∠OBA + ∠BOA = 180° (Equation 2)

Since AO and BO are the angle bisectors of ∠A and ∠B respectively.

Therefore,

∠A/2 = ∠BAO

∠B/2 = ∠OBA

Putting the values of ∠A/2 and ∠B/2 from equation 1 in equation 2;

67.5 + ∠AOB = 180°

∠AOB = 180-67.5 = 112.5°

Now,

∠AOB + ∠x = 360° (Complete angle)

Therefore,

112.5 + ∠x = 360

∠x = 360-112.5

= 247.5°

Hence, The value of ∠x is 247.5°

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