Question

In triangle abc ,if ab=ac and ab is produced to d such that bd=bc, find angle acd:angle adc

Answer 1

Solution:

Consider the given attachment.

In Δabc, side ab = ac. when side ab is produced to d such that bd = bc

Since, ab = ac so, ∠abc = ∠acb = x (say) ------by using equal sides correspond equal angle.

Similarly, in Δbdc, side bc = bd so, ∠bdc = ∠bcd = y (say) ------by using equal sides correspond equal angle.

Since, ∠abc is the exterior angle for triangle Δbdc. so,

∠abc = ∠bdc + ∠bcd

or, x = y + y

or, x = 2y -----------(1)

Now, we shall find the ratio of∠acd and ∠adc

or, ∠acd :∠adc = ( ∠acb + ∠bcd ) : ∠adc

or, = (x + y) : y

or, = (2y +y ): y ----------using equation (1)

or, = 3:1

Hence, the required ratio of ∠acd and ∠adc will be 3:1

Answer 2

Step-by-step explanation:

Here

In ∆ ABC

AB = AC

So, abc= acb = x ( angle opposite to equal side are equal)

Similarly, BD= BC

Therefore

Bcd= Bdc= y

Now,

Abc= bdc+bcd

( exterior angle property)

X= y+y

X= 2y

Therefore,

Acd:Adc= (acb+bcd) :Adc

= (X+y) : y

= (2y+y) : y (2y = x)

= 3y :y

= 3:1