Question

If in the given $△ABC,AB=BD=BC$, then find the ratio of $\angle ACD$ : $\angle ADC$

Answer 1

In $\Delta abc$, side $ab=ac$. when side ab is produced to $d$ such that $bd=bc$

Since, $ab=ac$ so, $\angle abc=\angle acb=x$ by using equal sides correspond equal angle.

Similarly, in $\Delta bdc,sidebc=bdso,\angle bdc=\angle bcd=y$by using equal sides correspond equal angle.

Since, $\angle abc$ is the exterior angle for triangle $\Delta bdc$. so,

$\angle abc=\angle bdc+\angle bcd$

or, $x=y+y$

or, $x=2y-----------\left(1\right)$

Now, we shall find the ratio of $\angle acd$ and $\angle adc$

or, $\angle acd:\angle adc=(\angle acb+\angle bcd):\angle adc$

or, $=(x+y):y$

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

or, = $3:1$

Hence, the required ratio of $\angle acd$ and $\angle adc$ will be $3:1$