Question

In the adjoining figure,\[AB = AC = CD,\angle ADC = {35^ \circ }\]. Calculate$:\left( 1 \right)\angle ABC$ $\left( 2 \right)\angle BEC$

Answer 1

Hint: In this question use angle sum property of a triangle i.e. sum of all the interior angles of triangle is ${180^ \circ }$.
          $\vartriangle ACD$ is an isosceles triangle with $AD$ as a base.
          $\vartriangle ABC$ is an isosceles triangle with $BC$ as a base.

Complete step-by-step answer: Given, $ABCE$ is a cyclic quadrilateral, $AE$ and $BC$ are produced to meet at a point $D$. $AC$ and $EB$ are joined.
Also, \[AB = AC = CD,\angle ADC = {35^ \circ }\]
To find out:
$\left( 1 \right)\angle ABC$
$\left( 2 \right)\angle BEC$
Now, in $\vartriangle ACD$
\[AC = CD\],\[\angle ADC = {35^ \circ }\]
So, \[\angle ADC = \angle CAD = {35^ \circ }\]($\vartriangle ACD$ is an isosceles triangle)
Now, by angle sum property
\[\angle ADC + \angle CAD + \angle ACD = {180^ \circ }\]
But \[\angle ADC = \angle CAD = {35^ \circ }\]
$ \Rightarrow $ \[\angle ACD + {70^ \circ } = {180^ \circ }\]
       \[\angle ACD = {180^ \circ } - {70^ \circ }\]
   $\therefore \angle ACD = {110^ \circ }$
Now, $\angle ACB = {180^ \circ } - \angle ACD = {180^ \circ } - {110^ \circ } = {70^ \circ }$(Linear pair)
          $\therefore \angle ACB = {70^ \circ }$ $........1$
We know that $\vartriangle ABC$ is an isosceles triangle.
$ \Rightarrow $$\angle ACB = \angle ABC$
So from equation $1$
$\therefore \angle ABC = {70^ \circ }$
So $\angle ABC = {70^ \circ }$.
Now, in $\vartriangle ABC$, Use angle sum property
$ \Rightarrow \angle BAC + \angle ABC + \angle ACB = {180^ \circ }$
But $\angle ACB = \angle ABC$$ = {70^ \circ }$
$ \Rightarrow $ $\angle BAC + {140^ \circ } = {180^ \circ }$
        $\angle BAC = {180^ \circ } - {140^ \circ }$
$\therefore $ $\angle BAC = {40^ \circ }$
Now, $\angle BAC$ and $\angle BEC$ have been subtended by the chord $BC$ to the circumference of the circle at $A$ and $E$.
$\therefore $ $\angle BAC = \angle BEC = {40^ \circ }$
So $\angle BEC = {40^ \circ }$

Note: In this type of questions you should know some properties of the cyclic quadrilateral, i.e. it is different from the simple quadrilateral. One should make mistakes while naming the angles. So be careful while doing this because making mistakes in naming angle will lead to the wrong answer.