Question

A villager Ram has a plot of land in the shape of concave quadrilateral ABCDFA as shown in the figure below. He himself decided to construct a health care centre for the villagers in the area FCD and keep the remaining with him.
(i)Find the value of x.
(ii)What values of Ram is he depicting while deciding to do so?

Answer 1

Hint: First we will find the value of $\angle B$ in the given figure. We know that the sum of all the angles of the triangle is equal to ${180^ \circ }$ so we can use it for triangle ABC as the value of angle C and A is already given in the figure. Put all the values in the formula and solve to find the value of angle B. Then again use this theorem to find angle D.

Complete step-by-step answer:
Given, Ram builds a health care centre in his village in the area FCD on the plot of his land.

Here It is given in the diagram that \[\angle ACB = {90^ \circ }\] ,$\angle BAC = {40^ \circ }$ and $\angle BED = {100^ \circ }$
i)We have to find x i.e. $\angle EDB$
We know that the sum of all the angles of the triangle is equal to ${180^ \circ }$
So In $\Delta ABC$, we can write-

$ \Rightarrow \angle BAC + \angle ACB + \angle ABC = {180^ \circ }$ -- (i)
Now, we know the values-\[\angle ACB = {90^ \circ }\] and $\angle BAC = {40^ \circ }$.On putting these values in eq. (i), we get-
$ \Rightarrow {40^\circ } + {90^\circ } + \angle ABC = {180^\circ }$
On addition, we get-
$ \Rightarrow {130^\circ } + \angle ABC = {180^\circ }$
On transferring the constant value of angle on the right side, we get-
$ \Rightarrow \angle ABC = {180^\circ } - {130^\circ }$
On subtracting the terms on the right side, we get-
$ \Rightarrow \angle ABC = {50^\circ }$
Now, In $\Delta DEB$

It is given that $\angle BED = {100^ \circ }$ and we have already calculated $\angle DBE$
Now, we know cc. Then we can write-
$ \Rightarrow \angle BED + \angle EDB + \angle DBE = {180^ \circ }$--- (ii)
On putting the given values in eq. (ii), we get-
$ \Rightarrow 100 + \angle EDB + 50 = {180^ \circ }$
On addition, we get-
$ \Rightarrow 150 + \angle EDB = {180^ \circ }$
On transferring ${150^\circ }$ on the right side, we get-
$ \Rightarrow \angle EDB = {180^ \circ } - {150^\circ }$
On subtraction, we get-
$ \Rightarrow \angle EDB = {30^\circ }$
Hence the value of x=${30^\circ }$.
Ram is depicting the values of benevolence and generosity towards the villagers.

Note: In this type of question we have to remember that the value depicted by the person is defined by his action. Also, here we have not used the theorem that the sum of all the angles of the triangle is equal to ${180^ \circ }$ for triangle FCD because we do not know the value of$\angle F$.