Question

Which of the following is the correct match with respect to the given question?
‘ABCD’ is a rhombus such that $\angle ACB={{40}^{\circ }}$ then $\angle ADB=\_\_\_\_\_\_\_\_\_$.

(A) $a\to (i)$
(B) $b\to (iii)$
(C) $c\to (iii)$
(D) $(d)\to (iv)$

Answer 1

Hint: Assume any rhombus ’ABCD’ and draw a rough diagram to get a clear idea about the question. Use the properties of triangles and rhombus like: diagonals of the rhombus intersect each other at right angle and sum of co-interior angles of the rhombus is ${{180}^{\circ }}$, to get the answer.

Complete step-by-step answer:

We can see in the figure that ABCD is a rhombus and we have been given $\angle ACB={{40}^{\circ }}$. We have to find the value of $\angle ADB$.

Now, we know that all the sides of a rhombus are equal. Therefore, in triangle BCD, CB = CD. Therefore, triangle BCD is an isosceles triangle.

Let us assume that the diagonals AC and BD of the rhombus intersect each other at O. Since, the diagonals of the rhombus intersect each other at right angles, therefore we can say that in triangle BCD, CO is perpendicular to BD.

We already know that, in an isosceles triangle the perpendicular is also the angle bisector.

$\therefore \angle BCD=2\times \angle ACB=2\times {{40}^{\circ }}={{80}^{\circ }}$

Now, $\angle BCD+\angle ADC={{180}^{\circ }}$ because the sum of co-interior angles of the rhombus is ${{180}^{\circ }}$.

$\Rightarrow \angle ADC={{180}^{\circ }}-{{80}^{\circ }}={{100}^{\circ }}$.

Now, since triangle ADC is also an isosceles triangle, therefore, DO is the angle bisector of angle ADC.

$\Rightarrow \angle ADB=\dfrac{{{100}^{\circ }}}{2}={{50}^{\circ }}$

Hence, option (C) is the correct answer.


Note: Always draw a rough diagram for the questions involving the concept of geometry. Here, there are many ways to get the answer but we have considered the options as the way to answer. We found the angles given in the options to determine the value of the angle. If we would have used another approach then we would not have found the angles given in the options.