Question

In the given figure, ABCD is a rhombus in which $\angle BCD = {110^0}$, find ( x + y ).

Answer 1

Hint: In this question, we will use the concept of angle sum property of a rhombus. The sum of all the angles of a rhombus is equal to ${360^0}$ and the opposite angles in a rhombus are equal to each other.

Complete step by step solution:
Rhombus is a type of quadrilateral which is a simple closed figure with four sides.
Properties of a Rhombus
1. All sides are congruent.
2. Opposite angles are congruent.
3. The diagonals are perpendicular to and bisect each other.
4. Adjacent angles are supplementary.
5. The diagonals of a rhombus are perpendicular to each other.
As we know that the opposite angles in a rhombus are equal to each other then from the figure,
We can say that
$ \Rightarrow \angle ABC = \angle CDA$ and $\angle DAB = \angle BCD$.
We also know that the sum of all angles of a rhombus is equal to ${360^0}$.
Then,
$ \Rightarrow \angle ABC + \angle BCD + \angle CDA + \angle DAB = {360^0}$
It can also be written as,
\[
   \Rightarrow 2\angle ABC + 2\angle BCD = {360^0} \\
   \Rightarrow \angle ABC + \angle BCD = \dfrac{{{{360}^0}}}{2} = {180^0} \\
\]
Now by putting the values of the angles given in the question, we get
$
   \Rightarrow (x + y) + {110^0} = {180^0} \\
   \Rightarrow (x + y) = {180^0} - {110^0} \\
   \Rightarrow (x + y) = {70^0} \\
$
Hence, we can say that $\angle ABC = {70^0}$ or $(x + y) = {70^0}$.

Note: whenever we ask this type of question, we have to remember some basic points of properties of a rhombus like angle sum property. First we have to find out the angles of the rhombus that are equal to each other and then by using the angle sum property we will find the other unknown values of angles. Through this we will get the answer.