Question

In each of the figures given below, ABCD is a rhombus. Find the value of x and y in each case.

Answer 1

ABCD is a rhombus and a rhombus is also a parallelogram. A rhombus has four equal sides.
(i)​ In ∆ABC, ∠​BAC = ∠BCA = $\frac{1}{2}\left(180-110\right)={35}^{\mathrm{o}}$
i.e., x = 35^o
Now, ∠​B + ∠C = 180^o (Adjacent angles are supplementary) ​
But ∠​C​ = x + y = 70^o
⇒​ y = 70^o − x
⇒​y = 70^o − 35^o = 35^o
Hence, x = 35^o; y = 35^o

(ii) The diagonals of a rhombus are perpendicular bisectors of each other.
So, in ∆​AOB, ​∠OAB = 40^o, ∠AOB = 90^o and ∠ABO = 180^o − (40^o + 90^o) = 50^o
∴ ​x = 50^o
In ∆​AB​D, AB = AD
So, ∠ABD = ​∠ADB = ​50^o
Hence, x = 50^o; y = 50^o


​(iii) ∠​BAC = ∠​DCA (Alternate interior angles)​
i.e., x = 62^o
In ∆BOC, ∠​BCO = 62^o [In ∆​ ABC, AB = BC, so ∠​BAC = ∠​ACB]
Also, ∠​BOC = 90^o
∴ ∠​OBC = 180^o − (90^o + 62^o) = 28^o
Hence, x = 62^o; y = 28^o