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

ABCD is a rhombus and a rhombus is also a parallelogram. A rhombus has four equal sides.

(i) $\ln \triangle A B C, \angle B A C=\angle B C A=\frac{1}{2}(180-110)=35^{\circ}$

i.e., x = 35o
Now, ∠B + ∠C = 180o (Adjacent angles are supplementary)
But ∠C = x + y = 70o
⇒ y = 70o − x
⇒y = 70o − 35o = 35o
Hence, x = 35o; y = 35o

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

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

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