**Question:**

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

**Solution:**

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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