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

(i) ABCD is a rhombus.


We know that rhombus is type of parallelogram whose all sides are equal.


In ∆ABC, BAC = BCA = (180° 110°) = 35°


Hence x = 35°


But AB || DC …opposite sides of rhombus are parallel


BAC = DCA for transversal AC


BAC = DCA = 35°


Hence, x = y = 35°


(ii) ABCD is a rhombus.


We know that the diagonals of a rhombus are perpendicular bisectors of each other.


in ∆AOB,


OAB = 40°, AOB = 90°


ABO = 180° − (40° + 90°) = 50°


Hence x = 50°


Now in ∆DAB,


AB = AD … as rhombus has all sides equal.


ie. AOB is isosceles triangle.


Also base angles of isosceles triangle are equal.


Hence, x = y = 50°


(iii) ABCD is a rhombus.


We know that rhombus is type of parallelogram whose all sides are equal.


So in ∆DCB,


DC = BC


CDB = CBD = y° base angles of isosceles triangle are equal.


Now, x = CAB …alternate angles with transversal AC


x = BAD


x = × 62°


x = 31°


In ∆DOC,


We know sum of angles of triangle is 180°


CDO + DOC + OCD = 180°


CDO + 90° + 31° = 180°


CDO = 59°


y = 59°


Hence, x = 31° and y = 59°


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