Question

In the following figures, ABCD is a rhombus. Find the values of x and y.

Answer 1

in a rhombus diagonals bisect the angles and the opposite angles measure the same . In a rhombus all sides measures the same 

so, 

     in 1)  in ΔABC
  
                ∠CAB = ∠BCA = 40


               ∠CAB+∠BCA+∠ABC= 180

                80 +∠ ABC = 180

                ∠ABC = 100

as we know that the diagonals of a Rhombus bisect the angles

                     ∠ABC/2 = x = 50

as we know that opposite angles of a rhombus are equal 
y = 50

2) ∠DAC=∠DCA=62

∠DAC/2= ∠DCB/2 = x = 31

 ∠CAB+∠ACB+∠ABC = 180

62 + ∠ABC = 180

∠ABC= 118
∠ABC/2 = y = 59

similarly in 3)
 ∠ABC=∠ADC
AB = BC
so, ∠BAC =∠ACB= y 

∠BAC+∠ACB +∠ABC = 180
2y + ∠ABC = 180
2y + 120 = 180

2y = 60
y = 30
 as we know the diagonals bisect the angles so X=Y

x=y=30

                    

Answer 2

Answer:

in a rhombus diagonals bisect the angles and the opposite angles measure the same . In a rhombus all sides measures the same  

so,  

    in 1)  in ΔABC

 

               ∠CAB = ∠BCA = 40

              ∠CAB+∠BCA+∠ABC= 180

               80 +∠ ABC = 180

               ∠ABC = 100

as we know that the diagonals of a Rhombus bisect the angles

                    ∠ABC/2 = x = 50

as we know that opposite angles of a rhombus are equal  

y = 50

2) ∠DAC=∠DCA=62

∠DAC/2= ∠DCB/2 = x = 31

∠CAB+∠ACB+∠ABC = 180

62 + ∠ABC = 180

∠ABC= 118

∠ABC/2 = y = 59

similarly in 3)

∠ABC=∠ADC

AB = BC

so, ∠BAC =∠ACB= y  

∠BAC+∠ACB +∠ABC = 180

2y + ∠ABC = 180

2y + 120 = 180

2y = 60

y = 30

as we know the diagonals bisect the angles so X=Y

x=y=30

Please next time, give us more points for long questions, not just 5.