Question

In each of the figures given below, ABD is a rectangle. Find the values of x and y in each case.

Answer 1

(i) ABCD is a rectangle.
The diagonals of a rectangle are congruent and bisect each other. Therefore, in​ ∆ AOB, we have:
OA = OB
∴​ ∠​OAB = ∠​OBA = 35^o
∴​ x = 90^o − 35^o = 55^o
And ∠AOB = 180^o − (35^o + 35^o) = 110^o
∴​ y = ∠AOB​ = 110^o​ [Vertically opposite angles]
Hence, x = 55^o and y = 110^o​​

(ii) In ∆AOB, we have:
OA = OB
Now, ∠​OAB = ∠OBA = $\frac{1}{2}\times \left(180°-110°\right)={35}^{\mathrm{o}}$
∴​ y = ∠BAC = 35^o [Interior alternate angles]
Also, x = 90^o − y [ ​∵∠C = 90^o = x + y ]
⇒​ x = 90^o − 35^o = 55^o
Hence, x = 55^o and y = 35^o​​