Q. 124.4( 10 Votes )

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

Answer :

(i) Here, ABCD is rectangle.

We know that the diagonals of a rectangle are congruent and bisect each other.

∴ In ∆ AOB, we have *OA = OB*

This means that ∆ AOB is isosceles triangle.

We know that base angles of isosceles triangle are equal.

∴ ∠*OAB =* ∠O*BA =* 35°

∴ ∴ *x* = 90° − 35° = 55°

Also, ∠*AOB* = 180° − (35° + 35°) = 110°

∴ *y* = ∠*AOB*** = 110° …Vertically opposite angles

Hence, *x* = 55° and *y* = 110°

(ii) Here, ABCD is rectangle.

We know that the diagonals of a rectangle are congruent and bisect each other.

∴ In ∆ AOB, we have *OA = OB*

This means that ∆ AOB is isosceles triangle.

We know that base angles of isosceles triangle are equal.

∴ ∠*OAB =* ∠O*BA =* *× (*180° − 110°) = 35°

∴ *y* = ∠BAC *=* 35° … alternate angles with transversal AC

Also, *x* = 90° – *y* … ∵∠C = 90° = *x* + *y*

∴ *x* = 90° − 35° = 55°

Hence, *x* = 55° and *y* = 35°

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