Rajbeer is a farmer. He has a plot of land in the shape of a quadrilateral ABCD as shown in the figure. In ABCD, AB ∥ CD and AD ∥ BC. He divided the field into two parts, viz, triangle BCE and trapezium CDAE by making an embankment CE such that CE = AD.

Based on the above information, answer the following questions :

1. ABCD is a :
(a) Rectangle
(b) Parallelogram
(c) Square
(d) Trapezium

2. ∠A is equal to :
(a) ∠D
(b) ∠B
(c) ∠C
(d) ∠E

3. ∠DCE is equal to :
(a) ∠E
(b) ∠A
(c) ∠D
(d) ∠B

4. Which of the following is correct?
(a) ΔAEC ≅ ΔEAD
(b) ΔACE ≅ ΔAED
(c) ΔCEA ≅ ΔEAD
(d) none of these

5. Diagonal DE is equal to :
(a) diagonal DB
(b) diagonal AC
(c) DC
(d) BC

1. Given,

AB ∥ CD and AD ∥ BC

Since both pairs of opposite sides are parallel, the quadrilateral ABCD is a parallelogram.

Hence, option (b) is the correct option.

2. Given,

AD = CE

In trapezium CDAE:

AD and CE are the non-parallel sides and they are equal.

∴ CDAE is an isosceles trapezium.

∠A = ∠E [Angles on the same base of an isosceles triangle area equal]

Hence, option (d) is the correct option.

3. Given,

AD = CE

∴ CDAE is an isosceles trapezium.

∠DCE = ∠ADC [Base angles of an isosceles trapezium are equal.]

Hence, option (c) is the correct option.

4. In ΔAEC and ΔAED,

AD = CE (Given)

AE = AE (common side)

∠AEC = ∠EAD [Base angles of an isosceles trapezium are equal]

∴ ΔAEC ≅ ΔEAD [By S.A.S. axiom]

Hence, option (a) is the correct option.

5. As ADCE is an isosceles trapezium and diagonals of an isosceles trapezium are equal in length.

Hence, option (b) is the correct option.