In Fig., ABCD is a parallelogram, CE bisects ∠C and AF bisect

1.	In Fig., ABCD is a parallelogram, CE bisects ∠C and AF bisects ∠A. In each of the following, if the statement is true, give a reason for the same:(i) ∠A = ∠C(ii) ∠FAB = ½ ∠A(iii) ∠DCE = ½ ∠C(iv) ∠CEB = ∠FAB(v) CE ∥ AF
Answer» (i) ∠A = ∠C True, Since ∠A =∠C = 55° [opposite angles are equal in a parallelogram] (ii) ∠FAB = ½ ∠A True, Since AF is the angle bisector of ∠A. (iii) ∠DCE= ½ ∠C True, Since CE is the angle bisector of angle ∠C. (iv) ∠CEB= ∠FAB True, Since ∠DCE = ∠FAB (opposite angles are equal in a parallelogram). ∠CEB = ∠DCE (alternate angles) ½ ∠C = ½ ∠A [AF and CE are angle bisectors] (v) CE \|\| AF True, since one pair of opposite angles are equal, therefore quad. AEFC is a parallelogram.

In Fig., ABCD is a parallelogram, CE bisects ∠C and AF bisects ∠A. In each of the following, if the statement is true, give a reason for the same:(i) ∠A = ∠C(ii) ∠FAB = ½ ∠A(iii) ∠DCE = ½ ∠C(iv) ∠CEB = ∠FAB(v) CE ∥ AF

Answer»

(i) ∠A = ∠C

True, Since ∠A =∠C = 55° [opposite angles are equal in a parallelogram]

(ii) ∠FAB = ½ ∠A

True, Since AF is the angle bisector of ∠A.

(iii) ∠DCE= ½ ∠C

True, Since CE is the angle bisector of angle ∠C.

(iv) ∠CEB= ∠FAB

True,

Since ∠DCE = ∠FAB (opposite angles are equal in a parallelogram).

∠CEB = ∠DCE (alternate angles)

½ ∠C = ½ ∠A [AF and CE are angle bisectors]

(v) CE || AF

True, since one pair of opposite angles are equal, therefore quad. AEFC is a parallelogram.