Question

In Fig. 17.31, 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) $\angle FAB=\frac{1}{2}\angle A$
(iii) $\angle DCE=\frac{1}{2}\angle C$
(iv) $\angle CEB=\angle FAB$
(v) CE || AF

Answer 1

(i) True, since opposite angles of a parallelogram are equal.
(ii) True, as AF is the bisector of $\angle$A.
(iii) True, as CE is the bisector of $\angle$C.
(iv) True
$\angle$CEB = $\angle$DCE........(i) (alternate angles)
$\angle$DCE= $\angle$ FAB.........(ii) (opposite angles of a parallelogram are equal)

From equations (i) and (ii):
$\angle$CEB = $\angle$FAB

(v) True, as corresponding angles are equal ($\angle$CEB = $\angle$FAB).