In the following Figure *ABCD* is a arallelogram, *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 *

#### Solution

(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).