91Ó°ÊÓ

The given diagram is:

It is being given that ABCD is a parallelogram.

In a parallelogram, both pairs of opposite sides are congruent and parallel.

Therefore, in the parallelogram ABCD, both pairs of opposite sides are congruent and parallel and both pairs of opposite angles are congruent.

Therefore, $\stackrel{¯}{AB}â‰\dots \stackrel{¯}{CD}$, $\stackrel{¯}{AD}â‰\dots \stackrel{¯}{BC}$, $\stackrel{¯}{AB}âˆ\yen \stackrel{¯}{CD}$and $\stackrel{¯}{AD}âˆ\yen \stackrel{¯}{BC}$, $∠BCDâ‰\dots ∠DAB$and $∠ADCâ‰\dots ∠CBA$.

Therefore, $AD=BC$, $\stackrel{¯}{AB}=\stackrel{¯}{CD}$,$∠BCD=∠DAB$ and $∠ADC=∠CBA$.

It is also being given that $\stackrel{¯}{AN}$bisects $∠DAB$and $\stackrel{¯}{CM}$bisects $∠BCD$.

As, $\stackrel{¯}{AN}$bisects $∠DAB$, therefore by using the definition of angle bisector it can be said that $∠DAN=\frac{1}{2}∠DAB$.

As, $\stackrel{¯}{CM}$bisects $∠BCD$, therefore by using the definition of angle bisector it can be said that $∠BCM=\frac{1}{2}∠BCD$.

Therefore, it can be noticed that:

$\begin{array}{c}∠BCD=∠DAB\\ \frac{1}{2}∠BCD=\frac{1}{2}∠DAB\\ ∠BCM=∠DAN\end{array}$

Therefore, $∠BCMâ‰\dots ∠DAN$.

In the triangles $â–³DAN$and $â–³BCM$, it can be noticed that $∠BCMâ‰\dots ∠DAN$, $AD=BC$and $∠ADC=∠CBA$.

Therefore, the triangles $â–³DAN$and $â–³BCM$are congruent by ASA congruence.

Therefore, by corresponding parts of congruent triangles, it can be said that $ANâ‰\dots CM$and $DNâ‰\dots BM$.

As, $\stackrel{¯}{AB}=\stackrel{¯}{CD}$, therefore it can be obtained that:

$\begin{array}{c}\stackrel{¯}{AB}=\stackrel{¯}{CD}\\ \stackrel{¯}{AB}−\stackrel{¯}{BM}=\stackrel{¯}{CD}−\stackrel{¯}{BM}\\ \stackrel{¯}{AB}−\stackrel{¯}{BM}=\stackrel{¯}{CD}−\stackrel{¯}{DN}\left(âˆ\mu \stackrel{¯}{BM}=\stackrel{¯}{DN}\right)\\ \stackrel{¯}{AM}=\stackrel{¯}{NC}\end{array}$

Therefore, it can be noticed that $ANâ‰\dots CM$and $AMâ‰\dots NC$.

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

As, $ANâ‰\dots CM$and $AMâ‰\dots NC$, therefore, the quadrilateral AMCN is a parallelogram.