Question

The diagonal $AC$ of a parallelogram $ABCD$ bisects angle $A$ Show that it bisects angle $C$ also.

Answer 1

Let $ABCD$ is a parallelogram in which $AC$ bisects angle $A$.All the angles are shown in the figure.

Here , $\angle 1=\angle 2[ACbiscetsangleA]............\left(i\right)$

$$\begin{gathered} \angle 1=\angle 4[Alternateangle]................\left(ii\right) \\ \angle 2=\angle 3[Alternateangle]...............\left(iii\right) \end{gathered}$$

From $\left(i\right),\left(ii\right)and\left(iii\right)$

$$\begin{gathered} \angle 1=\angle 2=\angle 3=\angle 4 \\ \Rightarrow \angle 3=\angle 4 \end{gathered}$$

Hence, $AC$ bisects angle $C$.