Question

ABCD is a rhombus. Show that diagonal AC bisects $\angle A$ as well as $\angle C$ and diagonal BD bisects $\angle B$ as well as $\angle D$.

Answer 1

Let us join AC.
In $\angle$ ABC,
BC=AB (Sides of a rhombus are equal to each other)
$\angle 1=\angle 2$ ( Angles opposite to equal sides of a triangle are equal)
However, $\angle 1=\angle 3$ (Alternate interior angles for parallel lines AB and CD)
$\angle 2=\angle 3$
Therefore, AC bisects $\angle C$.
Also, $\angle 2=\angle 4$ ( Alternate interior angles)
$\angle 1=\angle 4$
Therefore, AC bisects $\angle A$
Similarly, it can be proved that BD bisects $\angle$ B and $\angle$ D as well.