Question

ABCD is a rhombus. Among the following what can be said about the diagonals?

• A. diagonal AC bisects $\angle A$
• B. diagonal AC bisects $\angle C$
• C. diagonal BD bisects $\angle B$
• D. diagonal BD bisects $\angle D$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A diagonal AC bisects $\angle A$
B diagonal AC bisects $\angle C$
C diagonal BD bisects $\angle B$
D diagonal BD bisects $\angle D$


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 for || lines BC and DA)
$\angle 1=\angle 4$
Therefore, AC bisects $\angle A$
Similarly, it can be proved that BD bisects $\angle$ B and $\angle$ D as well.