# Ex.8.1 Q7 Quadrilaterals Solution - NCERT Maths Class 9

## 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 \(\angle D\).

## Text Solution

**What is known?**

ABCD is a rhombus.

**What is unknown?**

How we can show that diagonal \(AC\) bisects \(∠A\) as well as \(∠C\) and diagonal BD bisects \(∠B\) as well as \(∠D.\)

**Reasoning:**

We can use alternate interior angles property to show diagonal \(AC\) bisects angles \(A\) and \(C\), similarly diagonal \(BD\) bisects angles B and D.

**Steps:**

Let us join \(AC\).

In \(\Delta\)\(ABC\),

\[\begin{align}&BC=AB\\&\left( \begin{array} & \text{Sides of a rhombus are} \\ \text{ equal to each other} \\ \end{array} \right)\\\\&\angle1=\angle2\\&\left( \begin{array} & \text{Angles opposite to equal sides } \\ \text{ of a triangle are equal} \\

\end{array} \right)\end{align}\]

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.