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Ex.8.1 Q6 Quadrilaterals Solution - NCERT Maths Class 9

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Question

Diagonal \(AC\) of a parallelogram \(ABCD\) bisects \(\angle A\) (see the given figure). Show that

(i) It bisects \(\angle C\) also,

(ii) \(ABCD\) is a rhombus.

 

 Video Solution
Quadrilaterals
Ex 8.1 | Question 6

Text Solution

 

What is known?

The diagonal AC of a parallelogram \(ABCD\) bisects \(∠A.\)

What is unknown?

How we can show that (i) It bisects \(∠C\) also, (ii) ABCD is a rhombus.

Reasoning:

We can use alternate interior angles property to show diagonal \(AC\) bisects angle \(C\) also, by showing all sides equal it can be said rhombus.

Steps:

(i) \(ABCD\) is a parallelogram.

\(\begin{align} &\angle \mathrm{DAC}=\angle {BCA}\\&\text{(Alternate interior angles are equal)} \ldots(1)\end{align}\)

\(\begin{align}\text{And,} &\angle \mathrm{BAC}=\angle {DCA}\\&\text{(Alternate interior angles are equal)} \ldots(2)\end{align}\)

However, it is given that \(AC\) bisects \(\angle A\).

\(\angle \mathrm{DAC}=\angle \mathrm{BAC} \ldots(3)\)

From Equations (\(1\)),(\(2\)), and (\(3\)), we obtain 

\[\begin{align} {\angle D A C}&\!=\!\!{\angle B C A\!=\!\!\angle B A C\!=\!\!\angle D C A \ldots\!(\!4\!)} \\ {\angle D C A}&\!=\!\!{\angle B C A}\end{align}\]

Hence, \(AC\) bisects \(\angle C\).

(ii) From Equation (\(4\)), we obtain 

\(\angle {DAC}=\angle {DCA}\)

\({{DA}}={DC}\)(Side opposite to equal angles are equal)

However, \(DA = BC\) and \(AB = CD\)

(Opposite sides of a parallelogram)

\(AB = BC = CD = DA \)

Hence, \(ABCD\) is a rhombus.

 Video Solution
Quadrilaterals
Ex 8.1 | Question 6