Answer

Verified

401.4k+ views

**Hint:**Rhombus has all the four sides equal and opposite two sides of the rhombus are parallel to each other and the diagonals of the rhombus always intersect at right angles. Here first of all draw the figure of rhombus and diagonals.

**Complete step by step solution:**

Since in the above figure ABCD is a rhombus.

Therefore, all the four sides of the rhombus are equal.

$\therefore AB = BC = CD = DA$

In triangles $\Delta AOB$ and $\Delta COB$ ,

$OA = OC$ (Diagonals of the parallelogram bisect each other where bisection means two equal sections.)

$OB = OB$(Common Sides of the triangles)

$AB = CB$ (Sides of the rhombus are equal)

$\therefore \Delta AOB \cong \Delta COB$(By SSS congruence rule)

By using Congruent part of the congruent triangle,

$\angle AOB = \angle COB$….. (I)

Also, AC is a line,

$\angle AOB + \angle COB = 180^\circ $ (Linear pair)

By using the equation (I)

$\angle AOB + \angle AOB = 180^\circ $

Simplify the above equation,

$ \Rightarrow 2\angle AOB = 180^\circ $

Term multiplicative on one side is moved to the opposite side, it goes to the denominator.

$ \Rightarrow \angle AOB = \dfrac{{180^\circ }}{2}$

Common factors from the numerator and the denominator cancel each other.

$ \Rightarrow \angle AOB = 90^\circ $

From equation (I),

$\begin{gathered}

\angle AOB = \angle COB \\

\Rightarrow \angle AOB = 90^\circ \\

\end{gathered} $

Also,

Vertically opposite angles are equal to each other,

$\angle DOC = \angle AOB = 90^\circ $

$\angle AOD = \angle COB = 90^\circ $

Using above equations and its co-relations,

$\angle DOC = \angle AOB = \angle AOD = \angle COB = 90^\circ $

$ \Rightarrow AC \bot BD$

**Therefore, diagonals of the rhombus are perpendicular to each other.**

**Note:**Remember the difference between different types of the quadrilaterals and follow the properties accordingly. follow the different conditions of the congruence of the triangles to prove these types of solutions such as –

SSS criteria (Side - Side - Side)

SAS criteria (Side – Angle - Side)

ASA criteria (Angle – Side – Angle)

AAS criteria (Angle – Angle – Side)

RHS criteria (Right angle – Hypotenuse – Side)

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Which are the Top 10 Largest Countries of the World?

Difference Between Plant Cell and Animal Cell

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE