Prove diagonals of a rhombus are perpendicular

A rhombus is a parallelogram with four congruent sides.

So, all sides of rhombus ABCD are congruent.

i.e,

Also, we know that the diagonals of a parallelogram bisect each other.  

Since a rhombus is a parallelogram.

By property of rhombus , if point N is the intersection of the diagonals as shown in the figure, then

        .....[1]

In ΔJNM and ΔJNK

    [side]    [by (1)]

    [side]          [Given]

By reflexive property states that a segment is congruent to itself:

  [Side]                 [Reflexive Property]

SSS(Side-Side-Side) postulates states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

then by SSS congruence,

By CPCT [Corresponding Part of congruent triangles are congruent]

Since, JNM and JNK are corresponding angles therefore,

Linear pair theorem states that two angles that form a linear pair are supplementary.

By linear pair theorem, JNM and JNK are supplementary

this mean:

Since, the angles are congruent i.e,

so;

or

2

Simplify:

also;

therefore, the diagonals of JKLM are perpendicular to each other i,e

is only a line? i think so