Question

show that the diagonals of a rhombus are perpendicular to each other

Answer 1

CONSIDER RHOMBUS ABCD YOU KNOW THAT AB=BC=CD=AD NOW IN TRIANGLE AOD AND TRIANGLE CODOA=OC(DIAGONALS OF A //GM BISECT EACH OTHER) OD=OD (COMMON) AD=CDTHEREFORE,TRIANGLE AOD CONGRUENT TO TRIANGLE COD (SSS) THIS GIVES ANGLE AOD = ANGLE COD (CPCT) BUT, ANGLE AOD + ANGLE COD = 180 (LINEAR PAIR) SO, 2 ANGLE AOD=180 OR, ANGLE AOD =90 SO,THE DIAGONALS OF A RHOMBUS ARE PERPENDICULAR TO EACH OTHER  HENCE , PROVED

Answer 2

Hey mate here is your answer

Proofs

Proof that a rhombus is a parallelogram

All sides of a rhombus are congruent, so opposite sides are congruent, which is one of the properties of a parallelogram.
Or, there is always the longer way:
In rhombus all 4 sides are congruent (definition of a rhombus).
, and 
By the SSS Postulate, 
Corresponding parts of congruent triangles are congruent, so  and . The same can be done for the two other angles, so 
Convert the congruences into measures to get  and . Adding these two equations yields .
The interior angles of a quadrilateral add up to 360 degrees, so , or 
Substituting gives . When simplified, .
If two lines are cut by a transversal and same-side interior angles add up to 180 degrees, the lines are parallel. This means  The same can be done for the other two sides, and know we know that opposite sides are parallel. Therefore, a rhombus is a parallelogram.

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