SOLUTION: 1.how to prove if the diagonals in a parallelogram are perpendicular, then the parallelogram is a rhombus. 2. how to prove if the diagonals in a paralellogram are congruent, the

Outline for proof:

Since the diagonals of a parallelogram bisect each other,

BE and DE are congruent
AE is congruent to itself

We are given that all four angles at point E are 90° and 
are therefore congruent.

Triangles ABE and ADE are congruent by side-angle-side, 

Therefore AB and AD are congruent.

Since opposite sides of a parallelogram are congruent, 

AD and BC are congruent, and  AB and CD are congruent.

So all four sides are congruent, so parallelogram ABCD

is a rhombus.

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Outline:

We are given that diagonals AC and BD are congruent.
AB and DC are congruent because they are opposite sides of a parallelogram
AD is congruent to itself

Triangles ABD and DCA are congruent by side-side-side.

Angles BAD and CDA are congruent because they are corresponding
angles of congruent triangles.

Angles BAD and CDA are also supplementary because 
angles of a parallelogram are supplementary.

Angles BAD and CDA are right angles because they are congruent
and supplementary.  

Parallelogram ABCD is a rectangle because a parallelogram with one
right interior angle is a rectangle.

Edwin