SOLUTION: how do i prove if the diagnols of a parallelogram are congruent the parallelgram is a rectangle

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If it is a parallelogram, then the opposite sides are equal.
If the diagonals are also equal, then you have 4 congruent triangles.
Let your parallelogram be ABCD where:
A is top left corner.
B is top right corner.
C is bottom right corner.
D is bottom left corner.

Opposite sides are:
AB and CD
AD and BC

Diagonals are:
AC
BD

Triangles that are congruent are:

ADC with CBA by SSS because:
AD = BC
AC = AC
AB = DC

This means that angle ADC = angle ABC because congruent angles of congruent triangles are congruent.

BCD with DAB by SSS because:
AD = BC
DC = AB
BD = BD

This means that angle DAB equals angle BCD because congruent angles of congruent triangles are congruent.

Since the opposite angles of a parallelogram are supplementary (sum of their angles is equal to 180 degrees), this means that:

angle ADC = 90 degrees
angle ABC = 90 degrees
angle DAB = 90 degrees
angle BCD = 90 degrees

This means that parallelogram ABCD is a rectangle because a rectangle is a parallelogram with all angles being 90 degrees.