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Congruent triangles:

Take a parallelogram PQRS.

Draw in the diagonals PR and QS.

Let the point where the diagonals meet be M,

Consider one pair of the parallel sides, PS and QR, say.

Consider angles PSQ and RQS:

As PS and QR are parallel they are equal (Z- or alternate angles).

Now consider angles SPR and QRP:

As PS and QR are parallel they are equal (z- or alternate angles).

As PS and QR are opposite sides of a parallelogram they are equal in length; thus the triangles PMS and RMQ are congruent (Angle-Angle-Side).

As the two triangle are congruent, equivalent sides are equal in length.

Thus QM is the same length as MS and PM is the same length as MR

As QM is the same length as MS and QMS lie on a straight line, M must be the mid point of QS, ie the diagonal PQ bisects the diagonal QS

Similarly PM is the same length as MR and PMR lie on a straight line, thus M must be the mid point of PR, ie the diagonal QS bisects the diagonal PR

Therefore the diagonals of a parallelogram bisect each other.

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Not for every parallelogram. Only for a rhombus (diamond) or square will the diagonals bisect the opposite angles they connect, and diagonals are perpendicular. In rectangles, the diagonals do not bisect the angles and are notperpendicular, but they do bisect each other.

A quadrilateral whose diagonals bisect each other at right angles is a rhombus. each other at right angles at M. So AB = AD and by the first test above ABCD is a rhombus. 'If the diagonals of a parallelogram are perpendicular, then it is a rhombus

always

true

The missing word is "bisect".

Related questions

Yes, the diagonals of a parallelogram bisect each other.

Yes every parallelogram has bisecting diagonals

No.

Not for every parallelogram. Only for a rhombus (diamond) or square will the diagonals bisect the opposite angles they connect, and diagonals are perpendicular. In rectangles, the diagonals do not bisect the angles and are notperpendicular, but they do bisect each other.

The diagonals of a parallelogram are congruent (equal in length) and bisect each other.

A parallelogram.

Yes

Yes, they do.

Only if the parallelogram is in the form of a rhombus will its diagonals bisect each other at right angles

Yes the diagonals of a parallelogram have the same midpoint since they bisect each other.

True

yes

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