91Ó°ÊÓ

The theorem 5-7 states that if the diagonals of the quadrilateral bisect each other, then the quadrilateral is a parallelogram.

The labelled diagram is:

The given statements are: $BOâ‰\dots DO$and $AOâ‰\dots CO$.

To prove statements are: ABCD is a parallelogram.

It is being given that in the triangles, $â–³AOB$and $â–³COD$,$BOâ‰\dots DO$ and $AOâ‰\dots CO$.

In the triangles, $△AOB$and $△COD$, it can be noticed that the angles $∠AOB$and $△COD$are the vertically opposite angles.

Therefore, $∠AOBâ‰\dots ∠COD$.

Therefore, in the triangles $â–³AOB$and $â–³COD$, it can be noticed that $BOâ‰\dots DO$,$∠AOBâ‰\dots ∠COD$and $AOâ‰\dots CO$.

Therefore, the triangles $â–³ABC$and $â–³CDA$are congruent triangles by using the SAS postulate.

Therefore, by using CPCT, it can be said that $ABâ‰\dots CD$.

It is being given that in the triangles, $â–³AOD$and $â–³COB$,$BOâ‰\dots DO$ and $AOâ‰\dots CO$.

In the triangles, $△AOD$and $△COB$, it can be noticed that the angles $∠AOD$and $∠COB$are the vertically opposite angles.

Therefore, $∠AODâ‰\dots ∠COB$.

Therefore, in the triangles $â–³AOD$and $â–³COB$, it can be noticed that $BOâ‰\dots DO$, $∠AODâ‰\dots ∠COB$and $AOâ‰\dots CO$.

Therefore, the triangles $â–³AOD$and $â–³COB$are congruent triangles by using the SAS postulate.

Therefore, by using CPCT, it can be said that $ADâ‰\dots CB$.

Therefore, it can be noticed that $ABâ‰\dots CD$and $ADâ‰\dots CB$, therefore the pairs of opposite sides of the quadrilateral ABCD are congruent.

If both pairs of opposite sides of the quadrilateral are congruent, then the quadrilateral is a parallelogram.

As $ABâ‰\dots CD$and $ADâ‰\dots CB$, therefore ABCD is a parallelogram.

The two-column proof of the theorem is: