Question

Draw and label a diagram. List what is given and what is to be proved. Then write a two-column proof of the theorem.

Theorem 5-7.

Short answer

The labeled diagram is:

The given statements are: $\mathbf{BO}\cong \mathbf{DO}$and $\mathbf{AO}\cong \mathbf{CO}$.

To prove statements are: ABCD is a parallelogram.

The two-column proof of the theorem is:

Statements	Reasons
In $△AOB$and $△COD$, $BO\cong DO$and$AO\cong CO$	given
$\angle AOB\cong \angle COD$	Vertically opposite angles.
$△AOB\cong △COD$	SAS postulate
$AB\cong CD$	By CPCT
In $△AOD$and $△COB$, $BO\cong DO$and$AO\cong CO$	given
$\angle AOD\cong \angle COB$	Vertically opposite angles.
$△AOD\cong △COB$	SAS postulate
$AD\cong CB$	By CPCT
As, $AB\cong CD$and $AD\cong CB$, therefore ABCD is a parallelogram.	If both pairs of opposite sides of the quadrilateral are congruent, then the quadrilateral is a parallelogram.

Step-by-step solution

Step 1. Write the theorem 5-7.

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

Step 2. Draw a labelled diagram.

The labelled diagram is:

Step 3. Description of step.

The given statements are: $BO\cong DO$and $AO\cong CO$.

To prove statements are: ABCD is a parallelogram.

Step 4. Description of step.

It is being given that in the triangles, $△AOB$and $△COD$,$BO\cong DO$ and $AO\cong CO$.

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

Therefore, $\angle AOB\cong \angle COD$.

Therefore, in the triangles $△AOB$and $△COD$, it can be noticed that $BO\cong DO$,$\angle AOB\cong \angle COD$and $AO\cong CO$.

Therefore, the triangles $△ABC$and $△CDA$are congruent triangles by using the SAS postulate.

Therefore, by using CPCT, it can be said that $AB\cong CD$.

Step 5. Description of step.

It is being given that in the triangles, $△AOD$and $△COB$,$BO\cong DO$ and $AO\cong CO$.

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

Therefore, $\angle AOD\cong \angle COB$.

Therefore, in the triangles $△AOD$and $△COB$, it can be noticed that $BO\cong DO$, $\angle AOD\cong \angle COB$and $AO\cong CO$.

Therefore, the triangles $△AOD$and $△COB$are congruent triangles by using the SAS postulate.

Therefore, by using CPCT, it can be said that $AD\cong CB$.

Therefore, it can be noticed that $AB\cong CD$and $AD\cong CB$, 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 $AB\cong CD$and $AD\cong CB$, therefore ABCD is a parallelogram.

Step 6. Description of step.

The two-column proof of the theorem is:

Statements	Reasons
In $△AOB$and $△COD$, $BO\cong DO$and$AO\cong CO$	given
$\angle AOB\cong \angle COD$	Vertically opposite angles.
$△AOB\cong △COD$	SAS postulate
$AB\cong CD$	By CPCT
In $△AOD$and $△COB$, $BO\cong DO$and$AO\cong CO$	given
$\angle AOD\cong \angle COB$	Vertically opposite angles.
$△AOD\cong △COB$	SAS postulate
$AD\cong CB$	By CPCT
As, $AB\cong CD$and $AD\cong CB$, therefore ABCD is a parallelogram.	If both pairs of opposite sides of the quadrilateral are congruent, then the quadrilateral is a parallelogram.