Chapter 5.2, Problem 13WE

To determine

To write: a two column proof of the theorem.

Explanation of Solution

Given Information:

Given theorem,

If the diagonal of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Calculation:

Draw a figure,

  

Proof: the two column proof is given below.

Statements	Reasons
1. In $\Delta AOB$ and $\Delta BOC$ $\begin{array}{l}AO\cong OC\\ OB\cong OD\end{array}$	Given and it follows from the statement of the theorem.
2. $\angle AOB\cong \angle DOC$	Vertically opposite angles
3. $\Delta AOB\cong \Delta BOC$	S.A.S test
4. $\angle DCA\cong \angle BAC$	Corresponding parts of congruent triangles.
5. But $\angle DCA\cong \angle BAC$ Then $AB\parallel BC$	In a similar way, this can also be proved.
6. $AD\parallel BC$	In a similar way, this can also be proved.
7. $ABCD$ is a parallelogram	If opposite side in quadrilateral are parallel then that quadrilateral is parallelogram.