Theorem Of Parallelograms

Theorem 8

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

Theorem
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Given: A quadrilateral ABCD whose diagonals AC and BD intersect at O such that OA=OC and OB=OD.
To Prove: ABCD is a parallelogram.

Proof:
Statements
Reasons
In △OAB and △OCD
1. OA=OC
1. Given.
2. OB=OD
2. Given.
3. ∠AOB=∠COD 3. Vert. opp.∠s.
4. △OAB≌△OCD 4. SAS axiom of congruency
5. ∠AOB=∠OCD
⇒∠CAB=∠ACD
⇒ AB∥DC
5. c.p.c.t.
Alt.∠s are equal formed by lines AB,DC and transversal AC.
6. AB=CD
6. c.p.c.t.
7. ABCD is a parallelogram.
7. In quadrilateral ABCD, AB∥DC and AB=DC