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There are some properties which are always true for any parallelogram. 
Let's see those properties with explanation.
In a parallelogram, the following properties are true:
  1. the opposite sides are equal in length.
  2. the opposite angles are equal in measure.
  3. the adjacent angles are supplementary.
  4. the diagonals bisect each other.
Proof:
 
Let's prove (1) and (2).
 
Let ABCD be a parallelogram. 
 
Theory_2_1_1.png
 
Draw a diagonal BD and denote the interior angles as ∠1, ∠2, ∠3 and ∠4.
 
Since ABCD is a parallelogram, AD is parallel to BC and AB is parallel to CD.
 
Consider the parallel lines AD and BC and take the diagonal BD as transversal.
 
Here ∠2 = ∠4 by alternate interior angle property [Alternate interior angles are equal in measure].
 
Now consider parallel lines AB and CD and take the diagonal BD as transversal.
 
Here ∠1 = ∠3 by alternate interior angle property [Alternate interior angles are equal in measure].
 
So we have  ∠1 + ∠2 = ∠3+∠4. That is, ∠B = ∠D.
 
Consider the triangles DAB and BCD with the common side (BD = BD).
 
Now we have ∠2 = ∠4, ∠1 = ∠3 and the common side (BD = BD).
 
By ASA criterion of congruence, ΔDAB  ΔBCD.
 
That is, they are congruent triangles.
 
Therefore, AD = BC, AC = CD and ∠A= ∠C.
 
Hence, in parallelogram ABCD, we have AD = BC, AC = CD, ∠A = ∠C and ∠B = ∠D.
 
This proves (1) and (2).
 
From (2), it is obvious that ∠A = ∠C and ∠B = ∠D.
 
Let's prove (3).
 
Let the measure of angles are∠A = ∠C = x and ∠B = ∠D = y.
 
We have the property that sum of all interior angles of a quadrilateral is 360°.
 
That is, ∠A + ∠B +∠C + ∠D = 360°.
 
Substituting the taken values in the above equation.
 
x + y + x + y = 360°
 
2x +2y = 360°
 
2(x + y) = 360°
 
x + y  = 180°.
 
That is, we can write this as ∠A + ∠B = 180°, ∠B + ∠C = 180°, ∠C + ∠D = 180° and ∠A + ∠D = 180°.
 
It proves the property (3).
 
Let's prove (4).
 
Consider a parallelogram ABCD. Draw its diagonals AC and BD. Let the intersection point of the diagonals be O.
 
10.PNG
 
In triangle AOB and COD, we have:
 
AB = CD as opposite sides are equal in parallelogram.
 
∠AOB = ∠COD [Because vertically opposite angles are equal].
 
Here AB is parallel to CD, so ∠OAB = ∠DCO.
 
By AAS criterion of congruence, ΔOABΔOCD.
 
This implies, OA = OC and OB = OD.
 
As they are equal, diagonals of a parallelogram bisect each other.