Question

prove that the diagonal of a parallelogram bisect each other

Answer 1

Note: I recommend that this page be printed out, so that the instructions are easier to follow.

In order to successfully complete a proof, it is important to think of the definition and the construction of a parallelogram. 
In the following outline, I will provide the statements, you provide the reasons.

Prove: If a quadrilateral is a parallelogram, then the diagonals bisect each other.

Given: Parallelogram ABCD with diagonals BD



and AC intersecting at point M.



Proof:

Angle DBA is congruent to angle BDC.Angle CMD is congruent to angle AMB.Triangle CMD is congruent to triangle AMB.Segment AM is congruent to segment MC.M is the midpoint of segment AC.Segment BD bisects segment AC.Segment BM is congruent to segment MD.M is the midpoint of segment BD.Segment AC bisects segment BD.

Prove: Segment AC and BD bisect each other.

Consider how a parallelogram is constructed------parallel lines.Consider properties of parallel lines and vertical angles.The diagonals create 4 triangles.Consider triangle congruency properties.



Answer 2

Step-by-step explanation:

Let consider a parallelogram ABCD in which AB||CD and AD||BC.

In ∆AOB and ∆COD , we have

∠DCO=∠OAB (ALTERNATE ANGLE)

∠CDO= ∠OBA. (ALTERNATE ANGLE)

AB=CD. (OPPOSITE SIDES OF ||gram)

therefore , ∆ AOB ≅ ∆COD. (ASA congruency)

hence , AO=OC and BO= OD. (C.P.C.T)

$\sf \: hence \: proved....$