Properties of parallelogram — lesson. Mathematics CBSE, Class 8.

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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:

the opposite sides are equal in length.
the opposite angles are equal in measure.
the adjacent angles are supplementary.
the diagonals bisect each other.

Proof:

Let's prove (1) and (2).

Let

$ABCD$ be a parallelogram.

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

$(BD = BD)$ .

$ASA$ criterion of congruence,

$Δ$

$DAB$

$≅$

$Δ$

$BCD$ .

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

$O$ .

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,

$Δ$

$OAB$

$≅$

$Δ$

$OCD$ .

This implies,

$OA = OC$ and

$OB = OD$ .

As they are equal, diagonals of a parallelogram bisect each other.

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2. Properties of parallelogram

Theory: