**Proving Quadrilaterals are Parallelograms :**

In this section, you will learn how to prove that a quadrilateral is a parallelogram.

We can use the following Theorems to prove the quadrilateral are parallelograms.

**Theorem 1 : **

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

It has been illustrated in the diagram shown below.

In the diagram above,

AB ≅ DC

AD ≅ BC

**Theorem 2 : **

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

It has been illustrated in the diagram shown below.

In the diagram above,

m∠A ≅ m∠C

m∠B ≅ m∠D

**Theorem 3 :**

If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

It has been illustrated in the diagram shown below.

In the diagram above,

m∠A + m∠B = 180°

m∠B + m∠C = 180°

m∠C + m∠D = 180°

m∠A + m∠D = 180°

**Theorem 4 : **

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

It has been illustrated in the diagram shown below.

In the diagram above,

AM ≅ CM

BM ≅ DM

**Theorem 5 : **

If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

It has been illustrated in the diagram shown below.

In the diagram above,

AD ≅ BC

AD || BC

**Problem 1 :**

In the diagram given below, if AB ≅ CD, AD ≅ CB, then prove that ABCD is a parallelogram.

AB ≅ CD, AD ≅ CB aaaaaaa AC ≅ AC aaaaa aaaaaaaaaaaaaaaaaaaaaa ΔABC ≅ ΔCDA aa m∠BAC ≅ m∠DCA a aa m∠DAC ≅ m∠BCA aa aaaaaaaaaaaaaaaaaaaaaa aa AB || CD, AD || CB a aaaaaaaaaaaaaaaaaaaaa ABCD is a parallelogram |
Given Reflexive Property of Congruence SSS Congruence Postulate Corresponding parts of congruent triangles are congruent Alternate Interior Angles Converse Definition of parallelogram |

**Problem 2 :**

In the diagram given below, if BC || DA, BC ≅ DA, then prove that ABCD is a parallelogram.

BC || DA aa m∠DAC ≅ m∠BCA aa aaaaaaaaaaaaaaaaaaaa aaaaaaa AC ≅ AC aaaaa aaaaaaaaaaaaaaaaaaaaaa ΔABC ≅ ΔCDA BC ≅ DA ΔABC ≅ ΔCDA aaaaaa BC ≅ DA aaaaa aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaa ABCD is a parallelogram |
Given Alternate Interior Angles Converse Reflexive Property of Congruence SSS Congruence Postulate Given SAS Congruence Postulate Corresponding parts of congruent triangles are congruent If opposite sides of a quadrilateral are congruent, then it is a parallelogram |

**Problem 3 :**

Show that A(2, - 1), B(1, 3), C(6, 5) and D(7, 1) are the vertices of a parallelogram.

**Solution : **

Let us plot the given points in a coordinate plane as shown below.

**Solution : **

There are many ways to prove that the given points are the vertices of a parallelogram.

**Method 1 : **

Show that opposite sides have the same slope, so they are parallel.

Using slope formula to find the slopes of AB, CD, BC and DA.

Slope of AB = [3 - (-1)] / [1 - 2] = - 4

Slope of CD = [1 - 5] / [7 - 6] = - 4

Slope of BC = [5 - 3] / [6 - 1] = 2 / 5

Slope of DA = [- 1 - 1] / [2 - 7] = 2 / 5

AB and CD have the same slope. So they are parallel.

Similarly, BC and DA are parallel.

Because opposite sides are parallel, ABCD is a parallelogram.

**Method 2 :**

Show that opposite sides have the same length.

Using distance formula to find the lengths of AB, CD, BC and DA.

AB = √[(1 - 2)^{2} + (3 + 1)^{2}] = √17

CD = √[(7 - 6)^{2} + (1 - 5)^{2}] = √17

BC = √[(6 - 1)^{2} + (5 - 3)^{2}] = √29

DA = √[(2 - 7)^{2} + (- 1 - 1)^{2}] = √29

From the lengths AB, CD, BC and DA, it is clear that

AB ≅ CD and BC ≅ DA

Because both pairs of opposite sides are congruent, ABCD is a parallelogram.

**Method 3 :**

Show that one pair of opposite sides is congruent and parallel.

Find the slopes and lengths of AB and CD as shown in Methods 1 and 2.

Slope of AB = Slope of CD = - 4

Because AB and CD have the same slope, they are parallel.

AB = CD = √17

Because AB and CD have the same length, they are congruent.

So, ABCD is a parallelogram.

After having gone through the stuff given above, we hope that the students would have understood, "Proving quadrilaterals are parallelograms".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

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