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.

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