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. 

Theorems

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

Practice Problems

Problem 1 :

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

Problem 2 :

In the diagram given below, if BC || DA, BC  ≅  DA, then prove that ABCD 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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