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Proofs for Parallelograms

Instructor: Usha Bhakuni

Usha has taught high school level Math and has master's degree in Finance

In this lesson, you will learn about important theorems related to parallelograms. You will also explore their detailed proofs with the help of illustrations.

Parallelograms

A parallelogram is a quadrilateral that has two pairs of parallel sides. Simply put, a parallelogram looks like this:


Parallelograms with Two Pairs of Parallel Sides
Parallelogram


The perimeter of a parallelogram is calculated as the sum of the lengths of all the sides.

Let's take a look at some of the basic proofs for parallelograms, including those related to the opposite sides, angles and diagonals found in these geometric shapes.

Proofs for Parallelograms

Opposite Sides

The opposite sides of a parallelogram are equal.

Consider parallelogram ABCD, where side AB is parallel to side CD (AB||CD) and side AD is parallel to side BC (AD||BC).


Parallelogram ABCD: Opposite Sides Theorem
Parallelogram


Now, let's draw a line joining the points A and C. This line acts as a transversal for the pair of parallel lines AB and CD.


Effect of a Transversal on Parallelogram ABCD
Proof that opposite sides are equal


As you can see, line AC has created two triangles: ABC and CDA.


Parallelogram ABCD Contains Two Triangles
Proof that opposite sides are equal


In these two triangles, according to the alternate interior angles theorem of parallel lines,


Proof that opposite sides are equal


and they have side AC in common. Therefore, according to the angle-side-angle property of triangles, triangles ABC and CDA are congruent.


Congruent triangles


This means that the other two sides of these triangles are equal: AB = CD and BC = AD. Thus, the opposite sides in a parallelogram are equal.

Opposite Angles

The opposite angles in a parallelogram are also equal.

Consider the same parallelogram ABCD, where AB||CD and AD||BC.


Parallelogram ABCD: Opposite Angles Theorem
Parallelogram


Now, let's draw a line joining the points A and C. This line acts as a transversal for both pairs of parallel lines.


Joining of Points A and C
Proof that opposite angles are equal


Here, we can see that the alternate interior angles are equal.


Proof that opposite angles are equal


Therefore,


Sum of angles


Similarly, drawing another transversal between points B and D would prove that the other two angles are equal. So,


Opposite angles in a parallelogram are equal


Hence, we see that the opposite angles in a parallelogram are equal.


Equal Opposite Angles in Parallelogram ABCD
Opposite angles in a parallelogram are equal


Consecutive Angles

The sum of consecutive angles in a parallelogram is 180 degrees.

Consider parallelogram ABCD again, where AB||CD and AD||BC.


Parallelogram ABCD: Consecutive Angles Theorem
Parallelogram


Let's extend lines AB, CD and BC to form angle x.


Forming Angle x
Proof that sum of consecutive angles is 180 degrees


We designate angle ABC as 180 - x; its alternate interior angle would also be180 - x. This means angle BCD is:

180 - (180 - x) = x

So, the sum of angles ABC and BCD is 180 degrees. The same can be said for the other two angles.


Sum of consecutive angles


Diagonals

The diagonals of a parallelogram bisect, or divide each other.

Consider parallelogram ABCD, where AB||CD and AD||BC.


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