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Proofs for Rhombuses

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 rhombuses. You will also explore their detailed proofs with the help of illustrations.

What Is a Rhombus?

In geometry, a rhombus is a quadrilateral that has all equal sides, with opposite sides parallel to each other.

The quadrilateral ABCD is a rhombus, with AB = BC = CD = AD. AB is parallel to CD (AB||CD), and BC is parallel to AD (BC||CD).


Rhombus
Rhombus


Theorems Related to Rhombuses

Now that you understand what a rhombus is, let's explore some of the important properties and theorems related to it.

Opposite Angles

This theorem states that opposite angles in a rhombus are equal.

To prove this, let's consider a rhombus ABCD that has AB||CD and BC||CD.


Rhombus ABCD
Rhombus


If we draw a line joining the two points B and D, two triangles are created - DAB and BCD.


Diagonal in rhombus


Now, in these triangles, AB = CD, AD = BC, and side DB is common. Therefore, according to the side-side-side property of triangles, these are congruent.


Congruent triangles


Therefore, the corresponding angles would be equal.


Opposite angles


The same can be proved for the other two angles as well. Thus, the opposite angles in a rhombus are equal.


Opposite angles in a rhombus


Diagonals of a Rhombus

The diagonals of a rhombus bisect each other and are perpendicular to each other.

Consider the same rhombus ABCD that has AB||CD and BC||CD.


Rhombus ABCD
Rhombus


Let's draw two diagonals, AC and BD, interesecting each other at point O.


Diagonals in a rhombus


Consider the two triangles formed by this, AOD and COB. The line DB is a transversal to the parallel lines AD and BC. So, the angles ADO and CBO will be equal using the alternate interior angles theorem of parallel lines.


Alternate interior angles


Similarly, the line AC is a transversal to the parallel lines CD and AB. So, the angles DAO and BCO will be equal.


Alternate interior angles


Also, the sides AD and BC are equal to each other. So, using the angle-side-angle property of triangles, these triangles are congruent.


Congruent triangles


Therefore, the corresponding sides of these triangles would be equal.


Corresponding angles


This proves that the O is the midpoint of the lines AC and BD. Thus, the diagonals of a rhombus bisect each other.

Now, to prove that the diagonals are perpendicular at the point O, consider the triangles BOC and DOC.


Diagonals in a rhombus


In these triangles, we already proved that BO = OD. We know that BC = DC and OC is the common side. Therefore, using the side-side-side property, the triangles BOC and DOC are congruent.


Congruent triangles


Now, the corresponding angles, BOC and DOC, are equal.


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Also, the sum of angles BOC and DOC is 180 degrees because they are on a straight line.


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As these angles are equal, each of them will be 90 degrees. Therefore,


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