From Parallelogram to Rhombus

🏆Practice from a parallelogram to a rhombus

You will be able to determine that the parallelogram is a rhombus if at least one of the following conditions is met:

  1. If in the parallelogram there is a pair of adjacent equal sides - it is a rhombus.
  2. If in the parallelogram the diagonals bisect each other, forming angles of 90o 90^o degrees, that is, they are perpendicular - it is a rhombus.
  3. If in the parallelogram one of the diagonals is the bisector - it is a rhombus.
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Test yourself on from a parallelogram to a rhombus!

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Given the parallelogram:

AAABBBDDDCCC149149

Is this parallelogram a rhombus?

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From Parallelogram to Rhombus

How will you realize that the parallelogram in front of you is a rhombus?
We are here to teach you 3 criteria by which you can demonstrate, quickly and simply, that you have a rhombus in front of you.

First let's remember the definition of a rhombus.
The definition of a rhombus says the following: it is a parallelogram that has a pair of sides (or edges) adjacent that are equal.


First criterion - Adjacent sides equal

If the parallelogram has a pair of equal adjacent sides, it is a rhombus.
You can use this theorem to show that it is a rhombus without having to prove it.
However, for you to understand the logic behind it, we will demonstrate this theorem below.

Data:

Parallelogram ABCDABCD
AB=BCAB=BC

Equal contiguous sides

We have to prove that: ABCDABCD  is a rhombus

Solution:
Since we have that ABCDABCD  is a parallelogram, we deduce that:

AB=DCAB=DC
AD=BCAD=BC because, in the parallelogram, each pair of opposite sides are also equivalent.

Let's observe our data BC=ABBC=AB
Now, we can determine that all sides of the parallelogram are equivalent according to the transitive relation.
We obtain:
AB=DC=BC=ADAB=DC=BC=AD

Indeed, ABCDABCD   is a quadrilateral with all its sides equal, therefore, it is a rhombus.


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Second criterion - Perpendicular diagonals

If in the parallelogram the diagonals bisect each other forming angles of 90o 90^o degrees, that is, they are perpendicular, it is a rhombus.

You can use this theorem to show that it is a rhombus without having to prove it.
However, for you to understand the logic behind it, we will demonstrate this theorem below.

Data:
Parallelogram ABCDABCD
ACBDAC⊥BD

It must be demonstrated that: ABCDABCD  is a rhombus

Solution:
Based on the first criterion, we already know that, it is enough for us to see that in the parallelogram there are two adjacent equivalent sides for us to know that it is a rhombus.
We can see that there is in this parallelogram a pair of adjacent equivalent sides if we use the congruence of triangles ABE ABE   and BECBEC .
We will place them on top of each other:
Side AE=AE  AE=AE     common side, of the same length.
Angle AEB=BEC∢AEB=∢BEC  These angles measure 90º since we know that the diagonals that create them are perpendicular.
Side AE=CEAE=CE  in the parallelogram, the diagonals intersect

From this it follows that: ABE=BEC ⊿ABE=⊿BEC   according to SAS
And, consequently,
we will be able to determine that: AB=BC AB=BC 

According to the congruence, corresponding sides are equivalent.

We will notice that ABAB and BCBC are adjacent equivalent sides in the parallelogram and, therefore, we will be able to determine that ABCDABCD is a rhombus since a parallelogram with a pair of adjacent equivalent sides is a rhombus.


Third Criterion - Diagonal equal to bisector

If in the parallelogram one of the diagonals is the bisector - it is a rhombus.

You can use this theorem to show that it is a rhombus without having to prove it.
However, for you to understand the logic behind it, we will prove this theorem below.

Data:
Parallelogram ABCDABCD   
c1=c2∢c1=∢c2

We have to prove that: ABCDABCD  is a rhombus

Solution:
Let's remember, a parallelogram that has a pair of equal adjacent sides is a rhombus.
Let's start:
From the data we have we can deduce ABDCAB∥DC since in a parallelogram, the opposite sides are also parallel.
Then: BAC=C1∢BAC=∢C1, alternate angles between parallel lines are equivalent.
According to the transitive relation, BAC=C2∢BAC=∢C2.
Now we can deduce that: AB=BCAB=BC
In the triangle, opposite equivalent angles there are sides equal to each other.
Now we can determine that ABCDABCD is a rhombus.
We found in the parallelogram a pair of equal adjacent sides, therefore, it is a rhombus.

Suggestion:
To remember the three theorems that prove that a parallelogram is a rhombus, try to remember the three key terms: sides, diagonals, and angles.

Great!
Now you know all the criteria to prove that a parallelogram is a rhombus.


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