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IF ONE OF THE DIAGONAL OF RHOMBUS OS EQUAL TO ONE OF ITS SIDE WHAT IS THE VALUE
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If the length of one side of a rhombus is equal to the length of one diagonal, find the angles of the rhombus.

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Hint: If we have the rhombus as one side is equal to the diagonal then we have the rhombus divided into two equilateral triangles. So, other angles are equal as that is a rhombus and two adjacent angles will give us sum as 180∘180∘ .

Complete step-by-step answer:

One of the diagonals of a rhombus is equal to one of its sides.

We know that a rhombus has four equal sides. If one side has the same length as a diagonal, the diagonal is part of two equilateral triangles. Since a rhombus has four sides we can say ,

Rhombus ABCD where AB == diagonal of AC .

Since the triangle ABC is equilateral and so we have ∠ABC = 60∘∠ABC = 60∘ , ∠BCA = 60∘∠BCA = 60∘ , and ∠CAB = 60∘∠CAB = 60∘.

Also the triangle ADC is an equilateral triangle so we have ∠ADC = 60∘∠ADC = 60∘ , ∠DCA = 60∘∠DCA = 60∘ and ∠CAD = 60∘∠CAD = 60∘ .

Now, as we have ∠DAB = ∠DAC+∠CAB∠DAB = ∠DAC+∠CAB

On substituting the values of ∠DAC∠DAC and ∠CAB∠CAB, we get,

⇒∠DAB= 60∘+ 60∘⇒∠DAB= 60∘+ 60∘

On solving we get,

⇒∠DAB= 120∘⇒∠DAB= 120∘

Similarly we have, ∠DCB = ∠DCA+∠ACB∠DCB = ∠DCA+∠ACB

On substituting the values of ∠DCA∠DCA and ∠ACB∠ACB, we get,

⇒∠DCB= 60∘+ 60∘⇒∠DCB= 60∘+ 60∘

On solving we get,

⇒∠DCB= 120∘⇒∠DCB= 120∘

So, we have the angles as, ∠ABC = 60∘=∠ADC∠ABC = 60∘=∠ADC and ∠DCB= 120∘=∠DAB∠DCB= 120∘=∠DAB.