In a rhombus, the diagonals bisect each other at right angles. Given that one diagonal is equal to the side of the rhombus, we can use geometry to determine the angles formed by the diagonals.

Let the side of the rhombus be $a$, and let the diagonals be $p$ and $q$, where $p = a$. The diagonals form two congruent right triangles with hypotenuse $a$.

Using the Pythagorean theorem:

$\left(\frac{p}{2}\right)^2 + \left(\frac{q}{2}\right)^2 = a^2$

Since $p = a$, this becomes:

$\left(\frac{a}{2}\right)^2 + \left(\frac{q}{2}\right)^2 = a^2$

Simplify:

$\frac{a^2}{4} + \frac{q^2}{4} = a^2$

Multiply through by 4:

$a^2 + q^2 = 4a^2$

$q^2 = 3a^2$

$q = \sqrt{3}a$

Now, the angles formed by the diagonals can be calculated using trigonometry. The diagonals intersect at right angles, forming four angles. The acute angle can be determined using:

$\tan\theta = \frac{\text{shorter diagonal}}{\text{longer diagonal}} = \frac{\frac{p}{2}}{\frac{q}{2}} = \frac{p}{q} = \frac{a}{\sqrt{3}a} = \frac{1}{\sqrt{3}}$

$\theta = \arctan\left(\frac{1}{\sqrt{3}}\right) = 30^\circ$

Thus, the angles formed by the diagonals are $30^\circ$ and $60^\circ$, because they are complementary in a right angle.

Would you like me to delve further into the calculations or clarify anything?

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5 Related Questions:

1. How do the properties of a rhombus affect its diagonals?
2. Can you derive the relation between diagonals in a general rhombus?
3. How does the side length of the rhombus impact the angles formed?
4. Could you explore angles formed by diagonals in other quadrilaterals?
5. How do trigonometric ratios apply in solving geometric shapes?

Tip: Always check if the diagonals intersect at right angles when dealing with rhombuses!