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

Answer

To find the angles of a rhombus where one of the diagonals is equal to its sides, we can follow these steps:

### Step 1: Understand the properties of a rhombus
A rhombus is a type of quadrilateral where all sides are equal in length. The diagonals of a rhombus bisect each other at right angles.

### Step 2: Set up the problem
Let’s denote the rhombus as ABCD, where AC is one of the diagonals and BC is a side of the rhombus. According to the problem, we know that:
- \( BC = AB \) (since all sides of a rhombus are equal)
- \( AC = BC \) (since one of the diagonals is equal to its sides)

### Step 3: Establish relationships
From the information given, we can say:
- \( AB = BC = AC \)
This means triangle ABC is an equilateral triangle because all three sides are equal.

### Step 4: Determine the angles of triangle ABC
In an equilateral triangle, all angles are equal and sum up to 180 degrees. Therefore, each angle in triangle ABC is:
\[ \text{Angle ABC} = \text{Angle ACB} = \text{Angle CAB} = \frac{180^\circ}{3} = 60^\circ \]

### Step 5: Analyze triangle ADC
Since ABCD is a rhombus, the opposite angles are equal. Therefore, triangle ADC, which is also formed by the sides AD and DC (both equal to AB), will also be an equilateral triangle. Thus:
\[ \text{Angle ADC} = \text{Angle ACD} = \text{Angle CAD} = 60^\circ \]

### Step 6: Calculate the angles of the rhombus
Now, we can find the angles of the rhombus:
- Angle ABC = 60°
- Angle ADC = 60°

Since the angles at points B and D are supplementary to the angles at points A and C, we can find:
\[ \text{Angle A} = \text{Angle ABC} + \text{Angle ADC} = 60^\circ + 60^\circ = 120^\circ \]
Thus, the angles of the rhombus are:
- Angle A = 120°
- Angle B = 60°
- Angle C = 120°
- Angle D = 60°

### Final Answer
The angles of the rhombus are:
- 120°, 60°, 120°, and 60°.

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