The angles of a quadrilateral are in arithmetic progression and their common difference is `10^(@)`. Find the angles.

Explanation

To find the angles of a quadrilateral that are in arithmetic progression with a common difference of \(10^\circ\), we can follow these steps:

### Step 1: Define the angles
Let the four angles of the quadrilateral be:
- \(A - 3D\)
- \(A - D\)
- \(A + D\)
- \(A + 3D\)

Where \(A\) is the middle value and \(D\) is the common difference.

### Step 2: Write the equation for the sum of angles
The sum of the angles in a quadrilateral is \(360^\circ\). Therefore, we can write the equation:
\[ (A - 3D) + (A - D) + (A + D) + (A + 3D) = 360 \]

### Step 3: Simplify the equation
Combine like terms:
\[ 4A = 360 \]

### Step 4: Solve for \(A\)
Now, divide both sides by 4:
\[ A = \frac{360}{4} = 90 \]

### Step 5: Use the common difference
Given that the common difference \(D = 10\), we can now substitute \(A\) and \(D\) back into our angle definitions:
- \(A - 3D = 90 - 3(10) = 90 - 30 = 60^\circ\)
- \(A - D = 90 - 10 = 80^\circ\)
- \(A + D = 90 + 10 = 100^\circ\)
- \(A + 3D = 90 + 3(10) = 90 + 30 = 120^\circ\)

### Step 6: List the angles
Thus, the angles of the quadrilateral are:
- \(60^\circ\)
- \(80^\circ\)
- \(100^\circ\)
- \(120^\circ\)

### Final Answer
The angles of the quadrilateral are \(60^\circ\), \(80^\circ\), \(100^\circ\), and \(120^\circ\).

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