The angles of a quadrilateral are in A.P. with common difference `20^@.` Find its angles.

Answer

To find the angles of a quadrilateral that are in arithmetic progression (A.P.) with a common difference of \(20^\circ\), we can follow these steps:

### Step 1: Understand the properties of a quadrilateral
The sum of all the angles in a quadrilateral is \(360^\circ\).

### Step 2: Define the angles in terms of a variable
Let the first angle be \(x\). Since the angles are in A.P. with a common difference of \(20^\circ\), we can express the angles as:
- First angle: \(x\)
- Second angle: \(x + 20\)
- Third angle: \(x + 40\)
- Fourth angle: \(x + 60\)

### Step 3: Set up the equation for the sum of angles
According to the property of quadrilaterals, we can write the equation:
\[ x + (x + 20) + (x + 40) + (x + 60) = 360 \]

### Step 4: Simplify the equation
Combine like terms:
\[ x + x + 20 + x + 40 + x + 60 = 360 \]
This simplifies to:
\[ 4x + 120 = 360 \]

### Step 5: Solve for \(x\)
Now, isolate \(x\) by subtracting \(120\) from both sides:
\[ 4x = 360 - 120 \]
\[ 4x = 240 \]
Now, divide both sides by \(4\):
\[ x = 60 \]

### Step 6: Find the angles
Now that we have \(x\), we can find the four angles:
- First angle: \(x = 60^\circ\)
- Second angle: \(x + 20 = 60 + 20 = 80^\circ\)
- Third angle: \(x + 40 = 60 + 40 = 100^\circ\)
- Fourth angle: \(x + 60 = 60 + 60 = 120^\circ\)

### Conclusion
The angles of the quadrilateral are:
- \(60^\circ\)
- \(80^\circ\)
- \(100^\circ\)
- \(120^\circ\)