Angles of a quadrilateral are `(4x)^@, 5(x+2)^@, (7x-20)^@ and 6(x+3)^@`. Find:
(i) the value of x.
(ii) each angle of the quadrilateral.

Answer

To solve the problem, we will follow these steps:

### Step 1: Set up the equation for the sum of angles in a quadrilateral
The angles of the quadrilateral are given as:
1. \(4x\)
2. \(5(x + 2)\)
3. \(7x - 20\)
4. \(6(x + 3)\)

According to the property of quadrilaterals, the sum of the angles is \(360^\circ\). Therefore, we can set up the equation:
\[ 4x + 5(x + 2) + (7x - 20) + 6(x + 3) = 360 \]

### Step 2: Expand and simplify the equation
Now, we will expand the terms in the equation:
\[ 4x + 5x + 10 + 7x - 20 + 6x + 18 = 360 \]

Combine like terms:
- Combine the \(x\) terms: \(4x + 5x + 7x + 6x = 22x\)
- Combine the constant terms: \(10 - 20 + 18 = 8\)

So, the equation simplifies to:
\[ 22x + 8 = 360 \]

### Step 3: Solve for \(x\)
Next, we will isolate \(x\):
\[ 22x = 360 - 8 \]
\[ 22x = 352 \]
\[ x = \frac{352}{22} \]

Now, simplify:
\[ x = 16 \]

### Step 4: Find each angle
Now that we have \(x\), we can find each angle:

1. First angle:
\[ 4x = 4 \times 16 = 64^\circ \]

2. Second angle:
\[ 5(x + 2) = 5(16 + 2) = 5 \times 18 = 90^\circ \]

3. Third angle:
\[ 7x - 20 = 7 \times 16 - 20 = 112 - 20 = 92^\circ \]

4. Fourth angle:
\[ 6(x + 3) = 6(16 + 3) = 6 \times 19 = 114^\circ \]

### Summary of Results
- The value of \(x\) is \(16\).
- The angles of the quadrilateral are:
- \(64^\circ\)
- \(90^\circ\)
- \(92^\circ\)
- \(114^\circ\)