The angle of a hexagon are `x+10^@, 2x+20^@, 2x-20^@, 3x-50^@, x+40^@ and x+20^@`. Find x.

Answer

To solve the problem, we need to find the value of \( x \) given the angles of a hexagon. The angles are:

1. \( x + 10^\circ \)
2. \( 2x + 20^\circ \)
3. \( 2x - 20^\circ \)
4. \( 3x - 50^\circ \)
5. \( x + 40^\circ \)
6. \( x + 20^\circ \)

### Step 1: Find the sum of the interior angles of a hexagon.
The formula for the sum of the interior angles of a polygon is given by:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides. For a hexagon, \( n = 6 \).

Calculating the sum:
\[ \text{Sum} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \]

### Step 2: Set up the equation for the sum of the angles.
Now, we will add all the given angles and set them equal to the sum of the interior angles:
\[ (x + 10^\circ) + (2x + 20^\circ) + (2x - 20^\circ) + (3x - 50^\circ) + (x + 40^\circ) + (x + 20^\circ) = 720^\circ \]

### Step 3: Combine like terms.
Combining all the terms:
\[ x + 10 + 2x + 20 + 2x - 20 + 3x - 50 + x + 40 + x + 20 = 720 \]
This simplifies to:
\[ (1x + 2x + 2x + 3x + 1x + 1x) + (10 + 20 - 20 - 50 + 40 + 20) = 720 \]
\[ 10x + 20 = 720 \]

### Step 4: Solve for \( x \).
Now, we will isolate \( x \):
\[ 10x + 20 = 720 \]
Subtract 20 from both sides:
\[ 10x = 700 \]
Now, divide by 10:
\[ x = 70 \]

### Final Answer:
The value of \( x \) is \( 70^\circ \).

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