The angles of a polygon are given by 2x, 5x, x and 4x respectively. The value of x is

1. Analyze the question

We are given the interior angles of a polygon: $2x$, $5x$, $x$, and $4x$. We need to find the value of $x$. Since there are four angles listed, this is a quadrilateral (a 4-sided polygon).

2. Explain the concept

The sum of the interior angles of any polygon is calculated using the formula:

where $n$ is the number of sides.

For a quadrilateral ($n = 4$):

Therefore, the sum of the given angles must equal $360^\circ$.

3. Show the solution

• Step 1: Set up the equation by adding all the given angles and equating them to $360^\circ$. $$2x + 5x + x + 4x = 360^\circ$$

• Step 2: Combine the like terms. $$(2 + 5 + 1 + 4)x = 360^\circ$$ $$12x = 360^\circ$$

• Step 3: Solve for $x$ by dividing both sides by 12. $$x = \frac{360^\circ}{12}$$ $$x = 30^\circ$$

4. Identify the answer

Comparing our result to the options provided: A. 31° B. 30° C. 26° D. 48°

The calculated value matches option B.

5. Tip

Whenever you are given the interior angles of a polygon, always count the number of terms to determine $n$. Remember the "Sum of interior angles" table:

• Triangle ($n=3$): $180^\circ$
• Quadrilateral ($n=4$): $360^\circ$
• Pentagon ($n=5$): $540^\circ$
• Hexagon ($n=6$): $720^\circ$