The interior angles of a heptagon are x, 2x, 1.5x, 2.5x, 0.5x, 1.25x, and 1.25x. Find the value of x.

A heptagon has $n=7$ sides. The sum of the interior angles of a polygon is given by the formula $(n-2) \times 180^\circ$.

$$\text{Sum of angles} = (7-2) \times 180^\circ = 5 \times 180^\circ = 900^\circ$$

The given angles are $x, 2x, 1\frac{1}{2}x, 2\frac{1}{2}x, \frac{1}{2}x, \frac{1}{4}x$, and $1\frac{1}{4}x$.
Convert mixed numbers to improper fractions: $1\frac{1}{2}x = \frac{3}{2}x$, $2\frac{1}{2}x = \frac{5}{2}x$, $1\frac{1}{4}x = \frac{5}{4}x$.
Sum of angles:

$$x + 2x + \frac{3}{2}x + \frac{5}{2}x + \frac{1}{2}x + \frac{1}{4}x + \frac{5}{4}x = 900^\circ$$

Combine the coefficients of $x$:

$$\left(1 + 2 + \frac{3}{2} + \frac{5}{2} + \frac{1}{2} + \frac{1}{4} + \frac{5}{4}\right)x = 900^\circ$$ $$\left(3 + \frac{3+5+1}{2} + \frac{1+5}{4}\right)x = 900^\circ$$ $$\left(3 + \frac{9}{2} + \frac{6}{4}\right)x = 900^\circ$$ $$\left(3 + \frac{9}{2} + \frac{3}{2}\right)x = 900^\circ$$ $$\left(3 + \frac{12}{2}\right)x = 900^\circ$$ $$(3 + 6)x = 900^\circ$$ $$9x = 900^\circ$$ $$x = \frac{900^\circ}{9} = 100^\circ$$

• $x = 100^\circ$
• $2x = 2 \times 100^\circ = 200^\circ$
• $1\frac{1}{2}x = \frac{3}{2} \times 100^\circ = 150^\circ$
• $2\frac{1}{2}x = \frac{5}{2} \times 100^\circ = 250^\circ$
• $\frac{1}{2}x = \frac{1}{2} \times 100^\circ = 50^\circ$
• $\frac{1}{4}x = \frac{1}{4} \times 100^\circ = 25^\circ$
• $1\frac{1}{4}x = \frac{5}{4} \times 100^\circ = 125^\circ$

a) Calculate the values of the smallest and the biggest angles.
The angles are $100^\circ, 200^\circ, 150^\circ, 250^\circ, 50^\circ, 25^\circ, 125^\circ$.
The smallest angle is $25^\circ$.
The biggest angle is $250^\circ$.

$$\boxed{\text{Smallest angle} = 25^\circ, \text{Biggest angle} = 250^\circ}$$

b) Find the difference between the smallest and the biggest angles.

$$\text{Difference} = \text{Biggest angle} - \text{Smallest angle}$$ $$\text{Difference} = 250^\circ - 25^\circ = 225^\circ$$ $$\boxed{\text{Difference} = 225^\circ}$$