What is the sum of interior angles of a polygon and problems involving supplementary angles and complex numbers?

The sum of the interior angles of a polygon with $n$ sides is given by the formula:

We are given that the sum of the interior angles is $1080^\circ$.

Divide both sides by 180:

Add 2 to both sides:

A polygon with 8 sides is an octagon.
The correct option is A.
The name of the polygon is an $\boxed{\text{Octagon}}$.

48.

Let the angle be $\theta$.
The supplement of the angle is $180^\circ - \theta$.

The problem states that the supplement of an angle is twice as large as the angle.

Add $\theta$ to both sides of the equation:

Divide by 3:

The correct option is C.
The measure of the angle is $\boxed{60^\circ}$.

49.

The conjugate of $1+2i$ is $1-2i$.

Substitute $i^2 = -1$:

This is in the form $(a+bi)(a-bi) = a^2 + b^2$.

The correct option is A.
The simplified expression is $\boxed{-i}$.

50.

Subtract $c$ from both sides of the equation:

Divide both sides by $x$:

So, $m = \frac{y-c}{x}$.
The correct option is B.
Making $m$ the subject of the equation gives $\boxed{m = \frac{y-c}{x}}$.