The sum of the angles in a polygon is

$\displaystyle 180(n-3)$

For a pentagon, this equals 540. Since the first 3 angles add up to 240, the remaining 2 angles must add up to $\displaystyle 540-240=300^{\circ}$

The perimeter in this question is irrelevant. Use the interior angle formula to determine the total sum of the angles in a hexagon.

$\displaystyle \Theta =(n-2)\cdot180$

There are six interior angles in a hexagon.

$\displaystyle \Theta =(6-2)\cdot180 = 720^{\circ}$

Each angle will be a sixth of the total angle.

$\displaystyle \frac{720}{6}= 120^{\circ}$

Therefore, the sum of three angles in a hexagon is:

$\displaystyle 120^{\circ}+120^{\circ}+120^{\circ}=360^{\circ}$

Use the interior angle formula to find the total sum of angles in a pentagon.

$\displaystyle \sum \Theta=(n-2) \cdot 180$

$\displaystyle n=5$ for a pentagon, so substitute this value into the equation and solve:

$\displaystyle \sum \Theta=(5-2) \cdot 180 = 540^{\circ}$

Divide this number by 5, since there are five interior angles.

$\displaystyle \frac{540^{\circ}}{5}=108^{\circ}$

The sum of four interior angles in a regular pentagon is:

$\displaystyle 108^{\circ}+108^{\circ}+108^{\circ}+108^{\circ} =432^{\circ}$

The perimeter of a regular pentagon has no effect on the interior angles of the pentagon.

Use the following formula to solve for the sum of all interior angles in the pentagon.

$\displaystyle \sum\theta=(n-2)\cdot 180$

Since there are 5 sides in a pentagon, substitute the side length $\displaystyle n=5$.

$\displaystyle \sum\theta=(5-2)\cdot 180=540$

Divide this by 5 to determine the value of each angle, and then multiply by 2 to determine the sum of 2 interior angles.

$\displaystyle \frac{540}{5}=108$

$\displaystyle 108*2=216$

The sum of 2 interior angles of a pentagon is $\displaystyle 216$.

The pentagon has 5 sides. To find the value of the interior angle of a pentagon, use the following formula to find the sum of all interior angles.

$\displaystyle \sum\theta=(n-2)\cdot 180$

Substitute $\displaystyle n=5$.

$\displaystyle \sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each interior angle.

$\displaystyle \frac{540}{5}=108$

Every interior angle is 108 degrees.  The problem states that an interior angle is $\displaystyle 12x$.  Set these two values equal to each other and solve for $\displaystyle x$.

$\displaystyle 12x=108$

$\displaystyle x=9$

Area has no effect on the value of the interior angles of a pentagon. To find the sum of all angles of a pentagon, use the following formula, where $\displaystyle n$ is the number of sides:

$\displaystyle \sum\theta=(n-2)\cdot 180$

There are 5 sides in a pentagon.  

$\displaystyle \sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each angle.

$\displaystyle \frac{540}{5}=108^\circ$

A regular polygon with $\displaystyle N$ sides has $\displaystyle N$ congruent angles, each of which measures 

$\displaystyle \frac{(N-2)180^{\circ }}{N}$

Setting $\displaystyle N = 5$, the common angle measure can be calculated to be

$\displaystyle \frac{(5-2)180^{\circ }}{5} = \frac{3 \cdot 180^{\circ }}{5} = \frac{540^{\circ }}{5} = 108^{\circ }$

The statement is therefore false.

If one exterior angle is taken at each vertex of any polygon, and their measures are added, the sum is $\displaystyle 360^{\circ }$. Each exterior angle of a regular pentagon has the same measure, so if we let $\displaystyle t$ be that common measure, then

$\displaystyle 5t = 360 ^{\circ }$

Solve for $\displaystyle t$:

$\displaystyle 5t \div 5 = 360 ^{\circ } \div 5$

$\displaystyle t = 72 ^{\circ }$

The statement is true.

Suppose Pentagon $\displaystyle PENTA$ is regular. Each angle of a regular polygon of $\displaystyle N$ sides has measure

$\displaystyle \frac{(N-2)180^{\circ }}{N}$

A pentagon has 5 sides, so set $\displaystyle N = 5$; each angle of the regular hexagon has measure  

$\displaystyle \frac{(5-2)180^{\circ }}{5} = \frac{ 3 \cdot 180^{\circ }}{5} = 108^{\circ }$

Since one angle is given to be of measure $\displaystyle 108 ^{\circ }$, the pentagon might be regular - but without knowing more, it cannot be determined for certain. Therefore, the correct choice is "undetermined".