First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

• Vertical angles are congruent
• Corresponding angles are congruent
• Alternate interior angles are congruent
• Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\). 

For this problem, we want to find the angle that is congruent to angle \(\displaystyle \angle{\ a}\). Based on our answer choices, angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles; thus, both angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are congruent and equal \(\displaystyle 135^\circ\)

The provided triangle is a right triangle. We know this because the angle marker in the left corner of the triangle indicates that the triangle possesses a right or \(\displaystyle 90^\circ\) angle. When a triangle includes a right angle, the triangle is said to be a "right triangle." 

We can use the Pythagorean Theorem to help us solve this problem.

The Pythagorean Theorem states that for right triangles, the square of the hypotenuse is equal to the sum of the square of the other two sides. In other terms:

\(\displaystyle a^2+b^2=c^2\)

We can use the formula and substitute the known side lengths from the problem to solve for the missing side length:

\(\displaystyle 12^2+35^2=c^2\)

\(\displaystyle 144+1\textup,225=c^2\)

\(\displaystyle 1\textup,369=c^2\)

\(\displaystyle \sqrt{1\textup,369}=\sqrt{c^2}\)

\(\displaystyle 37=c\)

In order to solve this problem, we need to recall the formula used to calculate the volume of a cone:

\(\displaystyle V=\pi r^2\left ( \frac{h}{3} \right )\)

Now that we have this formula, we can substitute in the given values and solve:

\(\displaystyle V=\pi3^2\left ( \frac{6}{3} \right )\)

\(\displaystyle V=\pi9(2)\)

\(\displaystyle V=\pi18\)

\(\displaystyle V=56.55\textup{ in}^3\)