The purpose of this question is to understand the process of using geometry to find degree values of angles.

Since l and m are parallel lines, the respective angles on both lines will have equivalent degree values. This means that a and w, b and x, and so on will be the same degree amount. A straight line is 180 degrees, so you know that angle d is  degrees; and therefore angle z, which has the same value, is also 113 degrees.

Let's begin by observing the larger angle.  is cut into two 10-degree angles by . This means that angles  and  equal 10 degrees. Next, we are told that  bisects , which creates two 5-degree angles.   consists of , which is 10 degrees, and , which is 5 degrees. We need to add the two angles together to solve the problem.

The perimeter of the pentagonal track is one third of a mile; one mile is equal to 5,280 feet, so the perimeter is 

 feet.

Each side of the pentagon has length one fifth of its perimeter, or

 feet.

Adrianne runs three and one half sides, or

 feet.

This makes 1,200 feet the closest, and correct, choice.

Benny runs at a rate of eight miles an hour for ten minutes, or  hours. The distance he runs is equal to his rate multiplied by his time, so, setting in this formula:

 miles.

One mile comprises 5,280 feet, so this is equal to 

 feet.

Since each side of the track measures 264 feet, this means that Benny runs 

 sidelengths.

This means Benny runs around the track for 25 sidelengths, which is 5 complete times, back to Point A; he then runs one more complete sidelength to Point B; and, finally, he runs  of a sidelength, finishing closest to Point C.

To solve, simply use the formula to find the total degrees in a polygon, where n is thenumber of vertices.

In this particular case, a pentagon is a shape that has five sides and thus has five vertices.

Thus,

Alvin runs at a rate of seven miles an hour for twelve minutes, or  hours. The distance he runs is equal to his rate multiplied by his time, so, setting in this formula:

 miles.

One mile comprises 5,280 feet, so this is equal to 

 feet

Since each side of the track measures 264 feet, this means that Alvin runs 

 sidelengths.

,

which means that Alvin runs around the track four complete times, plus four more sides of the track. Alvin stops when he is at Point E.

Below is the figure referenced; note that the hexagon is divided by its diameters, and that an apothem—a perpendicular bisector from the center to one side—has been drawn.

The circle has circumference ; its radius, which coincides with the apothem of the hexagon,  is the circumference divided by :

The hexagon is divided into six equilateral triangles. One, , is divided by an apothem of the hexagon  - a radius of the circle - into two 30-60-90 triangles, one of which is . Since  has length 30, and it is a long leg of , then short leg  has length

 is the midpoint of , one of the six congruent sides of the hexagon, so

;

this makes the perimeter of the hexagon six times this, or 

.

A diagonal is a line segment joining two non-adjacent vertices of a polygon.  A regular hexagon has six sides and six vertices.  One vertex has three diagonals, so a hexagon would have three diagonals times six vertices, or 18 diagonals.  Divide this number by 2 to account for duplicate diagonals between two vertices. The formula for the number of vertices in a polygon is:

where .

A diagonal connects two non-consecutive vertices of a polygon.  A hexagon has six sides.  There are 3 diagonals from a single vertex, and there are 6 vertices on a hexagon, which suggests there would be 18 diagonals in a hexagon.  However, we must divide by two as half of the diagonals are common to the same vertices. Thus there are 9 unique diagonals in a hexagon. The formula for the number of diagonals of a polygon is:

where n = the number of sides in the polygon.

Thus a pentagon thas 5 diagonals.  An octagon has 20 diagonals.

Below is the referenced hexagon, with some additional segments constructed.

Note that the segments  and  have been constructed. Along with , they form right triangle  with hypotenuse .

 is the midpoint of , so

.

 has been divided by drawing the perpendicular from  to the segment and calling the point of intersection .  is a 30-60-90 triangle with hypotenuse , short leg , and long leg , so by the 30-60-90 Triangle Theorem,

and 

For the same reason, , so

By the Pythagorean Theorem,

 when rounded to the nearest tenth.