Geometry - Circle Properties

We can find the measures of each of the labeled angles using circle properties and basic geometric theorems.

First of all, since the arcs between each pair of adjacent vertices is equal, we know that we are looking at a regular pentagon inscribed in this circle. A pentagon can be made up of three triangles, thus the interior angles must add up to 3(180) degrees. Since it is a regular pentagon, we know that angle 1 is 3(180)/5 or 108 degrees.

Now, notice that line AB is parallel to line EF, so angle 1 is congruent to angle 8 by the Alternate Interior Angle Theorem. So angle 8 is also 108 degrees.

Angle 2 is supplementary to angle 8, so angle 2 must be 180-108=72 degrees.

Line BC is parallel to line AD, so we can use the Corresponding Angles theorem. Angle 2 must be congruent to the angle supplementary to angle 3 within the triangle also containing angles 4 and 5. Thus, angle 3 must be 180-72=108 degrees.

Now, let's just back up to the top of the circle. Line BF is tangent to the circle at point B. We know then that angle 7 must be half the measure of the arc between B and C. Thus, angle 7 is 72/2=36 degrees.

Angles 6, 7, and 8 make up a triangle, so they must add to be 180 degrees. Therefore, angle 6 must be 180-108-36=36 degrees.

Then, line BF is parallel to ED, so we know by the Alternate Interior Angle theorem that angle 6 must be congruent to the angle which, with angle 4, makes up the angle at point E. Remember that we know that the angle at point E must be 108 degrees, so angle 4 must be 108-36=72 degrees.

Finally, We know that angles 4 and 5 and the angle supplementary to 3 form a triangle and need to add to 180 degrees. So angle 5 must be 180-72-(180-108)=36 degrees.

So, we get that

angle 1=108 degrees
angle 2=72 degrees
angle 3=108 degrees
angle 4=72 degrees
angle 5=36 degrees
angle 6=36 degrees
angle 7=36 degrees
angle 8=108 degrees

using simple Euclidean geometry theorems (Alternate Interior Angles, Corresponding Angles, and Supplementary Angles) as well as properties in circle geometry (arc angles and tangent lines).