Problem 1, Part (a)
Draw a regular pentagon with all five sides the same length, and all interior angles the same.
Plot a point at the center of the pentagon.
From this center point, extend segments to the five vertices. This forms 5 isosceles triangles that are identical copies of each other.
The vertex angle is 360/n = 360/5 = 72 degrees.
The base angle is (180-72)/2 = 108/2 = 54
Two adjacent triangles have their adjacent base angles combine to form a single interior angle. So 2*54 = 108 is the interior angle measure.
An alternative approach is to use this formula
i = measure of interior angle
i = 180(n-2)/n
i = 180(5-2)/5
i = 180(3)/5
i = 540/5
i = 108
Answer: Each interior angle of a regular pentagon is 108 degrees.
=============================================================
Problem 1, Part (b)
E = exterior angle of a regular polygon with n sides
E = 360/n
E = 360/5
E = 72
This is equal to the measure of the central angle found in the previous part.
Alternatively we can use the result of part (a)
i = interior angle
E = exterior angle
i+E = 180
E = 180 - i
E = 180 - 108
E = 72
Answer: 72 degrees.
=============================================================
Problem 2, Part (a)
This is what the diagram looks like when we extend the exterior sides of the pentagon to form a star shaped pentagram
Notice the angles 108 and 72 show up, which were calculated back in the previous problem.
Add up the angles of the triangle to solve for x.
x+72+72 = 180
x+144 = 180
x = 180-144
x = 36
Answer: 36 degrees.
=============================================================
Problem 2, Part (b)
Erase the pentagon in the middle to get this
The red angle y is the reflex interior angle we want.
If we were to add the pentagon back in, and relevant angle measures
Then it's probably very clear that the red angle is:
y = 72+108+72
y = 252
Answer: 252 degrees