Answers:
x = 72
y = 54
z = 54
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The regular pentagon has all five sides the same length. It also has all interior angles the same measure. This leads to y = z. These two angles are the congruent base angles of the isosceles triangle. The base angles are opposite the congruent sides.
The fact that we have a regular pentagon also means that the five triangles shown are identical copies of one another (i.e. they are congruent triangles). Divide the full 360 degree rotation into 5 parts to get 360/5 = 72, which represents the measure of the central angle x.
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For any triangle, the three angles always add to 180. This means,
x+y+z = 180
72+y+z = 180 .... plug in x = 72
72+y+y = 180 ... recall that y = z
2y+72 = 180
2y = 180-72
2y = 108
y = 108/2
y = 54, which is also the measure of angle z as well
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As a quick check,
x+y+z = 72+54+54 = 180
which helps confirm the answer.