a regular pentagon is divided into 5 isoscleles triangles work out the size of angle x angle y angle z

x = 72

y = 54

z = 54

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.

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

As a quick check,

x+y+z = 72+54+54 = 180

which helps confirm the answer.