Find the angles of the pentagon $ABCDE$ in the figure below. Proposed by Morteza Saghafian

Notice that XYZ is equilateral and all the angles in the vertex X add to $360$ so each big angle is $(360-60)/4=75$ Then the angles in the popular isosceles triangle are $75,75, 30$ And the pentagon has angles $150,90, 150, 75, 75$

Note that, \[\triangle BCX \cong \triangle CXY \cong \triangle CYD \cong \triangle YDE \cong \triangle YZE \cong \triangle AXZ \cong ABX\]Hence, the angles are $\angle C = 90^{\circ}, \angle B = \angle D = 150^{\circ}, \angle A = \angle E = 75^{\circ}$

In the center of this hexagon we can see a triangle named as XYZ. We can find the angles. 60⁰, 60⁰ ,60⁰. Then we can name the angles like x. 《CBX=《CXB. All the triangles we can write this. So after that we will get 4x+60=360 and x=75⁰ 2x=150⁰ X=75⁰ 《C=30⁰+30⁰+30⁰=90⁰ Answer; 150⁰,90⁰,150⁰,75⁰,75⁰.