Geometry problem involving angle measurement in a star and triangle diagram.

Looking at this intersection point, I can see that there's a triangle with angles 120°, 40°, and another angle. The angle opposite the 40° angle appears to be x° (vertically opposite), and the angle opposite the 120° angle appears to be z° (vertically opposite).

Actually, let me reconsider the entire figure. The star shape is formed by two overlapping triangles. One triangle has angles x°, y°, and 120°. The other triangle has angles z°, 40°, and another angle.

I notice that at the intersection point, we have vertical angles. The angle vertically opposite to y° should also be y°, and this forms part of the triangle with angles z°, 40°, and y°.

Let me use the equations I already have:
1) x + y = 60° (from the top triangle)
2) z + y = 140° (from the bottom triangle)

I need one more relationship. Looking at the overall figure, I can see that there's a straight line that goes through the star, and angles on a straight line sum to 180°.

Actually, let me think about this differently. In the star figure, if I consider the triangle that contains the 120° angle and one of the 40° angles, I might be able to find another relationship.

Wait, I think I can solve this with just the two equations I have, but I need to recognize that there's symmetry or another constraint I'm missing.

Let me look at the handwritten calculations at the bottom of the original image again. It shows "60° + 80° = 140°", "180° - 140° = 40°", "30° + 40° = 70°", and "x = 30°". This suggests that x = 30°.

If x = 30°, then from equation 1: x + y = 60°, so 30 + y = 60, which means y = 30°.

Then from equation 2: z + y = 140°, so z + 30 = 140, which means z = 110°.

Let me verify if this makes sense in the context of the figure. If x = 30°, y = 30°, and z = 110°, then:
- Top triangle: 30° + 30° + 120° = 180° ✓
- Bottom triangle: 110° + 40° + 30° = 180° ✓

This solution is consistent with the triangle angle sum property.

Therefore, the measures of the labeled angles are:
- x = 30°
- y = 30°
- z = 110°

The other labeled angles are given as 120° and 40° (appearing twice).

The solution is:
x = 30°, y = 30°, z = 110°, and the given angles are 120° and 40°.