The image presents a geometric figure with several numbered angles and asks to find the measures of those angles. There are angles labeled as $\angle 1$, $\angle 2$, $\angle 3$, $\angle 4$, and $\angle 5$, along with the given angles of 80°, 60°, 40°, and 105°.

To find the unknown angles, we can apply basic principles of geometry, such as the fact that the sum of angles in a triangle is always 180°, and the angles on a straight line add up to 180°.

Let’s start by solving the angles step-by-step.

1. For $\angle 1$: The angle $\angle 1$ is part of the first triangle, and we know that the sum of angles in a triangle is 180°. So, we can set up the equation: $80^\circ + 60^\circ + \angle 1 = 180^\circ$ Solving for $\angle 1$: $\angle 1 = 180^\circ - 80^\circ - 60^\circ = 40^\circ$

2. For $\angle 2$: The angle $\angle 2$ is adjacent to $\angle 1$ along the straight line, and the angles on a straight line add up to 180°. Therefore: $\angle 1 + \angle 2 = 180^\circ$ Substituting $\angle 1 = 40^\circ$: $40^\circ + \angle 2 = 180^\circ$ $\angle 2 = 180^\circ - 40^\circ = 140^\circ$

3. For $\angle 3$: Angle $\angle 3$ is part of the second triangle, and we can use the fact that the sum of the angles in a triangle is 180°. So: $40^\circ + 105^\circ + \angle 3 = 180^\circ$ Solving for $\angle 3$: $\angle 3 = 180^\circ - 40^\circ - 105^\circ = 35^\circ$

4. For $\angle 4$: The angle $\angle 4$ is adjacent to $\angle 3$, forming a straight line. Thus: $\angle 3 + \angle 4 = 180^\circ$ Substituting $\angle 3 = 35^\circ$: $35^\circ + \angle 4 = 180^\circ$ $\angle 4 = 180^\circ - 35^\circ = 145^\circ$

5. For $\angle 5$: The angle $\angle 5$ is the last angle in the second triangle. Since we already know that the sum of the angles in a triangle is 180°, we can use: $\angle 4 + \angle 5 + 40^\circ = 180^\circ$ Substituting $\angle 4 = 145^\circ$: $145^\circ + \angle 5 + 40^\circ = 180^\circ$ $\angle 5 = 180^\circ - 145^\circ - 40^\circ = -5^\circ$ It appears there may be a mistake in the figure setup or a misinterpretation. The negative result suggests a problem with how the angles have been placed. Could you verify if the angles and the figure are correctly provided?