How many degrees are in the sum of the measures of the six numbered angles pictured? [asy]
draw((3,8)--(10,4)--(1,0)--cycle,linewidth(1));
draw((7,8)--(9,0)--(0,4)--cycle,linewidth(1));
label("1",(3,8),SSE);
label("2",(7,8),SSW);
label("3",(10,4),2W);
label("4",(9,0),NW+NNW);
label("5",(1,0),NE+NNE);
label("6",(0,4),2E);
[/asy]
The angles numbered 1, 3, and 5 are the three interior angles of a triangle, so they sum to $180^\circ$.  Similarly, the angles numbered 2, 4, and 6 are the three interior angles of a triangle, so they also sum to $180^\circ$.  Combining these, the sum of all six angle measures is $180^\circ + 180^\circ = \boxed{360^\circ}$.
360
Level 4
Prealgebra
Problem: How many degrees are in the sum of the measures of the six numbered angles pictured? [asy]
draw((3,8)--(10,4)--(1,0)--cycle,linewidth(1));
draw((7,8)--(9,0)--(0,4)--cycle,linewidth(1));
label("1",(3,8),SSE);
label("2",(7,8),SSW);
label("3",(10,4),2W);
label("4",(9,0),NW+NNW);
label("5",(1,0),NE+NNE);
label("6",(0,4),2E);
[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the sum of the measures of the six numbered angles, we first recognize that the angles are formed by the intersection of two triangles. Each triangle has a total internal angle sum of 180 degrees. 

Since there are two triangles in the figure, we calculate the total angle sum for both triangles: 

180 degrees (first triangle) + 180 degrees (second triangle) = 360 degrees.

Next, we consider the arrangement of the angles. The angles labeled 1 through 6 are the internal angles of the two triangles. Each angle contributes to the overall angle sum of the triangles. 

Furthermore, the angles at each vertex of the triangles are adjacent to each other, and they do not overlap. Therefore, the sum of the angles 1, 2, 3, 4, 5, and 6 must equal the total angle sum of the two triangles.

Thus, the final answer is 360.<|END_OF_LATENT|> The final answer is $360$.