And NO, assuming all the lines are straight is NOT unreasonable, it is the only way that the problem could ever possibly have a solution.

Wow, you got so close to my point but still fell short! My point is that you cannot reach a solution without making assumptions that fundamentally alter the solution. Your math is correct if and only if your straight line assumption is true. It may be a reasonable assumption, but that does not mean it must always be an accurate assumption.

I ask you to consider the following picture:

I tell you that the triangles are not to scale. We can definitively say that h = 80° and k = 90°. Note that h + k != 180°. Despite the strange and inconsistent scaling, this meets all requirements of triangles.

Now let me take away the defined 50° angle:

Once again, the triangles are not to scale. They are visually the same triangles. You might assume that h + k = 180°, yielding 40° for the missing angle above k. However, if I reveal to you that the missing angle is indeed 50° or 60° or ANY ANGLE (excl. 40°) such that the sum of angles can still be 180°, you and your assumption are suddenly wrong.

Perhaps consider nurturing your brain further before making such condescending remarks.

You’re overlooking a major assumption on your end. There is nothing in the image that suggests that the bottom of both triangles forms a straight line. The pair of bottom edges are two separate lines. They may or may not form a sum 180° angle. You are assuming the angles are supplementary. We know that the scale of the image is wrong, thus it is not safe to definitively say that the 80° angle’s neighbor is supplementary. They may be supplementary, or the triangles may share a consistently skewed scale, or the triangles may each have separately skewed scales.

This is a basic logical thought process and basic trigonometry.

Yes, I believe I implied this by suggesting that the sum of angles being 190° is absurd.

I see. I agree completely. The only place this belongs is as a thought experiment on making assumptions in geometry.

For the love of dog, you can’t solve this problem without making assumptions that fundamentally change the answer. People are too quick to spot the first error and then make assumptions that are conveniently consistent with the correction.

…what? I get that this drawing is very dysfunctional, but are you going to argue that a triangle within a plane can have a sum of angles of 190°?

You’re making the assumption that the straight line consisting of the bottom edge of both triangles is made of supplementary angles. This is not defined due to the nature of the image not being to scale.

We can’t assume that the straight line across the bottom is a straight line because the angles in the drawing are not to scale. Who’s to say that the “right angle” of the right side triangle isn’t 144°?

If the scale is not consistent with euclidian planar geometry, one could argue that the scale is consistent within itself, thus the right triangle’s “right angle” might also be 80°, which is not a supplement to the known 80° angle.

This is what I was thinking. The image is not to scale, so it is risky to say that the angles at the bottom center add up to 180, despite looking that way. If a presented angle does not represent the real angle, then presented straight lines might not represent real lines.