Sum of Angle A and B: Does Proportion Matter?

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The discussion revolves around the proof involving the sum of angles A and B, particularly in the context of triangles and their leg proportions. The confusion arises from substituting one triangle for another despite differing leg ratios, questioning whether this affects the validity of the proof. Participants clarify that the triangles' similarity allows for such substitutions without altering the result, as they essentially multiply by a factor of one. The conversation also touches on the necessity of including a right triangle in the diagram to accurately represent the sine function. Ultimately, the proof demonstrates that angle relationships hold true regardless of the triangle sizes involved.
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Ok.. I've been looking at the sum of angle A and angle B...
in the proof they substituted angle A ( adjacent to angle B ) for the upper right angle A.
It seems obvious that one can do this because they are the same exact angle, but what is confusing me is that the triangles do not have the same exact leg proportions... Does that not matter?
 

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It has to do with the geometric proof of a line intersecting two parallels. Notice that the central line in the middle shows two right angles, so you know they are parallel.

More specifically:

zn8wfp.jpg


Hope that is what you were asking? It was a bit unclear.
 
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I totally understand how they got the angles however I don't understand how they could exchange the triangle that I highlited in yellow for the triangle that I highlighted in orange... because after all when we use sine.. cos... tan... ect.. we are trying to find the leg ratios of a triangle... but if we use the smaller triangle that I highlited in orange.. it won't have the same ratio as the one in yellow..
The only way that I understand how they can use it is because it looks like they are multiplying by a factor of one.. so it wouldn't change the result...
 
I don't see any of the triangles highlighted? Could you re-upload the picture perhaps.
 
sorry... here it is..
 

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I see what you are trying to do. Is this the proof for sin(a+b) ? If so, then your diagram is missing a right triangle. The hyp of both yellow and orange triangles should be perpendicular (because the hyp of the orange triangle is supposed to represent the sine of angle B). Knowing that, we can determine that the bottom right angle of the orange triangle is equal to (90-a). The top angle a is unknown for now, we will call it x. The remaining angle is 90. The sum of all three angles must be 180, so we can write an equation such as:

90 + x + (90-a) = 180
180 + x - a = 180
x - a = 180 - 180
x - a = 0

In order for x-a to equal zero, x must be a.
 

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