Help finding the internal angles of my bike frame
I am a P.E. teacher and haven't done any trig in a decade... I am trying to find the internal angles of the frame of my bike to get a custom bag made for the frame. I have the measurements of the internal length of each tube, and I also have some kind of angle measurement the tube is sitting at, but not sure how to read it correctly, or use a cosine formula to find out the angles of each point. Again, it needs to be the INTERNAL angles.
(TT) Top Tube length: 19 1/8 inch OR 48.57 cm
(ST) Seat Tube length: 16 1/4 inch OR 41.275 cm
(DT) Down Tube length: 23 inches OR 58.42
Lastly, there is a gap in the triangle due to the steer tube and the size of the bike, so 2 points of the triangle are not exactly touching: 1 inch OR 2.54cm
Below is a picture of the bike, the labled tubes, and the lengths of each tube.
This is also the geometry of the bikes that has the ST angle but its on the wrong side... It is measured from the rear triangle not the main frame triangle. (SIZE 52)
I have tried using an online triangle calculator thing, and this was the output...
Side B is the Seat Tube (ST) Side A is the Top Tube (TT) and Side C is the Down Tube (DT). The unit of the numbers is in centimeters.
Let me know if I can provide any other information that would be helpful in solving this.
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MFH, Thank you very much for that! I really appreciate it. It was very helpful.

Thanks for the coffee ! :)

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The offered bounty is low for the amount of time one needs to write a good solution for this.
This is my first time posting here, I was suggested by a redditor. Please feel free to inform me as to what a fair bounty would be.
The measurement of the lengths only are not sufficient to determine the angles : imagine the pieces could move around the point where they are attached to each other. In particular the small one could "unfold" until it is completely aligned with the upper one, and also fold in until it is aligned : in these two cases the inner angle "at the bottom" is smaller, in the former case it is larger. However, the difference might be quite small.
I'm just looking for the answers (the internal angles of the bike frame) because I have tried a lot and have held off the product creator from starting this project for a while. I apologize if this is the wrong place to post for answers/solutions to the presented problem.
Yes but as I said the internal angles cannot be determined from the given information. The topmost angle could be anything between 0° and more than 180°, with the given lengths. We need to know at least one angle, or one more measurement, e.g., some diagonal (distance from one corner to the opposite one).
No need to apologize, this is actually the perfect place to ask this question As mentioned by M F H above, the angles can not be computed using the given information. Do you assume that the angle between TT and DT is 90 degrees?
From the picture we can see that none of the angles is 90°.
@Nirenberg The angle between the TT and DT is not. The chart for the bike geometry says the head tube angle is at 71 which the TT and DT are attached to. The diagram says the seat tube is at 74.5, but I read it as the seat tube is rising from the left side of the 180 flat plane. This is my first time posting here, I was suggested by a redditor. Please feel free to inform me as to what a fair bounty would be.
But probably we make a very small error assuming that the topmost angle is about 90°. OK, I can answer for that case and give the angles under various hypothetical values for that angle.
It looks like the angles on the bike (looking at the geometry chart) assumed to be on a 180 degree flat plane between the two axels of the bike (front and rear). Not sure if that observation will help.
Additionally, I used a angle measurement app on my iphone and found that the rise of the TT (or angle the TT is at in relation to 0 degree flat surface) is about 8.5 degrees. The DT angle by that same assumption (and measured by an iphone app) is about 48.8
Additionally, I used a angle measurement app on my iphone and found that the rise of the TT (or angle the TT is at in relation to 0 degree flat surface) is about 8.5 degrees. The DT angle by that same assumption (and measured by an iphone app) is about 48.8
That checks out with my result : The document tells us that ST makes an angle of (8) = 75° with the horizontal, if you add to this the 8.5 degrees you measured for TT you get 83.5° for angle(ST,TT), which compares extremely very well to the 83.6° degrees I find for this with gamma ~ 90°.
Similarly, if you compute 180° (blue horiz.line) 75° ("to the left of ST")  48.8° ("to the right of DT"= your measurement, if I understand well), then you get angle(ST,DT) = 56.2°, which should correspond to the 55.2° I get for this with gamma = 90°. But one would have to increase gamma up to ~ 106° to get 56.2°. So I'm inclined to think that this measurement of 48.8° is not very precise.
Wow, thank you very much! That is very helpful and informative. Out of curiosity would the head tube angle (HT, which is the tube that the TT and DT are attached too) being at 71 affect the other angles? I think you also labeled that as SS = 90. Would that matter in this situation or would we assum that the TT and DT are measured in realtion to an invisible 90 degree angle?
yes, indeed, my SS is the part of HT that touches the inside of the area you depicted ; HT is significantly longer. But yes, they have the same inclination w.r.t. the horizontal or vertical. You have the relation : (upper blue horizontal line =) 180°  71°  8.5° (= angle of TT w.r.t. horizontal you measured) = 100.5° = angle(TT, HT or SS) = gamma. So maybe gamma = 100° would be a better assumption? I will add the values in the answer cause not enough characters left here.