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I am trying to understand the following problem: State the basic angle theorem needed to prove $a\cong b$ in the following figure:

enter image description here

The answer is: Since $\overline{AB} \perp \overline{BC}$, $B$ is a right angle and hence, $b$ is the complement of $1$. Since $a$ is complement of $1$, $a\cong b$. Ie: Complements of the same angle are congruent.

I don't understand how $b$ being complement to $1$ forces it to be congruent to $a$.

Red Banana
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  • @Blue Sorry, I mixed up the concepts in my mind. – Red Banana Mar 20 '22 at 00:59
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    The given solution states the key result: "Complements of the same angle are congruent." After all, algebraically, if $\angle a$ and $\angle 1$ are complements, the $\angle a+\angle 1=90^\circ$; likewise, $\angle b+\angle 1=90^\circ$; therefore, $\angle a+\angle 1=\angle b+\angle 1$, and we can subtract $\angle 1$ from both sides. – Blue Mar 20 '22 at 01:03
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    BTW: Are there multiple users on this account? The question here is quite a departure from others asked under this name. – Blue Mar 20 '22 at 01:06
  • @Blue No. I am an undergraduate student of mathematics in a university and always had a personal project of learning Euclidean geometry but never quite found the time nor the right material. I just found a book about it now and am (finally) learning it! I plan to also read Altschiller-Court's College Geometry and Johnson's Advanced Euclidean Geometry as soon as I finish this one. :-) – Red Banana Mar 20 '22 at 01:09
  • @RedBanana You have a question about algebraic geometry and somehow you are just now learning Euclidean geometry. That doesn't add up. Seems there are multiple people using this account. – oliverjones Mar 20 '22 at 01:48
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    @oliverjones I guess you're making an unreasonable interpretation of what I just said: By "I am learning Euclidean geometry" I don't mean I don't know what a "circle" is, what "perpendicular" means, what "parallel" means, nor that I don't even know what "geometry" is. I meant that I am learning about all the other stuff I didn't learn before in Euclidean geometry. Geometric constructions, SSS, SAS, ASA theorems, etc. Obviously - as you noted - I knew enough geometry to pass a basic course in algebraic curves I did last semester. – Red Banana Mar 20 '22 at 02:24
  • You said you are "finally" learning it. Based on previous questions you've asked, it is puzzling that you would not understand $a \simeq b$ having already had so much mathematical experience. Your account is 10 years old, you asked about Pythagorean triples 8 years okay. To me, it seems as though there is more than one person using your account. – oliverjones Mar 20 '22 at 04:05
  • @oliverjones As you are American and you guys fail to perceive there could be other educational contexts in other places, I'll explain for you: I never learned basic geometry in school, at university, we learn "geometry with vectors" and we do stuff like: Given two vectors, compute the angle between them, find out (computing the dot product) if they are perpendicular, etc. Everything is "concrete" in the sense that a simple computation must be made so that I can check if something is true. This skill of reasoning through a diagram given only some properties is unknown to me and I got confused. – Red Banana Mar 20 '22 at 04:29
  • You are making assumptions about my nationality given my listed location. Furthermore, nothing I said indicates as though I have a narrow vision of what education can be. It is not unreasonable to be puzzled as to why someone with some advanced mathematical knowledge would be puzzled by congruent angles. I will not be replying further. – oliverjones Mar 20 '22 at 05:06
  • @oliverjones Now you're lying: You're not "puzzled as to why someone with some advanced mathematical knowledge would be puzzled by congruent angles" you made a direct accusation to me. You could at least be honest. – Red Banana Mar 20 '22 at 05:14
  • I guess I lied in that I said I wouldn't reply. Yeah I am puzzled hence why I said it seems there are multiple people using your account. Also to be clear I surmised there is multiple people that could be using your account. I never directly said multiple people are using your account. Now, I would like to end this. Good luck with your studies. – oliverjones Mar 20 '22 at 05:19

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