Suggestions for HS geometry book (proofs)

[Marin12]

Homework Statement: Can someone please recommend a geometry book for learning geometry proofs from ground up.
Relevant Equations: Lines, angles, poitn, plane...

Can someone please recommend a 9th grade high school geometry book that teaches how to write geometry proofs and that is not adding unnecessary steps or using undefined or not previously introduced concepts or unnecessary roundabouts to prove something.

For example of what irritates me is this kind of proof that proves that two adjacent angles on a straight line form a straight (180°) angle.

I am not sure if I am wrong about something, if this is the way it is supposed to be but I get very irritated with this Kind of complications.

If a straight angle is defined as two opposite rays forming 180 degre angle, and then we have this proof that proofs that if we have a ray on the line (which is not a curve, but a line) then the sum of adjacent angles is 180. And on top Of that ads the perpendicular to complicate things even further.

I want something simple, logical, with no unnecessary complications. Chat gpt recommend me Euclid elements with commentary for book 1 and after, Geometry and beyond by Robin Hartshorne or Geometry: A Comprehensive Course by Dan Pedoe.
It also advised me to avoid school textbooks because apparently they focus on test preparation instead of on actually learning geometry.

So I was curious if anyone has any other advices and recommendations.

Every help is appreciated.
Thank you.

 

[fresh_42]

This one looks nice:
https://www.math.ttu.edu/~rgelca/GEOMETRY.pdf

 

[FactChecker]

I have always been a big fan of the Schaum's Outline series. They have many worked problems and examples. You might take a look at this.

 

[symbolipoint]

To find a traditional highschool level Geometry book giving instruction about Geometry proofs should be very possible. Go to some local book sales. The proper kind of such book is easy to identify.

Here is one suggestion, and in case it is old, this should be no reason to worry:
Geometry (Prentice Hall Mathematics); Bass, charles, Johnson, Kennedy;
published some time before year 2003.

 

[FactChecker]

Can someone please recommend a 9th grade high school geometry book that teaches how to write geometry proofs and that is not adding unnecessary steps or using undefined or not previously introduced concepts or unnecessary roundabouts to prove something.

Wouldn't any good book satisfy that?

For example of what irritates me is this kind of proof that proves that two adjacent angles on a straight line form a straight (180°) angle.

If a straight angle is defined as two opposite rays forming 180 degre angle, and then we have this proof that proofs that if we have a ray on the line (which is not a curve, but a line) then the sum of adjacent angles is 180. And on top Of that ads the perpendicular to complicate things even further.

I agree. Is there some reason that the book did it the way it did?

 

[MidgetDwarf]

Depends on who the book is intended for.

Say for a student first learning geometry or someone who has done a bit more math, but not really upper division mathematics,

Then the Geometry book by Edwin E. Moise/Downs: Geometry is an excellent textbook. It does not contain the dreaded two column style proofs. It is clear and pedagogical.

Another book, which is good, but I personally believe Moise book is superior, is the Geometry book by Jacobs. Not sure which edition of Jacobs introduced the dreaded two column style proofs. Ie., was it the 2nd or 3rd, so do a bit of research to get the copy that has no 2 column proofs.

Now, if the the book is meant for a student who has done proof based mathematics, but wants a more rigorous presentation of geometry, then there is Geometry Illuminated.

Closer to how high school books present the material, there is Axiomatic Geometry by Lee.

I prefer Axiomatic Geometry by Lee, but both are excellent books.

 

[martinbn]

How about the book you took the picture form?

 

[TensorCalculus]

Homework Statement: Can someone please recommend a geometry book for learning geometry proofs from ground up.
Relevant Equations: Lines, angles, poitn, plane...

Can someone please recommend a 9th grade high school geometry book that teaches how to write geometry proofs and that is not adding unnecessary steps or using undefined or not previously introduced concepts or unnecessary roundabouts to prove something.

For example of what irritates me is this kind of proof that proves that two adjacent angles on a straight line form a straight (180°) angle.

I am not sure if I am wrong about something, if this is the way it is supposed to be but I get very irritated with this Kind of complications.

If a straight angle is defined as two opposite rays forming 180 degre angle, and then we have this proof that proofs that if we have a ray on the line (which is not a curve, but a line) then the sum of adjacent angles is 180. And on top Of that ads the perpendicular to complicate things even further.

I want something simple, logical, with no unnecessary complications. Chat gpt recommend me Euclid elements with commentary for book 1 and after, Geometry and beyond by Robin Hartshorne or Geometry: A Comprehensive Course by Dan Pedoe.
It also advised me to avoid school textbooks because apparently they focus on test preparation instead of on actually learning geometry.

So I was curious if anyone has any other advices and recommendations.

Every help is appreciated.
Thank you.

View attachment 363033

I will recommend https://amzn.in/d/2E7wcJI as a book to learn geometry, (don't know if you'll be able to get it outside of India though - it was difficult to find copies here in Britain) but I think that for learning to write clear step-by-step proofs, that comes with practice.

A lot of my close friends have spent hours and hours learning exactly this, since writing clear proofs is something absolutely crucial for their maths Olympiad endeavours. They have mastered this skill, but it was through practicing problems and then going up to their maths teachers to ask for feedback, and doing this again and again and again. Also looking at mistakes them and their peers have made in Olympiads, and keeping those in mind. A good first step would obviously be to learn notation such as the one in the screenshot provided, but that can be taught fairly swiftly from online rather than buying a book.
The book I've provided has good practice problems for geometry, but if you are looking for problems that will require longer/more rigorous proofs, check out https://ukmt.org.uk/competition-papers/jsf/jet-engine:free-past-papers/tax/challenge-type:69/ to start off, then https://ukmt.org.uk/competition-papers/jsf/jet-engine:free-past-papers/tax/challenge-type:73/, https://ukmt.org.uk/competition-papers/jsf/jet-engine:free-past-papers/tax/challenge-type:74/ and https://ukmt.org.uk/competition-papers/jsf/jet-engine:free-past-papers/tax/challenge-type:75/. Or, the maths Olympiad papers for your region. They will definitely have problems that require rigorous step-by-step proofs like this.

 

[mathwonk]

It is hard to know what to recommend, since you already have excellent recommendations, but are not taking them for some reason you do not mention. It would help to know why Euclid does not work for you if that is the case. I.e. the absolute best geometry book is Euclid, and although it can be challenging to get into, the book by Hartshorne is the perfect handbook and guide for that. So Euclid, green lion edition, plus Hartshorne, is my first recommendation. The illustration you provide is apparently not from those books, and indeed looks unappealing to me. Moise of course is very scholarly and excellent, but to me would be a choice more for a professional than a beginning student.

Still, even Euclid may be overwhelming to some people, who would benefit from a much gentler, fun based, approach. Harold Jacobs is the master of that, easy explanations, gentle challenges, and even cartoons for keeping the attention of kids. His first edition however is the only one, (or maybe the 2nd), that I recommend, not the third, which waters down the proof aspect. Here are some used copies.
https://www.thriftbooks.com/w/geome...mkoBoCkhUQAvD_BwE#idiq=887692&edition=2949690

But since Euclid is the best (I recommend one without the lengthy commentaries) I suggest you should try it first before trying others. The essay by Hartshorne on "teaching geometry according to Euclid" may be useful, as it was to me.

A remark: your request for a treatment that does not introduce undefined terms, raises the possibility that you have not understood a basic fact about axiomatic proofs, namely that undefined terms are necessary. It is impossible to begin anywhere and still define every notion in terms of previous ones. Indeed Euclid himself seems not to have granted this point, and one of the difficulties of getting into Euclid is that he does begin by trying to define the undefinable, by describing the meaning of point, line, etc...Nowadays we accept that the only meaning one can give to these terms is the meaning provided by the axioms they satisfy: e.g. that two distinct lines are either disjoint or meet in exactly one point.
To get started in Euclid one should glance at but not be bogged down in the definitions (at least the first nine or so vague ones) which are given first. Get on into the axioms and postulates and propositions.

Of course he does famously also violate one of your requests, in that he does try to prove Prop. 4 using rigid motions, which have not been stated as allowed. Hartshorne discusses this problem and its possible solutions, following Hilbert. There are also free notes on my webpage that I wrote for some brilliant children to whom I taught from Euclid.
https://www.math.uga.edu/sites/default/files/inline-files/10.pdf

Note: There are essentially two popular approaches to geometry, Euclid's original and Birkhoff's more modern one. Birkhoff tries to shortcut the difficulty of starting geometry from scratch, by assuming one already knows the properties of real numbers, thus assuming an even more difficult notion, in my opinion. I believe Harold Jacobs follows Hilbert's approach. The difference is that Euclid just assumes points, lines and planes, and equality, greater and less than, as basic notions, and Birkhoff also assumes rulers and protractors, i.e. measuring devices in terms of real numbers, which in my opinion most people do not actually understand. So in my opinion, Euclid's approach allows one to build up an understanding of numbers and even algebra, in terms of elementary geometry. This approach seems to be the historically correct one, but I am not a historian.

By the way, theorem 1 in your illustration is indeed "obvious" in Euclid's approach, since by definition "180 degrees", although not a term he uses, means for him a "straight" angle, i.e. formed by a straight line. If you are thinking of it this way too, that may be why the theorem seems unnecessary to you. I recall Euclid also has a postulate that all straight angles are equal, by the way. (Your book seems to be using Birkhoff's approach, in particular.) This statement however is also assumed to be obvious in at least one well regarded book using Birkhoff's approach, Planimetry, by Kiselev, p.15 "supplementary angles". What requires proof is heavily dependent on the choice and wording of definitions in each book, as observed below by vela. One also needs to know here that "addition" of angles is done by juxtaposing them. Note by contrast that even Prop. 1 in Euclid is already non obvious and interesting, how to construct an equilateral triangle with a "floppy" compass.

That reminds me, another good book is the Art of Problem Solving volume, with some flaws in my opinion, but with many good properties. As I recall, the treatment of similarity is incomplete there, and leads to a bit of logical circularity, but I may be wrong, as no one but me seems to have complained about it over many years.
https://www.abebooks.com/servlet/Bo...BOHhAqEfzvMexNrbdx4OAB65iqs6rHMRoCT6oQAvD_BwE

After perusing them, I agree that the notes linked by fresh_42 in post #2 are a nice possible choice. They are a sort of hybrid of the two approaches, Euclid and Birkhoff, as they start out more like Euclid, and then at a certain point introduce real numbers via the axioms of continuity. In fact I consider this the best way to see how an understanding of real numbers grows out of Euclidean geometry, rather than assuming real numbers as known first. These notes also avoid the logical gap in Euclid's own proof of SAS, by assuming a strong SAS type axiom, C6.
They will still require some effort to absorb the detailed axioms on betweenness. It may seem odd to prove, as they do, that sums of congruent objects are again congruent, a property Euclid takes as a "common notion". enjoy.

 

[vela]

I am not sure if I am wrong about something, if this is the way it is supposed to be but I get very irritated with this Kind of complications.

My guess is that you're not recognizing the subtleties the proofs are addressing. In the example you provided, for instance, the statement that angle ACB = 180 degrees and the statement that the sum of the angles ACD and BCD is 180 degrees are logically distinct. It may seem "obvious" to you that one follows from the other, but the thing with math is you need to learn to be wary of the seemingly obvious and not simply assume it's true.

You might question why the book's authors provided the proof they did if a result is so obvious, why they felt the supposedly obvious needed to be proved, etc.

 

[Marin12]

Thanks everyone for your time and advices. Iwill check all of the mentions. Just to add few things and explain myself. I have checked the books suggested by ChatGPT (Euclid was with commentary though so I was overwhelmed by it and it also started with triangles instead of with line, point, angle, etc. And I wanted to avoid that for no other reason but because this, point, line, angle feels more familiar right now since I came much further in the book, I just posted this example because it was easier to explain that one then the other harder ones) and the reason I asked for your opinion was because some of you are/were university/high school professors and tutors, so if somebody knows it should be you, I still trust uni prof. More than chatgpt so yeah, this is why I posted the question.

This is not the first book I used and some of them were not clear, had no examples, were poorly structured or felt like I was thrown in the deep while not knowing how to swim at all.

Also, I was writing in the moment of anger and frustration so I didn't really explain why I disliked this proof so much. The main reason is that I find it really unnecessary to prove that the sum of adjacent angle on a line is 180. But not even that is what I dislike. It's that it seems like I need to know in advance how to disassemble the angle (big one) into two smaller ones, and that I need to know how to build the other one (smaller one) in order for the algebra to cancel out and to be left with 90+90 (a lot of such examples and theorems in the book). If the perpendicular is standing on the line it is still a ray and just as right angle is defined as 90° angle so Is straight angle defined as 180° angle. Line is straight. So what is the point in prooving the definition of straight angle (180°) by using another definition of right angle (90°) . It is like saying outside is sunny, and that is true because in order to be sunny rain must not fall. While I haven't even looked out the Window and have no idea if it raining or not.

I came much further in the book and I don't want to say the book is bad but feels like decent amount of proofs and examples have been unclear and as I said felt like theorems have been proved by "trickery", backwards, allready knowing the solution and then just like, here is how it's done. But I uderstand that it is probably the best way to teach but still, I feel discouraged by that. It feels like I need to memorize entire proof to be able to understand it. It feels like if you miss the order of angle letters (formed by points) you can never solve the problem correctly. I would like to avoid that. Maybe I am asking for too much but just in case, it does not hurt to ask.

Again, thank you everyone for helping me

 

[martinbn]

Could take a look at this one.
https://link.springer.com/book/10.1007/978-1-4612-0961-4

 

[symbolipoint]

Also, I was writing in the moment of anger and frustration so I didn't really explain why I disliked this proof so much. The main reason is that I find it really unnecessary to prove that the sum of adjacent angle on a line is 180. But not even that is what I dislike. It's that it seems like I need to know in advance how to disassemble the angle (big one) into two smaller ones, and that I need to know how to build the other one (smaller one) in order for the algebra to cancel out and to be left with 90+90 (a lot of such examples and theorems in the book). If the perpendicular is standing on the line it is still a ray and just as right angle is defined as 90° angle so Is straight angle defined as 180° angle. Line is straight. So what is the point in prooving the definition of straight angle (180°) by using another definition of right angle (90°) . It is like saying outside is sunny, and that is true because in order to be sunny rain must not fall. While I haven't even looked out the Window and have no idea if it raining or not.

I came much further in the book and I don't want to say the book is bad but feels like decent amount of proofs and examples have been unclear and as I said felt like theorems have been proved by "trickery", backwards, allready knowing the solution and then just like, here is how it's done. But I uderstand that it is probably the best way to teach but still, I feel discouraged by that. It feels like I need to memorize entire proof to be able to understand

One must take much time, study regularly, and spend much effort. One needs to also willingly review more than once; or twice. Algebra 1&2 is/are different than "Geometry". Some people are algebra people, and some people are Geometry people. Geometry proofs require practice - regular practice.

A person may very well study Geometry first time, possibly struggling, and all the way through the "course" if possible. Person can later, maybe one or two years later, study Geometry again. Same book or different book does not matter - just need to use a standard good textbook. Person may again, later, maybe one or two years later, study Geometry yet again. Each time person review-studies Geometry, he should do better than the last time. Does all this sound scary? Maybe if one is a motivated high school student and just wants the course done-with, and wants to just graduate high school, possibly yes. But for student who just find Geometry difficult the first time through, then his study pathway will need to include another thorough restudy of Geometry and just try to look at the optimistic idea that he can learn better each time he studies through the course. TRUST THAT! I KNOW! I DID THAT!!

 

[mathwonk]

The link in #12 is to a nice little book written by my thesis advisor, together with his son, for a course offered, I believe, to the son's 8th grade class. I have taught from it in my college classes, and learned some new things from it myself. I still have it on my shelf. Very user friendly as I recall.

 

[Homelilly]

If it's for high school, why not take AoPS?
https://artofproblemsolving.com/store/book/intro-geometry