What Is a Proof in Geometry?
If you’re jumping into your first round of teaching geometry, a quick refresher – Yes, we are talking about the two column proofs that we learned while we were in school. But do you remember exactly why or what they were for?
Basically, a proof is an argument that begins with a known fact or a “Given.” From there, logical deductions are made through a series of conclusions based on facts, theorems and axioms. This will finally prove the proposition at hand, for example, the sum of the angle measures in a triangle equals 180˚. By writing out a proof, the answer is undeniable. Why Are They So Important?
Well, logical reasoning and deduction are central to understanding not only geometry, but mathematics as a whole. Being able to tell the difference between obvious mathematical concepts and ones that need to be justified is a new level of understanding in math. It shows comprehension of deductive logic and the ability to structure arguments to make mathematical conclusions. All of these skills are paramount to reaching a more mature and complete knowledge of geometry and arithmetic.
As powerful as our brains are, they can miss key facts and be fooled. There are times where things seem perfectly reasonable and they turn out to be wrong. That’s why we need to learn how to PROVE things. When you go through step by step, with the deductions laid out, you know what you’ve done is absolutely correct. When mathematicians first began to form rules to prove valid mathematical statements, they did so through trial and error. This allowed congruence in learning. One person could show another person a mathematical rule and prove it through reproduction, which in turn made it valid. However, proofs aren’t just ways to show that statements are true or valid. They help to confirm a student’s true understanding of axioms, rules, theorems, givens and hypotheses. And they confirm how and why geometry helps explain our world and how it works. What Harm Does It Do When Proofs Are Removed?
Limiting the amount of substantial and challenging proofs in a geometry curriculum pretty much defeats the purpose of the course. Now, that may sound a little exacting, but it is true. The reality is that geometry is different than other math courses.
All mathematics are rooted in problem sets, however the problems in geometry that require proofs of propositions do more than apply a theory. They are a part of it. When students learn how to postulate and prove concepts, they are tapping into a deeper stage of mathematics. Geometrical proofs offer students a clear introduction to logical arguments, which is central to all mathematics. They show the exact relationship between reason and equations. More so, since geometry deals with shapes and figures, it opens the student’s brains to visualizing what must be proven. It rounds out their knowledge, building upon the concepts of basic algebra. Students have to combine all of their acquired knowledge. They have to develop a mental list of steps that will lead from the given to the conclusion. Then, they have to find ways to show algebraically that it all works out while simultaneously following along in a diagram. They must combine two lines of logic to create a new one and flow from one step to another. It can take some deep planning and thinking for a more challenging proof. It’s a whole new way of thinking that develops entire new brain connections for them! Benefits Beyond the Classroom
Reasoning is a skill that has a multitude of applications. Whether you’re proving a geometric postulate, working through a detailed word problem, navigating facts in a debate, or even making a monthly budget, you will need reasoning. While we do learn reasoning outside of geometry, students that practice proofs strengthen that skill even more. You learn how to reason carefully and find links between facts. This is something that is important for everyone, not just mathematicians.
Basically, proofs do have a very important role in the geometry classroom. They offer:
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43 Comments
Michael Paul Goldenberg
3/4/2017 12:16:56 pm
1) Why proofs in geometry in K12 but not in virtually any other topic?
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3/4/2017 03:31:06 pm
Hi Michael,
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Lindsay
11/17/2017 01:18:41 am
As an geometry and AP calculus teacher, students absolutely use proofs in other courses besides Geometry  calculus students are expected to grapple with logical reasoning and justification everyday.
Big Lenny Jr
9/3/2018 05:17:39 pm
Stop with that cookie cutter conformist BS. Teach them proofs like a true freak would do!
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Joseph N Rushton
2/16/2019 10:48:32 pm
1) The reason proofs (as well as definitions and axioms) are emphasized geometry is historical rather than logical: it is because Euclid's _Elements_, which had a rigorous axiomdefinitionproof format, served as the standard geometry textbook in the Western world from the time of its writing through the 19'th century.
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John P.
10/8/2020 08:59:19 pm
I majored in business. My late mother was a high school math teacher before I was born. She had a BSE and an MS in Mathematics by in the 1960s at age 19! She really worked hard in her classes back when we did not have pocket calculators. Slide rules were the norm, and she never learned how to use the slide rule. She later returned to teaching when I was in high school. When I reached college, she got the opportunity to teach collegelevel math. She even went back and earned a PhD in Math Education. Cancer stuck her rather young. She died a few years after I finished college.
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Laura
10/9/2020 11:55:34 am
I have noticed with my own kids that teachers' (a) comfort level with math and (b) emphasis on form and process are inversely related. The ones who know less about math tend to overemphasize things like proofs. I absolutely think proofs should be thoroughly taught, but not to the degree you describe. They are a necessary first step in teaching the unnatural sort of abstract thinking that higher mathematics (and logic in general) requires. But they are only the first step. Unfortunately, they are easy to grade and a convenient crutch for someone who doesn't know or care about the endgame.
Karen Gold
3/5/2017 12:13:38 pm
I taught high school (not math) for 25 years. Now I'm retired and I teach basic math and reading to adults. About 25 years ago a girl in my homeroom came in and told me, gloomily, that she had a geometry test that day. I asked her what the problem was, since she was doing great in geometry, and she said, "But this test doesn't have any proofs!"
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3/5/2017 09:31:44 pm
That is awesome! :)
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3/2/2018 07:19:08 am
Awesome write up. You explained why we need proofs beautifully. I use this argument with my students as well. I always tell them to focus on the logical reasoning part of it. Once the kids get the hang of it they love it as well.
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3/2/2018 12:05:05 pm
Thanks so much, Vasudha!
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Mickey
10/13/2018 04:25:45 pm
Sorry, I just can't buy into this. I'll concede that one will need this to be an engineer or an architect, but I absolutely hated this stuff in high school. Frankly, knowing that alternate interior angles are equal never, I repeat...NEVER helped me ANYWHERE in real life. For me, mathematical proofs were worthless whether I worked in a supermarket, did taxes, or took the bar exam, or anything in between. And my high school's "learn it or else" only added to the worthlessness.
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Laura
10/16/2018 10:29:38 am
I presume you are a lawyer? I have found that logical reasoning (being able to parse something out and evaluate each point) helps enormously in negotiating and in trying to convince people to do/believe something without them figuring out my game. It also helps with not getting tricked or sidetracked myself. Aside from mathematical or career benefits, geometric proofs are a vehicle for learning the reasoning process, which is applicable in many fields, whether you are designing bridges and roof trusses, calculating astronomical trajectories, analyzing statistical claims, constructing a usable website, auditing corporate finances, or deciding whether you should buy alkaline water and refuse to vaccinate your kids. We do something similar with literature. Nobody needs to analyze the symbolism of the white whale, and most people hate doing it. We make kids do it anyway so they can practice grammar, argumentation and expressing themselves in a coherent manner. You may have hated doing proofs. (So did I, BTW. Trigonometry was like a breath of fresh air!) But I guarantee that unless you bombed the class you learned something worthwhile that has enabled you to be successful in things like law school. You could not have learned it any other way, IMO. Anything else, including logic classes and noncalculusbased sciences, would have been only memorization and not deep understanding.
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Mickey
10/21/2018 09:42:48 pm
Laura: With all due respect, I cannot disagree more. There is absolutely nothing...repeat...NOTHING that a mathematical proof ever did for me in everyday life, logically or otherwise. Recalling that vertical angles are equal somehow didn't help me figure out the marital deduction on my Estate Planning final exam while I was in law school. Nor did knowing the area of a triangle ever come up in any job interview I never had. Admittedly, if someone had ever taken the time to explain the real life relevance, I could at least see the point of it all. But to have it shoved down my throat and told to just "LEARN IT OR ELSE." is forever unconscionable. Now that it's been almost 40 years since my escape from high school, the fact that proving two triangles are "Side Angle Side" congruent is just worthless to me...now as then.
Jalal
5/26/2021 09:04:40 am
@ Mickey 
Alex
11/11/2019 02:57:58 am
“Frankly, knowing that alternate interior angles are equal never, I repeat...NEVER helped me ANYWHERE in real life. For me, mathematical proofs were worthless whether I worked in a supermarket, did taxes, or took the bar exam, or anything in between. And my high school's "learn it or else" only added to the worthlessness.”
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Mickey
11/14/2019 10:52:35 am
Hi Alex.
Jenny
2/6/2019 12:44:17 am
I guess this was posted a while ago, but I only just saw it.
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2/6/2019 02:56:12 pm
Hi Jenny,
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Jenny
2/6/2019 05:26:34 pm
My daughter is a freshman, so it is hard to say what she will study, but for years she has wanted to be a bryologist. On the other hand, this year she discovered engineering class. So who know? I suspect there will be a lot of math, but it doesn't seem likely to be her major.
Mickey
2/6/2019 08:49:05 pm
Hi Math Giraffe.
Mickey
2/6/2019 08:52:33 pm
Hi Jenny.
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Jenny
2/7/2019 01:28:31 am
Thanks for the kind words, Mickey. I get what you mean about not seeing the point of proofs. I mentioned a couple of things that I certainly didn’t see the point of, and I still feel that way. I like mathematician Paul Lockhat’s explanation for the purpose of math:
Pearl
10/13/2019 01:23:31 am
I am designing a lesson on quadrilaterals and triangles for 9th graders in India. Most of the questions are proofs and a lot of proofs require students to construct (a line segment, extend a side etc) to solve the problem. How can I help students identify the need for a construction and the kind of construction? Any tips will be deeply appreciated.
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10/15/2019 11:20:53 am
Hi Pearl,
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Bean
9/23/2020 11:20:55 am
Hi, I am in HS right now and we are "learning" geometric proofs, and right now I can not be more frustrated at this article. I can not see why we are made to learn this, in any job that uses geometry, architecture and fabrication for example. If it is for logic and reasoning then it should be in a english class, it would make later english classes more useful instead of writing multiple page essays and trying to figure out what a quote from a book means, and then forcing one persons take of it down our throats.
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9/23/2020 01:33:49 pm
Hi,
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Michael P Goldenberg
9/23/2020 02:42:59 pm
I suspect that this discussion would work better if people all recognized that geometry and algebra are two major flavor groups in high school (and higher) mathematics and that math education researchers probably would tell you that a LOT of people preferred one or the other in school. Assuming they haven't become professional research mathematicians, it's not terribly likely that they've changed their preferences over time.
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Jarom
7/7/2021 08:32:56 pm
This. Oh my gosh, this. Thank you. I hate that we have somehow gotten the idea that at least until college, geometry is the only branch of mathematics that uses proof. Mathematics, as an axiomatic system, relies on proof to avoid reinventing the wheel for every problem. Many don't realize that all of the formulae we use in mathematics (e.g., the quadratic formula) have all been incontrovertibly proven to yield consistent solutions when properly applied. Otherwise we wouldn't teach them.
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an anonymous geometry student
9/29/2020 05:26:04 pm
I'm a geometry student and its kind of insulting when you say that proofs teach us how to reason. I know how to reason and I'm pretty sure that everyone else in my class does too. Teaching us to memorize theorems doesn't help us to understand math. instead, it gives us shortcuts so that we don't have to understand it. This works in the short term, but in the long term this will make us hate math because if we don't understand it in geometry we're not going to understand it in the future either. Instead of teaching proofs we should teach kids to try to find the answer before teaching it to them. This helps them to find their own way of doing it that they understand more. Of coarse, they shouldn't be expected to be able to find out how to do it. Although, they should be given the opportunity to look for patterns that lead to the answer. I have done this in almost every math lesson in stead of listening to the teacher and have been given the math award for the advanced math class 3 years out of the 3 years i could have received one. I've also come up with a new formula for almost every math lesson. One of the most useful is when i found out that when looking for a missing endpoint and you have an endpoint (e) and a midpoint (m) you can use my formula 2me for the x coordinates and the y coordinates individually or you can use an extremely long formula using the Pythagorean theorem.
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an anonymous geometry student
9/29/2020 06:52:48 pm
No matter how long I look for people that agree with me, I still have to go back to lesson 32 in geometry and write my proof about angles made by a transversal. :(
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Bailey
10/2/2020 09:34:08 am
I think that proofs are important because you need to prove that things and equations can be true. Without proofs we would never know if a lot of things are true.
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an anonymous math student
10/2/2020 03:36:07 pm
There are other ways to prove things than to do a two column proofs and it seems like more of an English lesson than a math lesson to teach how to prove things.
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10/26/2020 09:23:12 pm
I completely agree with this. How the heck does this crap teach us “logical reasoning”. My sense of logical reasoning tells me that we shouldn’t have to waste our time by explaining why we need “reflexive property” to solve certain equations. It seems like Math Giraffe just doesn’t want to hear us though, or just doesn’t care.
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10/26/2020 09:24:30 pm
I meant that I agree with “anonymous Math Student”
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10/27/2020 10:53:04 am
Some Geometry teachers do not go as deep the logic and deductive reasoning skills. Sounds like you may be missing some of that, and it's not feeling like much logical reasoning. That can be frustrating, and I can see how it can feel like a waste of time when the math is disconnected from the logic. I would definitely encourage you to explore some more logic. Check out symbolic logic. You may like that better since you can see more clearly why it's so important to know, and how it applies to real life. (less silly and pointless feeling than the old Reflexive Property)... If p implies q, and q implies r, then p implies r... etc. That stuff is really interesting, and then you can look into logical fallacies. Just pull out a few news articles and hunt for those logical fallacies! :) You'll spot a lot right away once you get familiar with them. Two column proofs are just a structure that helps some students to get started with this, but don't give up! Move on and dive deeper into some more applicable logic if math is not your favorite path for it. Reason is still critical! :) Good luck.
Herbert
12/7/2022 01:56:11 am
The fact you call it ‘crap’ tells me you were very poorly taught at school. It’s a shame to see people later on in life reject certain parts of their education because of their experiences as a child. As an example I was taught history badly, why were we learning this ‘crap’, who cares what the romans did? Later in life I revisited history and saw something different. In fact the collapse of the Roman Empire turns out to have so many lessons that apply today. There are a lot of things we learn in school that appear to have no practical value, geography, history, mathematics, science (the vast majority of people won’t become scientists so why learn it?), music, Shakespeare, sports (I don’t care about sports at all, yet I was forced to take part in it), the list goes on. Part of the reason for teaching this range of subjects is to create, hopefully an informed and educated society. Republics and democracies tend not to survive long when the population is uneducated.
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7/5/2021 01:21:13 pm
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Joel Campbell
2/3/2022 12:53:39 pm
The big problem with this is it’s a required course and many kids that age haven’t yet developed the ability to think abstractly. As a result it’s nothing but pure torture for many of these kids. Even at the university level the only people concerned with this are pure mathematicians  not scientists and engineers, business majors, etc. So why subject kids to something like this when most of them will neither need more want it? If this is such an important topic offer an analysis class as an elective for those who are interested in it and leave everybody else alone.
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2/14/2022 09:48:47 am
Hi Joel,
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Mickey
10/14/2022 07:16:07 pm
Hi Math Giraffe.
Mickey
10/14/2022 07:02:07 pm
Jalal: The problem, as I saw it in high school is that no one ever took the time to explain WHY the process is the way it is. Alternate interior angles are equal? Vertical angles are equal? Great. Why is this relevant??? Why do I need to know this??? I believe that if teachers took the time to really connect the dots and show WHY something is relevant to the process instead of just showing how to it, maybe so much niformation would not go to waste.
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Herbert
12/7/2022 01:27:31 am
I’ve been rediscovering Euclid and I’m loving it. I like the idea of deducing ‘truths’ based on reason and a set of elementary definitions and axioms. I think geometry is great for learning this mode of thinking because it’s so visual and for those more practically mined it has immediate practical applications. Whether one needs to use the two column format is another matter. I think there is some truth however to say that learning how to reason using say geometry, does seep into one’s subconscious so that there is a tendency to start reasoning automatically in other areas of your life. As for learning things in school that one will never need, imagine we only taught basic arithmetic, basic reading skills and perhaps some elementary civics. Imagine what our society would be like within a generation or two?
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