definition for hypothesis in geometry

A statement that could be true, which might then be tested.

Example: Sam has a hypothesis that "large dogs are better at catching tennis balls than small dogs". We can test that hypothesis by having hundreds of different sized dogs try to catch tennis balls.

Sometimes the hypothesis won't be tested, it is simply a good explanation (which could be wrong). Conjecture is a better word for this.

Example: you notice the temperature drops just as the sun rises. Your hypothesis is that the sun warms the air high above you, which rises up and then cooler air comes from the sides.

Note: when someone says "I have a theory" they should say "I have a hypothesis", because in mathematics a theory is actually well proven.

Mathematical Mysteries

Revealing the mysteries of mathematics

Axiom, Corollary, Lemma, Postulate, Conjectures and Theorems

“Lions and tigers, and bears, oh my!” ~ Dorothy in Wizard of Oz

Or should we say axioms, corollaries, lemmas, postulates, conjectures and theorems, oh my!

There are certain elementary statements, which are self evident and which are accepted without any questions. These are called  axioms.

Axiom 1: Things which are equal to the same thing are equal to one another.

For example:

Draw a line segment AB of length 10cm. Draw a second line CD having length equal to that of AB, using a compass. Measure the length of CD. We see that, CD = 10cm.

We can write it as, CD = AB and AB = 10cm implies CD = 10cm.

Arif, View. 2016. “Axioms, Postulates And Theorems – Class VIII”.  Breath Math . https://breathmath.com/2016/02/18/axioms-postulates-and-theorems-class-viii/ .

A statement that is taken to be true, so that further reasoning can be done.

It is not something we want to prove.

Example: one of Euclid’s axioms (over 2300 years ago!) is: “If A and B are two numbers that are the same, and C and D are also the same, A+C is the same as B+D”

“Definition Of Axiom”. 2021.  mathsisfun.Com . https://www.mathsisfun.com/definitions/axiom.html .

In mathematics an axiom is something which is the starting point for the logical deduction of other theorems. They cannot be proven with a logic derivation unless they are redundant. That means every field in mathematics can be boiled down to a set of axioms. One of the axioms of arithmetic is that a + b = b + a. You can’t prove that, but it is the basis of arithmetic and something we use rather often.

“Theorems, Lemmas And Other Definitions | Mathblog”. 2011.  mathblog.dk . https://www.mathblog.dk/theorems-lemmas/ .

In math it is known that you can’t prove everything. So, in order to lay a ground work for proving things, there is a list of things we “take for granted as true”. These things are either very basic definitions such as “point” “line”, or facts assumed to be true without proof that are very very simple. Then with these an accepted rules, one can prove other statements are true. The assumed facts are called “axioms” or sometimes “postulates”. The most famous are five postulates/axioms that Euclid’s geometry takes for granted. There are the following:

• A straight  line segment  can be drawn joining any two points.
• Any straight  line segment  can be extended indefinitely in a straight  line .
• Given any straight  line segment , a  circle  can be drawn having the segment as  radius  and one endpoint as center.
• All  right angles  are  congruent .
• If two lines are drawn which  intersect  a third in such a way that the sum of the inner angles on one side is less than two  right angles , then the two lines inevitably must  intersect  each other on that side if extended far enough. This postulate is equivalent to what is known as the  parallel postulate .

The fifth postulate is perhaps the most “famous” as it is complex and people wanted to prove it from the first four, but couldn’t, and then it was discovered that there were systems in which the first four were true but the fifth wasn’t. These are called “non-Euclidean” geometries. Of course, here we take for granted what a point, line segment, line, circle, angle, and radius are at least as well.

Farris, Steven. “I don’t understand the concept of an axiom in mathematics. What is an axiom? How would you introduce or explain this concept to a 10-year-old?”. 2023.  Quora . https://qr.ae/pyVTM1 .

An  axiom  is just any concept or statement that we take as being true, without any need for a formal proof. It is usually something very fundamental to a given field, very well-established and/or self-evident. A non-mathematical example might be a simple statement of an observed truth, such as “the Sun rises in the East.” In math, such things as “a line can be extended to infinity” or “a point has no size” might be good examples. An axiom differs from a  postulate  in that an axiom is typically more general and common, while a postulate may apply only to a specific field. For instance, the difference between Euclidean and non-Euclidean geometries are just changes to one or more of the postulates on which they’re based. Another way to look at this is that a postulate is something we assume to be true only within that specific field.

Myers, Bob. “I don’t understand the concept of an axiom in mathematics. What is an axiom? How would you introduce or explain this concept to a 10-year-old?”. 2023.  Quora . https://qr.ae/pyVTwW .

It’s not so much that they don’t  require  proof, it’s that they can’t be proven. Axioms are  starting assumptions .

Everything that is proven is based on axioms, theorems, or definitions. You can’t prove an axiom without already having something to base your proof on, because deductive reasoning always needs a starting place. You have to start with good assumptions, and hope they’re true, or at least useful in the type of math you wish to create. (Don’t forget that math is just a human construct!)

That doesn’t mean that axioms come out of thin air. Some axioms are developed because if they don’t exist, the math doesn’t model the way we want it to. If you put 3 apples in your grocery cart, then put 4 more in, you have 7. But it works the same if you put 3 in, then 4. Now you have the commutative property of addition. You can’t  prove  addition works this way, but you need to set it up so that it does.

Often axioms are demonstrable. Try to draw two non-congruent triangles with sides of length 3, 4, and 5 units. You can’t. But you haven’t  proved  it using deductive reasoning. You’ve made a conjecture using inductive reasoning.

McClung, Carter. “Why don’t axioms require proofs?”. 2023.  Quora . https://qr.ae/pyVTO4 .

The axioms or postulates are the assumptions that are obvious universal truths, they are not proved. Euclid has introduced the geometry fundamentals like geometric shapes and figures in his book elements and has stated 5 main axioms or postulates. Here, we are going to discuss the definition of euclidean geometry, its elements, axioms and five important postulates. [4]

A theorem that  follows on  from another theorem.

Example: there is a  Theorem  that says: two angles that together form a straight line are “supplementary” (they add to 180°).

A  Corollary  to this is the “Vertical Angle Theorem” that says: where two lines intersect, the angles opposite each other are equal (a=c and b=d in the diagram).

Proof that a=c: Angles a and b are on a straight line, so: ⇒ angles a + b = 180° and so a = 180° − b Angles c and b are also on a straight line, so: ⇒ angles c + b = 180° and so c = 180° − b So angle a = angle c

“Corollary Definition (Illustrated Mathematics Dictionary)”. 2021.  mathsisfun.com . https://www.mathsisfun.com/definitions/corollary.html .

A corollary of a theorem or a definition is a statement that can be deduced directly from that theorem or statement. It still needs to be proved, though.

A simple example: Theorem: The sum of the angles of a triangle is pi radians.

Corollary: No angle in a right angled triangle can be obtuse.

Or: Definition: A prime number is one that can be divided without remainder only by 1 and itself.

Corollary: No even number > 2 can be prime.

A corollary is a theorem that can be proved from another theorem. For example: If two angles of a triangle are equal, then the sides opposite them are equal . A corollary would be: If a triangle is equilateral, it is also equiangular.

“What Are The Examples Of Corollary In Math? – Quora”. 2021.  quora.com . https://www.quora.com/What-are-the-examples-of-corollary-in-math .

Lemmas and corollaries are theorems themselves. It’s really not necessary to have different names for them. A corollary is a theorem that “easily” follows from the preceding theorem. For example, after proving the theorem that the sum of the angles in a triangle is 180°, an easy theorem to prove is that the sum of the angles in a quadrilateral is 360°. The proof is just to cut the quadrilateral into two triangles. So that theorem could be called a corollary. [2]

There is not formal difference between a theorem and a lemma.  A lemma is a proven proposition just like a theorem. Usually a lemma is used as a stepping stone for proving something larger. That means the convention is to call the main statement for a theorem and then split the problem into several smaller problems which are stated as lemmas.  Wolfram  suggest that a lemma  is a short theorem used to prove something larger.

Breaking part of the main proof out into lemmas is a good way to create a structure in a proof and sometimes their importance will prove more valuable than the main theorem.

Like a Theorem, but not as important. It is a minor result that has been proved to be true (using facts that were already known). [3]

Lemmas and corollaries are theorems themselves. It’s really not necessary to have different names for them. A lemma is a theorem that’s mentioned primarily because it’s used in one or more following theorems, but it’s not so interesting in itself. Sometimes lemmas are just minor observations, but sometimes they’ve got detailed proofs. [2]

Postulates  in geometry are very similar to axioms, self-evident truths, and beliefs in logic, political philosophy and personal decision-making.

Geometry postulates, or axioms, are accepted statements or facts. Thus, there is no need to prove them.

Postulate 1.1, Through two points, there is exactly 1 line. Line t is the only line passing through E and F.

In geometry, “ Axiom ” and “ Postulate ” are essentially interchangeable. In antiquity, they referred to propositions that were “obviously true” and only had to be stated, and not proven. In modern mathematics there is no longer an assumption that axioms are “obviously true”. Axioms are merely ‘background’ assumptions we make. The best analogy I know is that axioms are the “rules of the game”. In Euclid’s Geometry, the main axioms/postulates are:

• Given any two distinct points, there is a line that contains them.
• Any line segment can be extended to an infinite line.
• Given a point and a radius, there is a circle with center in that point and that radius.
• All right angles are equal to one another.
• If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (The  parallel postulate ).

A  theorem  is a logical consequence of the axioms. In Geometry, the “propositions” are all theorems: they are derived using the axioms and the valid rules. A “Corollary” is a theorem that is usually considered an “easy consequence” of another theorem. What is or is not a corollary is entirely subjective. Sometimes what an author thinks is a ‘corollary’ is deemed more important than the corresponding theorem. (The same goes for “ Lemma “s, which are theorems that are considered auxiliary to proving some other, more important in the view of the author, theorem).

A “ hypothesis ” is an assumption made. For example, “If xx is an even integer, then x2x2 is an even integer” I am not asserting that x2x2 is even or odd; I am asserting that if  something  happens (namely, if xx happens to be an even integer) then  something else  will also happen. Here, “xx is an even integer” is the hypothesis being made to prove it.

Gordon Gustafson, and Arturo Magidin. 2010. “Difference Between Axioms, Theorems, Postulates, Corollaries, And Hypotheses”.  Mathematics Stack Exchange . https://math.stackexchange.com/questions/7717/difference-between-axioms-theorems-postulates-corollaries-and-hypotheses .

In geometry, a postulate is a statement that is assumed to be true based on basic geometric principles. An example of a postulate is the statement “exactly one line may be drawn through any two points.” A long time ago, postulates were the ideas that were thought to be so obviously true they did not require a proof. [1]

An axiom is a statement, usually considered to be self-evident, that assumed to be true without proof. It is used as a starting point in mathematical proof for deducing other truths.

Classically, axioms were considered different from postulates. An axiom would refer to a self-evident assumption common to many areas of inquiry, while a postulate referred to a hypothesis specific to a certain line of inquiry, that was accepted without proof. As an example, in Euclid’s Elements, you can compare “common notions” (axioms) with postulates.

In much of modern mathematics, however, there is generally no difference between what were classically referred to as “axioms” and “postulates”. Modern mathematics distinguishes between logical axioms and non-logical axioms, with the latter sometimes being referred to as postulates.

Postulates are assumptions which are specific to geometry but axioms are assumptions are used thru’ out mathematics and not specific to geometry.

“What is the difference between an axiom and postulates”. 2023.  BYJUs . https://byjus.com/question-answer/what-is-the-difference-between-an-axiom-and-postulates/ .

Hint: First you need to define both the terms, axiom and postulates. Examples of both can be stated. The main difference is between their application in specific fields in mathematics.

An axiom is a statement or proposition which is regarded as being established, accepted, or self-evidently true on which an abstractly defined structure is based. More precisely an axiom is a statement that is self-evident without any proof which is a starting point for further reasoning and arguments.

Postulate verbally means a fact, or truth of (something) as a basis for reasoning, discussion, or belief. Postulates are the basic structure from which lemmas and theorems are derived.

Nowadays ‘axiom’ and ‘postulate’ are usually interchangeable terms. One key difference between them is that postulates are true assumptions that are specific to geometry. Axioms are true assumptions used throughout mathematics and not specifically linked to geometry.

“What is the difference between an axiom and a postulate?”. 2023. Vedantu . https://www.vedantu.com/question-answer/difference-between-an-axiom-and-a-post-class-10-maths-cbse-5efeafa98c08f1791a1cc34a .

A  conjecture  is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many  cases . However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases. Conjectures must be proved for the mathematical observation to be fully accepted. When a conjecture is rigorously proved, it becomes a theorem.

“Conjectures | Brilliant Math & Science Wiki”. 2022.  brilliant.org . https://brilliant.org/wiki/conjectures/ .

“The Subtle Art Of The Mathematical Conjecture | Quanta Magazine”. 2019.  Quanta Magazine . https://www.quantamagazine.org/the-subtle-art-of-the-mathematical-conjecture-20190507/ .

A result that has been  proved to be true  (using operations and facts that were already known).

Example: The “Pythagoras Theorem” proved that a 2  + b 2  = c 2  for a right angled triangle.

A Theorem is a major result, a minor result is called a Lemma.

“Theorem Definition (Illustrated Mathematics Dictionary)”. 2021.  mathsisfun.Com . https://www.mathsisfun.com/definitions/theorem.html .

“Theorems, Corollaries, Lemmas”. 2021.  mathsisfun.com . https://www.mathsisfun.com/algebra/theorems-lemmas.html .

A statement that is proven true using postulates, definitions, and previously proven theorems.

A theorem is a mathematical statement that can and must be proven to be true. You may have been first exposed to the term when learning about the Pythagorean Theorem . Learning different theorems and proving they are true is an important part of Geometry. [1]

[1] “4.1 Theorems and Proofs”. 2022. CK-12 Foundation . https://flexbooks.ck12.org/cbook/ck-12-interactive-geometry-for-ccss/section/4.1/primary/lesson/theorems-and-proofs-geo-ccss/ .

[2] Joyce, David . “Can a theorem be proved by a corollary?”. 2023.  Quora . https://qr.ae/pybAMq .

Yes, a theorem can be proved by a corollary just so long as the corollary is proved first. You might have a sequence of theorems in logical order like this: Theorem 1, Corollary 2, Lemma 3, Theorem 4, Theorem 5. Each one is proved from those that precede it, but Theorem 5 could depend only on Corollary 2 and Lemma 3. Sometimes theorems are presented in a different order than the logical order, and sometimes even in reverse logical order, but whatever order they’re presented, it is necessary that there is no circular logic.

[3] “Definition Of Lemma”. 2021. mathsisfun.com . https://www.mathsisfun.com/definitions/lemma.html .

[4] “Euclidean Geometry (Definition, Facts, Axioms and Postulates)”. 2021. BYJUS . BYJU’S. September 20. https://byjus.com/maths/euclidean-geometry/ .

Additional Reading

“Basic Math Definitions”. 2021.  mathsisfun.com . https://www.mathsisfun.com/basic-math-definitions.html .

Browning, Wes . “Can a theorem be proved by another theorem?”. 2023.  Quora . https://qr.ae/pybAUz .

Sure. Sometimes the second theorem is called a “corollary.” Sometimes the first theorem is called a “lemma” and the second is called a theorem implied by the lemma. Or they’re both called theorems. The choice of names is up to the author of the exposition and is meant to clarify the logical flow. You may occasionally also see the term “ porism ” used. After a theorem has been proved, a porism is another theorem that can be proved by essentially the same proof as the first, usually by obvious modifications. I had a professor in math grad school who loved to trot porisms out after proving a theorem in his classes.

“Byrne’s Euclid”. 2021.  C82.Net . https://www.c82.net/euclid/ .

THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID WITH COLOURED DIAGRAMS AND SYMBOLS A reproduction of Oliver Byrne’s celebrated work from 1847 plus interactive diagrams, cross references, and posters designed by Nicholas Rougeux

“Definitions. Postulates. Axioms: First Principles Of Plane Geometry “. 2021.  themathpage.com . https://themathpage.com/aBookI/first.htm#post .

“Geometry Postulates”. 2021.  basic-mathematics.com . https://www.basic-mathematics.com/geometry-postulates.html .

Mystery, Mike the. 2024. “Is George Orwell Right About 2+2=4 in Maths?”  Medium . Medium. March 12. https://medium.com/@Mike_Meng/is-george-orwell-right-about-2-2-4-in-maths-3bb0f6d5dd88 .

Freedom is the freedom to say that two plus two makes four. ——George Orwell, Nineteen Eighty-Four. When I first read George Orwell’s great “1984”, the above sentence left an indelible impact on me. It is worth mentioning that my first reaction to this quote was why Orwell used 2+2=4 instead of 1+1=2. And that’s exactly the first time I realized I was pedantic enough to get a maths degree in future. Ok, so why 2+2=4 is true? Before directly into the topic, i need to introduce some basic rules that we use to calculate numbers every single day. The rule is actually called Peano axioms, which is a logic system about natural numbers proposed by the 19th-century mathematician Giuseppe Peano . And we can establish an arithmetic system by these sets of axioms, which is also known as the Peano arithmetic system.

“Zermelo-Fraenkel Set Theory (ZFC)”. 2023.  Mathematical Mysteries . https://mathematicalmysteries.org/zermelo-fraenkel-set-theory-zfc/ .

Zermelo–Fraenkel set theory  (abbreviated  ZF ) is a system of  axioms  used to describe  set theory . When the  axiom of choice  is added to ZF, the system is called  ZFC . It is the system of axioms used in set theory by most mathematicians today.

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• Hypothesis | Definition & Meaning

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Explanation of Hypothesis

Contradiction, simple hypothesis, complex hypothesis, null hypothesis, alternative hypothesis, empirical hypothesis, statistical hypothesis, special example of hypothesis, solution part (a), solution part (b), hypothesis|definition & meaning.

A hypothesis is a claim or statement  that makes sense in the context of some information or data at hand but hasn’t been established as true or false through experimentation or proof.

In mathematics, any statement or equation that describes some relationship between certain variables can be termed as hypothesis if it is consistent with some initial supporting data or information, however, its yet   to be proven true or false by some definite and trustworthy experiment or mathematical law. 

Following example illustrates one such hypothesis to shed some light on this very fundamental concept which is often used in different areas of mathematics.

Figure 1: Example of Hypothesis

Here we have considered an example of a young startup company that manufactures state of the art batteries. The hypothesis or the claim of the company is that their batteries have a mean life of more than 1000 hours. Now its very easy to understand that they can prove their claim on some testing experiment in their lab.

However, the statement can only be proven if and only if at least one batch of their production batteries have actually been deployed in the real world for more than 1000 hours . After 1000 hours, data needs to be collected and it needs to be seen what is the probability of this statement being true .

The following paragraphs further explain this concept.

As explained with the help of an example earlier, a hypothesis in mathematics is an untested claim that is backed up by all the known data or some other discoveries or some weak experiments.

In any mathematical discovery, we first start by assuming something or some relationship . This supposed statement is called a supposition. A supposition, however, becomes a hypothesis when it is supported by all available data and a large number of contradictory findings.

The hypothesis is an important part of the scientific method that is widely known today for making new discoveries. The field of mathematics inherited this process. Following figure shows this cycle as a graphic:

Figure 2: Role of Hypothesis in the Scientific Method 

The above figure shows a simplified version of the scientific method. It shows that whenever a supposition is supported by some data, its termed as hypothesis. Once a hypothesis is proven by some well known and widely acceptable experiment or proof, its becomes a law. If the hypothesis is rejected by some contradictory results then the supposition is changed and the cycle continues.

Lets try to understand the scientific method and the hypothesis concept with the help of an example. Lets say that a teacher wanted to analyze the relationship between the students performance in a certain subject, lets call it A, based on whether or not they studied a minor course, lets call it B.

Now the teacher puts forth a supposition that the students taking the course B prior to course A must perform better in the latter due to the obvious linkages in the key concepts. Due to this linkage, this supposition can be termed as a hypothesis.

However to test the hypothesis, the teacher has to collect data from all of his/her students such that he/she knows which students have taken course B and which ones haven’t. Then at the end of the semester, the performance of the students must be measured and compared with their course B enrollments.

If the students that took course B prior to course A perform better, then the hypothesis concludes successful . Otherwise, the supposition may need revision.

The following figure explains this problem graphically.

Figure 3: Teacher and Course Example of Hypothesis

Important Terms Related to Hypothesis

To further elaborate the concept of hypothesis, we first need to understand a few key terms that are widely used in this area such as conjecture, contradiction and some special types of hypothesis (simple, complex, null, alternative, empirical, statistical). These terms are briefly explained below:

A conjecture is a term used to describe a mathematical assertion that has notbeenproved. While testing   may occasionally turn up millions of examples in favour of a conjecture, most experts in the area will typically only accept a proof . In mathematics, this term is synonymous to the term hypothesis.

In mathematics, a contradiction occurs if the results of an experiment or proof are against some hypothesis.  In other words, a contradiction discredits a hypothesis.

A simple hypothesis is such a type of hypothesis that claims there is a correlation between two variables. The first is known as a dependent variable while the second is known as an independent variable.

A complex hypothesis is such a type of hypothesis that claims there is a correlation between more than two variables.  Both the dependent and independent variables in this hypothesis may be more than one in numbers.

A null hypothesis, usually denoted by H0, is such a type of hypothesis that claims there is no statistical relationship and significance between two sets of observed data and measured occurrences for each set of defined, single observable variables. In short the variables are independent.

An alternative hypothesis, usually denoted by H1 or Ha, is such a type of hypothesis where the variables may be statistically influenced by some unknown factors or variables. In short the variables are dependent on some unknown phenomena .

An Empirical hypothesis is such a type of hypothesis that is built on top of some empirical data or experiment or formulation.

A statistical hypothesis is such a type of hypothesis that is built on top of some statistical data or experiment or formulation. It may be logical or illogical in nature.

According to the Riemann hypothesis, only negative even integers and complex numbers with real part 1/2 have zeros in the Riemann zeta function . It is regarded by many as the most significant open issue in pure mathematics.

Figure 4: Riemann Hypothesis

The Riemann hypothesis is the most well-known mathematical conjecture, and it has been the subject of innumerable proof efforts.

Numerical Examples

Identify the conclusions and hypothesis in the following given statements. Also state if the conclusion supports the hypothesis or not.

Part (a): If 30x = 30, then x = 1

Part (b): if 10x + 2 = 50, then x = 24

Hypothesis: 30x = 30

Conclusion: x = 10

Supports Hypothesis: Yes

Hypothesis: 10x + 2 = 50

Conclusion: x = 24

All images/mathematical drawings were created with GeoGebra.

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Definition Of Hypothesis

Hypothesis is the part of a conditional statement just after the word if.

Examples of Hypothesis

In the conditional, "If all fours sides of a quadrilateral measure the same, then the quadrilateral is a square" the hypothesis is "all fours sides of a quadrilateral measure the same".

Video Examples: Hypothesis  

Solved Example on Hypothesis

Ques:  in the example above, is the hypothesis "all fours sides of a quadrilateral measure the same" always, never, or sometimes true.

A. always B. never C. sometimes Correct Answer: C

Step 1: The hypothesis is sometimes true. Because, its true only for a square and a rhombus, not for the other quadrilaterals rectangle, parallelogram, or trapezoid.

Related Worksheet

• Identifying-and-Describing-Right-Triangles-Gr-4
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Professor: Erika L.C. King Email: [email protected] Office: Lansing 304 Phone: (315)781-3355

The majority of statements in mathematics can be written in the form: "If A, then B." For example: "If a function is differentiable, then it is continuous". In this example, the "A" part is "a function is differentiable" and the "B" part is "a function is continuous." The "A" part of the statement is called the "hypothesis", and the "B" part of the statement is called the "conclusion". Thus the hypothesis is what we must assume in order to be positive that the conclusion will hold.

Whenever you are asked to state a theorem, be sure to include the hypothesis. In order to know when you may apply the theorem, you need to know what constraints you have. So in the example above, if we know that a function is differentiable, we may assume that it is continuous. However, if we do not know that a function is differentiable, continuity may not hold. Some theorems have MANY hypotheses, some of which are written in sentences before the ultimate "if, then" statement. For example, there might be a sentence that says: "Assume n is even." which is then followed by an if,then statement. Include all hypotheses and assumptions when asked to state theorems and definitions!

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Terminology: Difference between Lemma, Theorem, Definition, Hypothesis, Postulate and a Proposition [duplicate]

Based on observation after reading few books and papers, I think that

Lemma : Lemma contains some information that is commonly used to support a theorem. So, a Lemma introduces a Theorem and comes before that Theorem. The information contained in the Lemma is generally used in the proof of the Theorem.

Q1: What is a Proposition ?

Q2: What is the difference between Proposition and Theorem ? A Proposition can also be proved, in the same way as a Theorem is proven.

Hypothesis : A hypothesis is like a statement for a guess, and we need to prove that analytically or experimentally.

Q3: What is the difference between Theorem and Hypothesis , for example Null hypothesis in statistics? In general, if a Theorem is always proven to be true then it no longer becomes a Hypothesis? Am I correct?

Q4: What is the difference between Postulate and Theorem ?

Q5 : What is the difference between a proposition and a definition ?

Looking for easy to remember answers. Thank you for help.

• terminology

• 1 $\begingroup$ A definition is just a 'naming' of an object and not like a theorem at all... a proposition is a theorem... usually an 'easy theorem' setting out some basic facts. $\endgroup$ –  JP McCarthy Commented Apr 22, 2015 at 17:38

I'm not the authority on this, but this is how I interpret all of these words in math literature:

Definition - This is an assignment of language and syntax to some property of a set, function, or other object. A definition is not something you prove, it is something someone assigns. Often you will want to prove that something satisfies a definition. Example: We call a mapping $f:X\to Y$ injective if whenever $f(x) = f(y)$ then $x=y$.

Proposition - This is a property that one can derive easily or directly from a given definition of an object. Example: the identity element in a group is unique.

Lemma - This is a property that one can derive or prove which is usually technical in nature and is not of primary importance to the overall body of knowledge one is trying to develop. Usually lemmas are there as precursors to larger results that one wants to obtain, or introduce a new technique or tool that one can use over and over again. Example: In a Hausdorff space, compact subsets can be separated by disjoint open subsets.

Theorem - This is a property of major importance that one can derive which usually has far-sweeping consequences for the area of math one is studying. Theorems don't necessarily need the support of propositions or lemmas, but they often do require other smaller results to support their evidence. Example: Every manifold has a simply connected covering space.

Corollary - This is usually a result that is a direct consequence of a major theorem. Often times a theorem lends itself to other smaller results or special cases which can be shown by simpler methods once a theorem is proven. Example: A consequence to the Hopf-Rinow theorem is that compact manifolds are geodesically complete.

Conjecture - This is an educated prediction that one makes based on their experience. The difference between a conjecture and a lemma/theorem/corollary is that it is usually an open research problem that either has no answer, or some partial answer. Conjectures are usually only considered important if they are authored by someone well-known in their respective area of mathematics. Once it is proven or disproven, it ceases to be a conjecture and either becomes a fact (backed by a theorem) or there is some interesting counterexample to demonstrate how it is wrong. Example: The Poincar$\acute{\text{e}}$ conjecture was a famous statement that remained an open research problem in topology for roughly a century. The claim was that every simply connected, compact 3-manifold was homeomorphic to the 3-sphere $\mathbb{S}^3$. This statement however is no longer a conjecture since it was famously proven by Grigori Perelman in 2003.

Postulate - I would appreciate community input on this, but I haven't seen this word used in any of the texts/papers I read. I would assume that this is synonymous with proposition.

• 5 $\begingroup$ I know Postulate is a synonym of axiom. Very used word in italian, but more in physics than mathematics. See wikipedia en.wikipedia.org/wiki/Axiom $\endgroup$ –  user3621272 Commented Apr 23, 2015 at 9:24
• $\begingroup$ @Mnifldz and user3621272: Thank you very much for the answers; it is easy to understand and clear. $\endgroup$ –  SKM Commented Apr 23, 2015 at 17:01

Not the answer you're looking for? Browse other questions tagged terminology definition .

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A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false.

In statistics, a hypothesis (sometimes called a statistical hypothesis) refers to a statement on which hypothesis testing will be based. Particularly important statistical hypotheses include the null hypothesis and alternative hypothesis .

In symbolic logic , a hypothesis is the first part of an implication (with the second part being known as the predicate ).

In general mathematical usage, "hypothesis" is roughly synonymous with " conjecture ."

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Angle Properties, Postulates, and Theorems

In order to study geometry in a logical way, it will be important to understand key mathematical properties and to know how to apply useful postulates and theorems. A postulate is a proposition that has not been proven true, but is considered to be true on the basis for mathematical reasoning. Theorems , on the other hand, are statements that have been proven to be true with the use of other theorems or statements. While some postulates and theorems have been introduced in the previous sections, others are new to our study of geometry. We will apply these properties, postulates, and theorems to help drive our mathematical proofs in a very logical, reason-based way.

Before we begin, we must introduce the concept of congruency. Angles are congruent if their measures, in degrees, are equal. Note : “congruent” does not mean “equal.” While they seem quite similar, congruent angles do not have to point in the same direction. The only way to get equal angles is by piling two angles of equal measure on top of each other.

We will utilize the following properties to help us reason through several geometric proofs.

Reflexive Property

A quantity is equal to itself.

Symmetric Property

If A = B , then B = A .

Transitive Property

If A = B and B = C , then A = C .

Addition Property of Equality

If A = B , then A + C = B + C .

Angle Postulates

Angle addition postulate.

If a point lies on the interior of an angle, that angle is the sum of two smaller angles with legs that go through the given point.

Consider the figure below in which point T lies on the interior of ?QRS . By this postulate, we have that ?QRS = ?QRT + ?TRS . We have actually applied this postulate when we practiced finding the complements and supplements of angles in the previous section.

Corresponding Angles Postulate

If a transversal intersects two parallel lines, the pairs of corresponding angles are congruent.

Converse also true : If a transversal intersects two lines and the corresponding angles are congruent, then the lines are parallel.

The figure above yields four pairs of corresponding angles.

Parallel Postulate

Given a line and a point not on that line, there exists a unique line through the point parallel to the given line.

The parallel postulate is what sets Euclidean geometry apart from non-Euclidean geometry .

There are an infinite number of lines that pass through point E , but only the red line runs parallel to line CD . Any other line through E will eventually intersect line CD .

Angle Theorems

Alternate exterior angles theorem.

If a transversal intersects two parallel lines, then the alternate exterior angles are congruent.

Converse also true : If a transversal intersects two lines and the alternate exterior angles are congruent, then the lines are parallel.

The alternate exterior angles have the same degree measures because the lines are parallel to each other.

Alternate Interior Angles Theorem

If a transversal intersects two parallel lines, then the alternate interior angles are congruent.

Converse also true : If a transversal intersects two lines and the alternate interior angles are congruent, then the lines are parallel.

The alternate interior angles have the same degree measures because the lines are parallel to each other.

Congruent Complements Theorem

If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

Right Angles Theorem

All right angles are congruent.

Same-Side Interior Angles Theorem

If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.

Converse also true : If a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

The sum of the degree measures of the same-side interior angles is 180°.

Vertical Angles Theorem

If two angles are vertical angles, then they have equal measures.

The vertical angles have equal degree measures. There are two pairs of vertical angles.

(1) Given: m?DGH = 131

Find: m?GHK

First, we must rely on the information we are given to begin our proof. In this exercise, we note that the measure of ?DGH is 131° .

From the illustration provided, we also see that lines DJ and EK are parallel to each other. Therefore, we can utilize some of the angle theorems above in order to find the measure of ?GHK .

We realize that there exists a relationship between ?DGH and ?EHI : they are corresponding angles. Thus, we can utilize the Corresponding Angles Postulate to determine that ?DGH??EHI .

Directly opposite from ?EHI is ?GHK . Since they are vertical angles, we can use the Vertical Angles Theorem , to see that ?EHI??GHK .

Now, by transitivity , we have that ?DGH??GHK .

Congruent angles have equal degree measures, so the measure of ?DGH is equal to the measure of ?GHK .

Finally, we use substitution to conclude that the measure of ?GHK is 131° . This argument is organized in two-column proof form below.

(2) Given: m?1 = m?3

Prove: m?PTR = m?STQ

We begin our proof with the fact that the measures of ?1 and ?3 are equal.

In our second step, we use the Reflexive Property to show that ?2 is equal to itself.

Though trivial, the previous step was necessary because it set us up to use the Addition Property of Equality by showing that adding the measure of ?2 to two equal angles preserves equality.

Then, by the Angle Addition Postulate we see that ?PTR is the sum of ?1 and ?2 , whereas ?STQ is the sum of ?3 and ?2 .

Ultimately, through substitution , it is clear that the measures of ?PTR and ?STQ are equal. The two-column proof for this exercise is shown below.

(3) Given: m?DCJ = 71 , m?GFJ = 46

Prove: m?AJH = 117

We are given the measure of ?DCJ and ?GFJ to begin the exercise. Also, notice that the three lines that run horizontally in the illustration are parallel to each other. The diagram also shows us that the final steps of our proof may require us to add up the two angles that compose ?AJH .

We find that there exists a relationship between ?DCJ and ?AJI : they are alternate interior angles. Thus, we can use the Alternate Interior Angles Theorem to claim that they are congruent to each other.

By the definition of congruence , their angles have the same measures, so they are equal.

Now, we substitute the measure of ?DCJ with 71 since we were given that quantity. This tells us that ?AJI is also 71° .

Since ?GFJ and ?HJI are also alternate interior angles, we claim congruence between them by the Alternate Interior Angles Theorem .

The definition of congruent angles once again proves that the angles have equal measures. Since we knew the measure of ?GFJ , we just substitute to show that 46 is the degree measure of ?HJI .

As predicted above, we can use the Angle Addition Postulate to get the sum of ?AJI and ?HJI since they compose ?AJH . Ultimately, we see that the sum of these two angles gives us 117° . The two-column proof for this exercise is shown below.

(4) Given: m?1 = 4x + 9 , m?2 = 7(x + 4)

In this exercise, we are not given specific degree measures for the angles shown. Rather, we must use some algebra to help us determine the measure of ?3 . As always, we begin with the information given in the problem. In this case, we are given equations for the measures of ?1 and ?2 . Also, we note that there exists two pairs of parallel lines in the diagram.

By the Same-Side Interior Angles Theorem , we know that that sum of ?1 and ?2 is 180 because they are supplementary.

After substituting these angles by the measures given to us and simplifying, we have 11x + 37 = 180 . In order to solve for x , we first subtract both sides of the equation by 37 , and then divide both sides by 11 .

Once we have determined that the value of x is 13 , we plug it back in to the equation for the measure of ?2 with the intention of eventually using the Corresponding Angles Postulate . Plugging 13 in for x gives us a measure of 119 for ?2 .

Finally, we conclude that ?3 must have this degree measure as well since ?2 and ?3 are congruent . The two-column proof that shows this argument is shown below.

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If-then statement

• Logical correct I
• Logical correct II

When we previously discussed inductive reasoning we based our reasoning on examples and on data from earlier events. If we instead use facts, rules and definitions then it's called deductive reasoning.

We will explain this by using an example.

If you get good grades then you will get into a good college.

The part after the "if": you get good grades - is called a hypotheses and the part after the "then" - you will get into a good college - is called a conclusion.

Hypotheses followed by a conclusion is called an If-then statement or a conditional statement.

This is noted as

$$p \to q$$

This is read - if p then q.

A conditional statement is false if hypothesis is true and the conclusion is false. The example above would be false if it said "if you get good grades then you will not get into a good college".

If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional.

Our conditional statement is: if a population consists of 50% men then 50% of the population must be women.

If we exchange the position of the hypothesis and the conclusion we get a converse statemen t: if a population consists of 50% women then 50% of the population must be men.

$$q\rightarrow p$$

If both statements are true or if both statements are false then the converse is true. A conditional and its converse do not mean the same thing

If we negate both the hypothesis and the conclusion we get a inverse statemen t: if a population do not consist of 50% men then the population do not consist of 50% women.

$$\sim p\rightarrow \: \sim q$$

The inverse is not true juest because the conditional is true. The inverse always has the same truth value as the converse.

We could also negate a converse statement, this is called a contrapositive statemen t:  if a population do not consist of 50% women then the population do not consist of 50% men.

$$\sim q\rightarrow \: \sim p$$

The contrapositive does always have the same truth value as the conditional. If the conditional is true then the contrapositive is true.

A pattern of reaoning is a true assumption if it always lead to a true conclusion. The most common patterns of reasoning are detachment and syllogism.

If we turn of the water in the shower, then the water will stop pouring.

If we call the first part p and the second part q then we know that p results in q. This means that if p is true then q will also be true. This is called the law of detachment and is noted:

$$\left [ (p \to q)\wedge p \right ] \to q$$

The law of syllogism tells us that if p → q and q → r then p → r is also true.

This is noted:

$$\left [ (p \to q)\wedge (q \to r ) \right ] \to (p \to r)$$

If the following statements are true:

If we turn of the water (p), then the water will stop pouring (q). If the water stops pouring (q) then we don't get wet any more (r).

Then the law of syllogism tells us that if we turn of the water (p) then we don't get wet (r) must be true.

Video lesson

Write a converse, inverse and contrapositive to the conditional

"If you eat a whole pint of ice cream, then you won't be hungry"

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Hypothesis Definition

In Statistics, the determination of the variation between the group of data due to true variation is done by hypothesis testing. The sample data are taken from the population parameter based on the assumptions. The hypothesis can be classified into various types. In this article, let us discuss the hypothesis definition, various types of hypothesis and the significance of hypothesis testing, which are explained in detail.

Hypothesis Definition in Statistics

In Statistics, a hypothesis is defined as a formal statement, which gives the explanation about the relationship between the two or more variables of the specified population. It helps the researcher to translate the given problem to a clear explanation for the outcome of the study. It clearly explains and predicts the expected outcome. It indicates the types of experimental design and directs the study of the research process.

Types of Hypothesis

The hypothesis can be broadly classified into different types. They are:

Simple Hypothesis

A simple hypothesis is a hypothesis that there exists a relationship between two variables. One is called a dependent variable, and the other is called an independent variable.

Complex Hypothesis

A complex hypothesis is used when there is a relationship between the existing variables. In this hypothesis, the dependent and independent variables are more than two.

Null Hypothesis

In the null hypothesis, there is no significant difference between the populations specified in the experiments, due to any experimental or sampling error. The null hypothesis is denoted by H 0 .

Alternative Hypothesis

In an alternative hypothesis, the simple observations are easily influenced by some random cause. It is denoted by the H a or H 1 .

Empirical Hypothesis

An empirical hypothesis is formed by the experiments and based on the evidence.

Statistical Hypothesis

In a statistical hypothesis, the statement should be logical or illogical, and the hypothesis is verified statistically.

Apart from these types of hypothesis, some other hypotheses are directional and non-directional hypothesis, associated hypothesis, casual hypothesis.

Characteristics of Hypothesis

The important characteristics of the hypothesis are:

• The hypothesis should be short and precise
• It should be specific
• A hypothesis must be related to the existing body of knowledge
• It should be capable of verification

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Last modified on August 3rd, 2023

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Hypotenuse of a triangle, what is the hypotenuse of a triangle.

A hypotenuse is the longest side of a right triangle. It is the side opposite the right angle (90°). The word ‘hypotenuse’ came from the Greek word ‘hypoteinousa’, meaning ‘stretching under’, where ‘hypo’ means ‘under’, and ‘teinein’ means ‘to stretch’.

How to Find the Hypotenuse of a Right Triangle

a) When Base and Height are Given

To calculate the hypotenuse of a right or right-angled triangle when its corresponding base and height are known, we use the given formula.

By Pythagorean Theorem,

(Hypotenuse) 2 = (Base) 2 + (Height) 2

Hypotenuse = √(Base) 2 + (Height) 2

Thus, mathematically, hypotenuse is the sum of the square of base and height of a right triangle.

The above formula is also written as,

c = √a 2 + b 2 , here c = hypotenuse, a = height, b = base

Let us solve some problems to understand the concept better.

Problem: Finding the hypotenuse of a right triangle, when the BASE and the HEIGHT are known.

What is the length of the hypotenuse of a right triangle with base 8m and height 6m.

As we know, c = √a 2 + b 2 , here a = 6m, b = 8m = √(6) 2 + (8) 2 = √36 + 64 = √100 = 10m

b) When Length of a Side and its Opposite Angle are Given

To find the hypotenuse of a right triangle when the length of a side and its opposite angle are known, we use the given formula, which is called the Law of sines.

c = a/sin α = b/sin β, here c = hypotenuse, a = height, b = base, α = angle formed between hypotenuse and base, β = angle formed between hypotenuse and height

Problem: Finding the hypotenuse of a right triangle, when the LENGTH OF A SIDE and its OPPOSITE ANGLE is known.

Here, we will use the Law of sines formula, c = a/sin α, here a = 12, α = 30° = 12/ sin 30° = 12 x 2 = 24 units

Using the Law of sines formula, c = b/sin β, b = 4, β = 60° = 4/ sin 60° = 8/√3 units

c) When the Area and Either Height or Base are Known

To determine the hypotenuse of a right triangle when the height or base is known, we use the Pythagorean Theorem to derive the formula as shown below:

As we know from the Pythagorean Theorem

c = √(a) 2 + (b) 2 …..(1), here c = hypotenuse, a = height, b = base

Area of right triangle (A) = a x b/2

 b = area x 2/a …… (2)

a = area x 2/b …… (3)

Putting (2) in (1) we get,

c = √(a 2 + (area x 2/a) 2 )

Putting (3) in (1) we get,

c = √(b 2 + (area x 2/b) 2 )

Problem: Finding the hypotenuse of a right triangle, when the AREA and one SIDE are known.

What is the length of the hypotenuse of a right triangle with area 20m 2 and height 6m.

As we know, c = √(a 2 + (area x 2/a) 2 ), here area = 20m 2 , a = 6m  = √6 2 + (20 x 2/6) 2 )  =√80.35 = 8.96 m

What is the length of the hypotenuse of a right triangle with area 14cm 2 and base 9cm.

As we know, c = √(b 2 + (area x 2/b) 2 ), here area = 14cm 2 , b = 9cm = √9 2 + (14 x 2/9) 2 ) = √45.67 = 6.75 m

How to Find the Hypotenuse of a Right Isosceles Triangle

To derive the formula for finding the hypotenuse of a right isosceles triangle we use the Pythagorean Theorem.

As we know,

c = √a 2 + b 2

Let the length of the two equal sides be x, such that (a = b = x)

c =√x 2 + x 2

What is the length of the hypotenuse of a right isosceles triangle with two equal sides measuring 5.5 cm each.

As we know, c = √2x, here x = 5.5 = √2 x 5.5 = 7.77 cm

Find the measure of the length of the hypotenuse of a 45-45-90 triangle with one of the two equal sides measuring 9 cm.

As a 45-45-90 triangle is a right isosceles triangle, we can apply the formula of right isosceles triangle for calculation of area As we know, c = √2x, here x = 9 cm =√2 x 9 = 12.72 cm

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Definition of a Hypothesis

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A hypothesis is a prediction of what will be found at the outcome of a research project and is typically focused on the relationship between two different variables studied in the research. It is usually based on both theoretical expectations about how things work and already existing scientific evidence.

Within social science, a hypothesis can take two forms. It can predict that there is no relationship between two variables, in which case it is a null hypothesis . Or, it can predict the existence of a relationship between variables, which is known as an alternative hypothesis.

In either case, the variable that is thought to either affect or not affect the outcome is known as the independent variable, and the variable that is thought to either be affected or not is the dependent variable.

Researchers seek to determine whether or not their hypothesis, or hypotheses if they have more than one, will prove true. Sometimes they do, and sometimes they do not. Either way, the research is considered successful if one can conclude whether or not a hypothesis is true. 

Null Hypothesis

A researcher has a null hypothesis when she or he believes, based on theory and existing scientific evidence, that there will not be a relationship between two variables. For example, when examining what factors influence a person's highest level of education within the U.S., a researcher might expect that place of birth, number of siblings, and religion would not have an impact on the level of education. This would mean the researcher has stated three null hypotheses.

Alternative Hypothesis

Taking the same example, a researcher might expect that the economic class and educational attainment of one's parents, and the race of the person in question are likely to have an effect on one's educational attainment. Existing evidence and social theories that recognize the connections between wealth and cultural resources , and how race affects access to rights and resources in the U.S. , would suggest that both economic class and educational attainment of the one's parents would have a positive effect on educational attainment. In this case, economic class and educational attainment of one's parents are independent variables, and one's educational attainment is the dependent variable—it is hypothesized to be dependent on the other two.

Conversely, an informed researcher would expect that being a race other than white in the U.S. is likely to have a negative impact on a person's educational attainment. This would be characterized as a negative relationship, wherein being a person of color has a negative effect on one's educational attainment. In reality, this hypothesis proves true, with the exception of Asian Americans , who go to college at a higher rate than whites do. However, Blacks and Hispanics and Latinos are far less likely than whites and Asian Americans to go to college.

Formulating a Hypothesis

Formulating a hypothesis can take place at the very beginning of a research project , or after a bit of research has already been done. Sometimes a researcher knows right from the start which variables she is interested in studying, and she may already have a hunch about their relationships. Other times, a researcher may have an interest in ​a particular topic, trend, or phenomenon, but he may not know enough about it to identify variables or formulate a hypothesis.

Whenever a hypothesis is formulated, the most important thing is to be precise about what one's variables are, what the nature of the relationship between them might be, and how one can go about conducting a study of them.

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Hypothesis | Definition, Meaning and Examples

Hypothesis is a hypothesis is fundamental concept in the world of research and statistics. It is a testable statement that explains what is happening or observed. It proposes the relation between the various participating variables.

Hypothesis is also called Theory, Thesis, Guess, Assumption, or Suggestion . Hypothesis creates a structure that guides the search for knowledge.

In this article, we will learn what hypothesis is, its characteristics, types, and examples. We will also learn how hypothesis helps in scientific research.

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What is Hypothesis?

Characteristics of hypothesis, sources of hypothesis, types of hypothesis, functions of hypothesis, how hypothesis help in scientific research.

Hypothesis is a suggested idea or an educated guess or a proposed explanation made based on limited evidence, serving as a starting point for further study. They are meant to lead to more investigation.

It’s mainly a smart guess or suggested answer to a problem that can be checked through study and trial. In science work, we make guesses called hypotheses to try and figure out what will happen in tests or watching. These are not sure things but rather ideas that can be proved or disproved based on real-life proofs. A good theory is clear and can be tested and found wrong if the proof doesn’t support it.

Hypothesis Meaning

A hypothesis is a proposed statement that is testable and is given for something that happens or observed.

• It is made using what we already know and have seen, and it’s the basis for scientific research.
• A clear guess tells us what we think will happen in an experiment or study.
• It’s a testable clue that can be proven true or wrong with real-life facts and checking it out carefully.
• It usually looks like a “if-then” rule, showing the expected cause and effect relationship between what’s being studied.

Here are some key characteristics of a hypothesis:

• Testable: An idea (hypothesis) should be made so it can be tested and proven true through doing experiments or watching. It should show a clear connection between things.
• Specific: It needs to be easy and on target, talking about a certain part or connection between things in a study.
• Falsifiable: A good guess should be able to show it’s wrong. This means there must be a chance for proof or seeing something that goes against the guess.
• Logical and Rational: It should be based on things we know now or have seen, giving a reasonable reason that fits with what we already know.
• Predictive: A guess often tells what to expect from an experiment or observation. It gives a guide for what someone might see if the guess is right.
• Concise: It should be short and clear, showing the suggested link or explanation simply without extra confusion.
• Grounded in Research: A guess is usually made from before studies, ideas or watching things. It comes from a deep understanding of what is already known in that area.
• Flexible: A guess helps in the research but it needs to change or fix when new information comes up.
• Relevant: It should be related to the question or problem being studied, helping to direct what the research is about.
• Empirical: Hypotheses come from observations and can be tested using methods based on real-world experiences.

Hypotheses can come from different places based on what you’re studying and the kind of research. Here are some common sources from which hypotheses may originate:

• Existing Theories: Often, guesses come from well-known science ideas. These ideas may show connections between things or occurrences that scientists can look into more.
• Observation and Experience: Watching something happen or having personal experiences can lead to guesses. We notice odd things or repeat events in everyday life and experiments. This can make us think of guesses called hypotheses.
• Previous Research: Using old studies or discoveries can help come up with new ideas. Scientists might try to expand or question current findings, making guesses that further study old results.
• Literature Review: Looking at books and research in a subject can help make guesses. Noticing missing parts or mismatches in previous studies might make researchers think up guesses to deal with these spots.
• Problem Statement or Research Question: Often, ideas come from questions or problems in the study. Making clear what needs to be looked into can help create ideas that tackle certain parts of the issue.
• Analogies or Comparisons: Making comparisons between similar things or finding connections from related areas can lead to theories. Understanding from other fields could create new guesses in a different situation.
• Hunches and Speculation: Sometimes, scientists might get a gut feeling or make guesses that help create ideas to test. Though these may not have proof at first, they can be a beginning for looking deeper.
• Technology and Innovations: New technology or tools might make guesses by letting us look at things that were hard to study before.
• Personal Interest and Curiosity: People’s curiosity and personal interests in a topic can help create guesses. Scientists could make guesses based on their own likes or love for a subject.

Here are some common types of hypotheses:

Simple Hypothesis

Complex hypothesis, directional hypothesis.

• Non-directional Hypothesis

Null Hypothesis (H0)

Alternative hypothesis (h1 or ha), statistical hypothesis, research hypothesis, associative hypothesis, causal hypothesis.

Simple Hypothesis guesses a connection between two things. It says that there is a connection or difference between variables, but it doesn’t tell us which way the relationship goes. Example: Studying more can help you do better on tests. Getting more sun makes people have higher amounts of vitamin D.

Complex Hypothesis tells us what will happen when more than two things are connected. It looks at how different things interact and may be linked together. Example: How rich you are, how easy it is to get education and healthcare greatly affects the number of years people live. A new medicine’s success relies on the amount used, how old a person is who takes it and their genes.

Directional Hypothesis says how one thing is related to another. For example, it guesses that one thing will help or hurt another thing. Example: Drinking more sweet drinks is linked to a higher body weight score. Too much stress makes people less productive at work.

Non-Directional Hypothesis

Non-Directional Hypothesis are the one that don’t say how the relationship between things will be. They just say that there is a connection, without telling which way it goes. Example: Drinking caffeine can affect how well you sleep. People often like different kinds of music based on their gender.

Null hypothesis is a statement that says there’s no connection or difference between different things. It implies that any seen impacts are because of luck or random changes in the information. Example: The average test scores of Group A and Group B are not much different. There is no connection between using a certain fertilizer and how much it helps crops grow.

Alternative Hypothesis is different from the null hypothesis and shows that there’s a big connection or gap between variables. Scientists want to say no to the null hypothesis and choose the alternative one. Example: Patients on Diet A have much different cholesterol levels than those following Diet B. Exposure to a certain type of light can change how plants grow compared to normal sunlight.

Statistical Hypothesis are used in math testing and include making ideas about what groups or bits of them look like. You aim to get information or test certain things using these top-level, common words only. Example: The average smarts score of kids in a certain school area is 100. The usual time it takes to finish a job using Method A is the same as with Method B.

Research Hypothesis comes from the research question and tells what link is expected between things or factors. It leads the study and chooses where to look more closely. Example: Having more kids go to early learning classes helps them do better in school when they get older. Using specific ways of talking affects how much customers get involved in marketing activities.

Associative Hypothesis guesses that there is a link or connection between things without really saying it caused them. It means that when one thing changes, it is connected to another thing changing. Example: Regular exercise helps to lower the chances of heart disease. Going to school more can help people make more money.

Causal Hypothesis are different from other ideas because they say that one thing causes another. This means there’s a cause and effect relationship between variables involved in the situation. They say that when one thing changes, it directly makes another thing change. Example: Playing violent video games makes teens more likely to act aggressively. Less clean air directly impacts breathing health in city populations.

Hypotheses have many important jobs in the process of scientific research. Here are the key functions of hypotheses:

• Guiding Research: Hypotheses give a clear and exact way for research. They act like guides, showing the predicted connections or results that scientists want to study.
• Formulating Research Questions: Research questions often create guesses. They assist in changing big questions into particular, checkable things. They guide what the study should be focused on.
• Setting Clear Objectives: Hypotheses set the goals of a study by saying what connections between variables should be found. They set the targets that scientists try to reach with their studies.
• Testing Predictions: Theories guess what will happen in experiments or observations. By doing tests in a planned way, scientists can check if what they see matches the guesses made by their ideas.
• Providing Structure: Theories give structure to the study process by arranging thoughts and ideas. They aid scientists in thinking about connections between things and plan experiments to match.
• Focusing Investigations: Hypotheses help scientists focus on certain parts of their study question by clearly saying what they expect links or results to be. This focus makes the study work better.
• Facilitating Communication: Theories help scientists talk to each other effectively. Clearly made guesses help scientists to tell others what they plan, how they will do it and the results expected. This explains things well with colleagues in a wide range of audiences.
• Generating Testable Statements: A good guess can be checked, which means it can be looked at carefully or tested by doing experiments. This feature makes sure that guesses add to the real information used in science knowledge.
• Promoting Objectivity: Guesses give a clear reason for study that helps guide the process while reducing personal bias. They motivate scientists to use facts and data as proofs or disprovals for their proposed answers.
• Driving Scientific Progress: Making, trying out and adjusting ideas is a cycle. Even if a guess is proven right or wrong, the information learned helps to grow knowledge in one specific area.

Researchers use hypotheses to put down their thoughts directing how the experiment would take place. Following are the steps that are involved in the scientific method:

• Initiating Investigations: Hypotheses are the beginning of science research. They come from watching, knowing what’s already known or asking questions. This makes scientists make certain explanations that need to be checked with tests.
• Formulating Research Questions: Ideas usually come from bigger questions in study. They help scientists make these questions more exact and testable, guiding the study’s main point.
• Setting Clear Objectives: Hypotheses set the goals of a study by stating what we think will happen between different things. They set the goals that scientists want to reach by doing their studies.
• Designing Experiments and Studies: Assumptions help plan experiments and watchful studies. They assist scientists in knowing what factors to measure, the techniques they will use and gather data for a proposed reason.
• Testing Predictions: Ideas guess what will happen in experiments or observations. By checking these guesses carefully, scientists can see if the seen results match up with what was predicted in each hypothesis.
• Analysis and Interpretation of Data: Hypotheses give us a way to study and make sense of information. Researchers look at what they found and see if it matches the guesses made in their theories. They decide if the proof backs up or disagrees with these suggested reasons why things are happening as expected.
• Encouraging Objectivity: Hypotheses help make things fair by making sure scientists use facts and information to either agree or disagree with their suggested reasons. They lessen personal preferences by needing proof from experience.
• Iterative Process: People either agree or disagree with guesses, but they still help the ongoing process of science. Findings from testing ideas make us ask new questions, improve those ideas and do more tests. It keeps going on in the work of science to keep learning things.

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Hypothesis is a testable statement serving as an initial explanation for phenomena, based on observations, theories, or existing knowledge . It acts as a guiding light for scientific research, proposing potential relationships between variables that can be empirically tested through experiments and observations.

The hypothesis must be specific, testable, falsifiable, and grounded in prior research or observation, laying out a predictive, if-then scenario that details a cause-and-effect relationship. It originates from various sources including existing theories, observations, previous research, and even personal curiosity, leading to different types, such as simple, complex, directional, non-directional, null, and alternative hypotheses, each serving distinct roles in research methodology .

The hypothesis not only guides the research process by shaping objectives and designing experiments but also facilitates objective analysis and interpretation of data , ultimately driving scientific progress through a cycle of testing, validation, and refinement.

Hypothesis – FAQs

What is a hypothesis.

A guess is a possible explanation or forecast that can be checked by doing research and experiments.

What are Components of a Hypothesis?

The components of a Hypothesis are Independent Variable, Dependent Variable, Relationship between Variables, Directionality etc.

What makes a Good Hypothesis?

Testability, Falsifiability, Clarity and Precision, Relevance are some parameters that makes a Good Hypothesis

Can a Hypothesis be Proven True?

You cannot prove conclusively that most hypotheses are true because it’s generally impossible to examine all possible cases for exceptions that would disprove them.

How are Hypotheses Tested?

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data

Can Hypotheses change during Research?

Yes, you can change or improve your ideas based on new information discovered during the research process.

What is the Role of a Hypothesis in Scientific Research?

Hypotheses are used to support scientific research and bring about advancements in knowledge.

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The hypotenuse is the largest side of a right triangle. It is a side opposite to the right angle in a right triangle. The Pythagoras theorem defines the relationship between the hypotenuse and the other two sides of the right triangle, the base, and the perpendicular side. The square of the hypotenuse is equal to the sum of the squares of the base and the perpendicular side of the right triangle.

The Pythagoras theorem has given the Pythagorean triplets and the largest value in Pythagorean triplets is the hypotenuse. Let us learn more about the hypotenuse in this article.

What is a Hypotenuse?

The hypotenuse is the longest side of a right-angled triangle . It is represented by the side opposite to the right angle. It is related to the other sides of the right triangle by the Pythagoras theorem . The square of the measure of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle. The hypotenuse can be easily recognized in a right triangle as the largest side.

Hypotenuse Definition: In a right-angled triangle, the longest side or the side opposite to the right angle is termed hypotenuse . The hypotenuse is related to the base and the altitude of the triangle , by the formula: Hypotenuse 2 = Base 2 + Altitude 2 . Let us look at the below real-world examples of a hypotenuse in right triangle-shaped objects.

Hypotenuse Equation

To derive an equation or a formula of the hypotenuse , years ago there was an interesting fact revealed about triangles. Hypotenuse equation : The fact states that with a right-angled triangle or a triangle with a 90º angle, squares can be framed using each of the three sides of the triangle. After putting squares against each side, it was observed that the biggest square has the exact same area as the other two squares. To simplify the whole observation, it was later put in a short equation that can also be called a hypotenuse equation.

So, the hypotenuse equation = a 2 + b 2 = c 2 , where c is the length of the hypotenuse and a and b are the other two sides of the right-angled triangle.

Now, look at the image given below to understand the derivation of the above formula. Here we have a = Perpendicular, b = Base, c = Hypotenuse.

Tips and Tricks on Hypotenuse:

The following points will help you to get a better understanding of the hypotenuse and its relation to the other two sides of the right triangle.

• The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides (base and perpendicular ).
• This is represented as: Hypotenuse 2 = Base 2 + Perpendicular 2 .
• Hypotenuse equation is a 2 + b 2 = c 2 . Here, a and b are the legs of the right triangle and c is the hypotenuse.
• The hypotenuse leg theorem states that two triangles are congruent if the hypotenuse and one leg of one right triangle are congruent/equal to the other right triangle's hypotenuse and leg side.

How to Find Hypotenuse?

To find the length of the hypotenuse of a triangle, we will be using the above equation. For that, we should know the values of the base and perpendicular of the triangle. For example, in a right triangle, if the length of the base is 3 units, and the length of the perpendicular side is 4 units, then the length of the hypotenuse can be found by using the formula Hypotenuse 2 = Base 2 + Perpendicular 2 . By substituting the values of the base and perpendicular, we get, Hypotenuse 2 = 3 2 + 4 2 = 9 + 16 = 25. This implies that the length of the hypotenuse is 5 units. This is how we can easily find the length of the hypotenuse by using the hypotenuse equation.

Follow the steps given below to find the hypotenuse length in a right-angled triangle:

• Step 1: Identify the values of base and perpendicular sides.
• Step 2: Substitute the values of base and perpendicular in the formula: Hypotenuse 2 = Base 2 + Perpendicular 2 .
• Step 3: Solve the equation and get the answer.

Let us consider one more example to find the hypotenuse of a triangle. The longest side of the triangle is the hypotenuse and the other two sides of the right triangle are the perpendicular side with a measure of 8 inches, and the base with a measure of 6 inches.

The following formula is helpful to calculate the measure of the hypotenuse → (Hypotenuse) 2 = (Base) 2 + (Perpendicular) 2 = 6 2 + 8 2 = 36 + 64 = 100. This implies, Hypotenuse = √100 = 10 inches. Also, any of the other two sides, the base or the perpendicular side can be easily calculated for the given value of the hypotenuse using the same equation.

Hypotenuse Theorem

The hypotenuse can be related to the other two sides of the right-angled triangle by the Pythagoras theorem. The Pythagoras theorem states that the square of the hypotenuse is equal to the sum of the squares of the base of the triangle, and the square of the altitude of the triangle. Among the three sides of the right triangle, the hypotenuse is the largest side, and Hypotenuse 2 = Base 2 + Altitude 2 . This is known as the hypotenuse theorem . The lengths of the hypotenuse, altitude, and base of the triangle, are together defined as a set called the Pythagorean triplets. A few examples of Pythagorean triples are (5, 4, 3), (10, 8, 6), and (25, 24, 7).

Challenging Questions:

Having understood the concepts related to the hypotenuse of a triangle, now try out these two challenging questions.

• A 5 meters ladder stands on horizontal ground and reaches 3 m up a vertical wall. How far is the foot of the ladder from the wall?
• Town B is 9 km north and 16 km west of town A. What is the shortest distance to go from town A to town B?

► Related Topics:

Check these articles related to the concept of the hypotenuse of a triangle.

• Hypotenuse Calculator
• Area of Right Triangle
• Properties of Triangle

Hypotenuse Examples

Example 1: Find the value of the longest side of a bread slice that is in the shape of a right-angle triangle with a given perpendicular height of 12 inches and the base of 5 inches.

Given dimensions are perpendicular (P) = 12 inches, and base (B) = 5 inches. Putting the given dimensions in the formula H 2 = B 2 + P 2 , we get, H 2 = 5 2 + 12 2 H = √{25+144} = √169 inches H = 13 inches. Therefore the length of the hypotenuse (longest side) of the bread slice is 13 inches.

Example 2: In a right triangle, the hypotenuse is 5 units, and the perpendicular is 4 units. Find the measure of the base of the triangle.

Given dimensions are perpendicular (P) = 4 units, and hypotenuse (H) = 5 units. We know that (H) 2 = (B) 2 + (P) 2 ⇒ (B) 2 = (H) 2 - (P) 2 . Putting the given dimensions in the formula, we get, B 2 = (5) 2 - (4) 2 B = √{25-16} B = √9 = 3 units Therefore, the length of the base is 3 units.

Example 3: How to find the missing hypotenuse of a triangle with base = 7 units and perpendicular = 24 units?

Given dimensions are base (B) = 7 units and perpendicular (P) = 24 units. To find the hypotenuse (H), we will use the equation: (H) 2 = (B) 2 + (P) 2 . Putting the given dimensions in the equation, we get, H 2 = (7) 2 + (24) 2 B = √{49+576} B = √625 = 25 units Therefore, the length of the hypotenuse is 25 units.

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Practice Questions on Hypotenuse

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FAQs on Hypotenuse

What is the meaning of hypotenuse.

In mathematics, the hypotenuse of a triangle is defined as the longest side of a right triangle. It is the side opposite to the 90-degree angle . It is equal to the square root of the sum of the squares of the other two sides.

What is the Length of the Hypotenuse?

The length of the hypotenuse is greater than the lengths of the other two sides of a right triangle. The square of the hypotenuse length is equal to the sum of squares of the other two sides of the triangle. Mathematically, it can be expressed in the form of an equation as Hypotenuse 2 = Base 2 + Perpendicular 2 .

What is the Hypotenuse Leg Theorem?

The hypotenuse leg theorem states that two right triangles are congruent if the lengths of the hypotenuse and any one of the legs of a triangle are equal to the hypotenuse and the leg of the other triangle.

How to Find the Missing Hypotenuse?

The missing hypotenuse can be easily known if we know the lengths of the other two sides by using the hypotenuse equation: Hypotenuse 2 = Base 2 + Perpendicular 2 . For example, if the base and perpendicular of a right triangle measure 6 units and 8 units respectively, then the hypotenuse is equal to:

Hypotenuse 2 = 6 2 + 8 2

Therefore, hypotenuse = 10 units.

How do you Find the Hypotenuse of a Triangle?

By using the Pythagorean theorem (Hypotenuse) 2 = (Base) 2 + (Altitude) 2 , we can calculate the hypotenuse. If the values of the other two sides are known, the hypotenuse can be easily calculated with this formula.

How do you Find the Longest Side of a Triangle?

The hypotenuse is termed as the longest side of a right-angled triangle. To find the longest side we use the hypotenuse theorem, (Hypotenuse) 2 = (Base) 2 + (Altitude) 2 . For example, a bread slice is given in the shape of a right-angled triangle. If the base is 4 inches and the height is 3 inches, then the hypotenuse is (H) 2 = (4) 2 + (3) 2 = √{16+9} = √25 = 5 inches.

How to Find Hypotenuse with Angle and Side?

If an angle and a side are known, then we can calculate hypotenuse by applying the formula of trigonometric ratios . If A is the angle known, then we have,

• sin A = Perpendicular/Hypotenuse
• cos A = Base/Hypotenuse

So, if the length of the base is given, then the cos formula can be used and if height is known then the sin formula can be used to find the hypotenuse length.

What is the Difference between the Hypotenuse and Other Sides of a Triangle?

The hypotenuse is the largest side of the triangle. The other two sides are the base and the altitude of the right triangle. These are related to each other with the formula (Hypotenuse) 2 = (Base) 2 + (Altitude) 2 .

How is the Hypotenuse Related to the Right Angle?

The hypotenuse is the side opposite to the right angle. The hypotenuse is the largest side of a right triangle and is drawn opposite to the largest angle, which is the right angle.

Can a Hypotenuse be Drawn for Any Triangle?

The hypotenuse can be drawn only for a right triangle, and not for any other triangle. The side opposite to the 90° angle is the hypotenuse. And since a right angle is there in a right triangle, it has a hypotenuse.

How to Calculate Hypotenuse?

The formula to calculate the hypotenuse is (Hypotenuse) 2 = (Base) 2 + (Altitude) 2 . The largest side of the right triangle is the hypotenuse, and it can be calculated if the other two sides are known.