Notes UNIT 1:
Throughout the unit, we learn many new things, with the helpful information from our notes. The following contains the notes I have taken and retained throughout Unit 1 of this class. Enjoy!
~ Aniket Matharasi- 8th grader at Fowler Middle School
~ Aniket Matharasi- 8th grader at Fowler Middle School
Unit 1A Notes
Lesson 1- Points,Lines,Rays, & Planes
Our learning experiences began on the very first day of school, diving right in.
Almost always the first thing you learn in a subject is a creator. In geometry, that person is Euclid.
We began at lesson 1 with a short study of Euclid and his contributions to the world of geometry. Euclid is known as the Father of Geometry. He is so synonymous with the study of geometry that the basic subject of geometry is known as 'Euclidean Geometry'. He originates geometry through three terms that are the basis of all geometric studies- point,line, and plane. These are known as undefined as the definitions are left to be understood by the student or user. Then as we moved past the origin of geometry, we learned about the fundamentals. Our first lesson included points,lines,planes,collinear points,coplanar points,segments,endpoints,rays,opposite rays, and intersections. These terms also brought along quite some postulates.
Postulates~
Through any two points there is exactly one line.
Through any 3 non-collinear points there is exactly one plane.
If two points lie in a plane, then the line containing those points lies in the plane.
If two unique lines intersect, then they intersect at exactly one point.If two unique planes intersect, they intersect at exactly one line.
This is connected from the origins and beginnings of geometry.Points, rays, lines, and planes are the very foundations of geometry.
Lesson 2-Inductive Reasoning and Use of Postulates and Diagrams
As we began to get into the groove of a new school year, we were introduced to a new aspect of geometry-logic. This warrants the use of inductive and deductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because of specific cases. This part of geometry also includes conjectures which is a statement believed to be true based on this inductive reasoning. Another key vocabulary term of inductive reasoning is a counterexample which is a specific example that shows why a certain conjecture or statement is false, and it basically gives a real world or general example of why it is wrong. For example, if I gave the conjecture that 'If you are in geometry, then you are in 10th grade', then all of us in this class would be an example of a counterexample since we are all in geometry but in 8th grade rather than 10th grade. In order to prove a conjecture false, only one counterexample is required. We can also use inductive reasoning to find the rule of a sequence of numbers or figures. The rule would therefore be the conjecture.
On the other hand, deductive reasoning is the process of using logic to draw conclusions based on facts, definitions, and properties. This utilizes the Law of Detachment and the Law of Syllogism. The first one states that 'If If p → q is a true statement and p is true, then q is true.' On the other hand, the law of syllogism states that 'If p → q and q → r are true statements, then p → r is a true statement.' These are both forms of deductive reasoning. In order to differentiate the two, you can go back to the definitions through you head. Inductive reasoning is logic based on specific examples. Using a counterexample to disprove a conjecture is an example of inductive reasoning. Deductive reasoning is logic based on given facts, definitions, and properties. Conclusions made based on known definitions, research, or data are examples of deductive reasoning.
Practice Problem: There is a myth that tapping the side of a carbonated beverage will prevent it from foaming over. However, if you were to shake it, tap it, and then open it, you would see that this myth is false. What type of reasoning is this?
In this example, the conclusion is that the myth is false. This conclusion is based on observation of a specific example, opening a carbonated beverage after it is shaken and tapped, which is a counterexample to the given conditional. Therefore, the type of reasoning used is inductive reasoning.
The connection to geometry here is the real world application. You can find conjectures,hypothesis and conclusions everywhere. This relates to the origins of geometry as everyone began to use logic when building on top of the basics.
Lesson 3-Analyze Conditional Statements
Euclid began to use a formal structure to write relationship-packed statements called theorems and postulates. These statements are sometimes called “IF-THEN statements”. Understanding how they are written will help you to understand how identifying relationships can save you time and effort when trying to solve a problem. The big idea here is to learn that knowing one relationship can be a shortcut for you discovering another new relationship. That can be very powerful in anything you do – especially if you see yourself in a science- or math-related profession. 4 types of IF-THEN statements – the conditional, the converse, the inverse, and the contrapositive. Pay close attention to the pattern that emerges regarding the truth value of these statements. That pattern will become very helpful to you in the future as you begin to study very complicated IF-THEN statements. The hypothesis of an If-Then Statement would be the part after the if but before the then, and the conclusion would be after the then.
Here's a practice problem to figure out hypotheses and conclusions.
If Ed has a test, then he will study.
Since the hypothesis is the part after the 'if', then the hypothesis would be 'Ed has a test'. On the other hand, the conclusion-the part after the 'then' would be 'Ed will study'.
The 4 types of conditional statements are built on negation and exchanges or in a simpler form 'nots' and 'flips'.
The types are
Converse is the statement formed by exchanging the hypothesis and conclusion. In other words, this would be a 'flip'!
Inverse is the statement formed by negating the hypothesis and conclusion. In other words, this would be a 'not'!
Lastly, the contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. In other words, the contrapositive would be the 'flip' and the 'not'!
Here's another problem that was quite mind boggling and took some thought.
Consider the true statement, “If it’s snowing, then it’s cold outside.”
1. Suppose it’s snowing. Does that mean it’s cold outside?
2. Suppose it’s cold outside. Does that mean it’s snowing?
For number one, if it's snowing, it must b cold. The statement states that as the conclusion of the hypothesis that it is snowing. Since it is true it must be cold outside.
For number 2 on the other hand, just because it is cold doesn't mean it is snowing. It can be 40 degrees and be cold, but still not necessarily be snowing which makes the second conjecture false. The example I gave for the 40 degrees was a counterexample!
The last type of statement is a bi-conditional statement. This is in the form of "p if and only if q." This means "if p then q" and "if q then p."
This is connected through different conditional statements. This relates to the previous lesson but in-depth.
Lesson 4
The problem is 2(3x+5)=40. We are now going to take that problem and show every single step and justify the reason why we are allowed to take that step.
First, you should have distributed the 2 over the parenthesis to get 6x+10=40. You are allowed to do that step because of the Distributive Property.
The second step is to undo the addition of 10 on the left by adding a negative 10 to both sides. Some of you may have learned to subtract 10 from both sides. Both ways are correct and both can be used in this course as long as you specify the property that you are using. You should then have 6x=30. You are allowed to do that step because of the Addition Property of Equality (or the Subtraction Property of Equality). It’s called a property of equality because we are working with equations, which have equal signs.
Next you should now undo the multiplication of six by dividing both sides by six. You should get x=5. Again, you may have been taught to multiply by the reciprocal of six, which is 1/6th. You are allowed to do this step because of the Division Property of Equality (or the Multiplication Property of Equality). Again, the word equality is used because we are working with equations.
Overall, you’ve used 3 properties to solve this simple equation: Distributive, Addition (or Subtraction), and finally, Division (or Multiplication) to come up with an answer of 5.
It is important to note at this point that it is sometimes possible to solve an equation in more than one way. For instance, you could divide both sides of the equation by 2 using the Division Property on the first step. The result would be 3x + 5 = 20. Then, subtract 5 from both sides (Subtraction Property) to get 3x = 15 and finally, divide both sides by 3 (Division Property) to get the same answer of 5. One way is not more correct than the other. The focus this lesson will be on justifying the steps that you do with a property of mathematics.
The above is a way to use algebra to reason and solve equations. We have already learned algebra so it should be fairly easy and mostly a review to do the basics.
The 8 main properties of equality are
Properties of Congruence- There are three Properties of Congruence: Reflexive, Symmetric, and Transitive.
The following is a practice problem for proofs.
-8=5n+2
The variable in this equation is n. In the original equation, 2 is added to 5n. So, use the Subtraction Property of Equality and subtract 2 from both sides of the equation. After the equation is simplified, n is multiplied by 5. So, use the Division Property of Equality and divide both sides of the equation by 5. After the equation is simplified again, the result is –2 = n. Since the variable n is isolated on one side of the equation, the equation is solved. However, the Symmetric Property of Equality can be applied to write the equation with n on the left side, as equations are more commonly written.
-8=5n+2 Given
-10=5n Subtraction POE
-10=5n Simplify
-2=n Division POE
Simplify
n=-2 Symmetric POE
This relates to last year's lessons of Algebra. This connected to the later lesson of geometric proofs later.
Lesson 5
This lesson was a review unit so I think this is a great time for a practice problem.
If I am in athletics, then I play basketball.
Using this conditional statement, show the 5 statements and their truth vlue. If it is false, give a counterexample.
Conditional- If I am in Athletics, then I play basketball. This is false as you can be in athletics and play either football or volleyball.
Converse- If I play basketball, then I am in athletics. This statement is also false, since you can play basketball for fun or play outside of school athletics.
Inverse- If I am not in Athletics, then I do not play basketball. This is a false conjecture, since I can still play basketball even though I am not in athletics.
Contrapositive- If I do not play basketball, then I am not in Athletics. This is false since I can be in Athletics even though I don't play basketball as I would be able to play another sport.
Bi-conditional- I am in athletics if and only if I play basketball. This is false, since you can be in basketball and not play basketball.
Since the conditional contains the hypothesis of being in athletics, this gives us multiple choices. You can be in numerous sports and be in athletics. Since the conclusion is playing basketball, the group of people in athletics is not selective to those in basketball. As a result all of these statements were false. In addition, every statement resulted in the possibility of their outcomes. If you weren't in athletics, you could still play basketball.
Unit 1B Notes
Lesson 6
Another area of geometry that is vitally important is the drawings that accompany the concepts, proofs, and problems that you will encounter. These drawings are called constructions. Constructions are drawn with a compass and straight edge – no rulers and no protractors. Using a ruler or protractor to measure creates greater human error than a compass and straight edge. Because the point is to be accurate, you will learn how to properly use these two tools as you navigate through the course.
Use a compass and a straight edge in such a way that your work will be draftsman quality. That means that your constructions will never be hand-drawn or sketched. You will always use a straight edge and you will always measure precisely with arc marks from your compass. We will start with the simplest of constructions today – copying a segment and bisecting a segment. As a big part of geometry, this is essential forever and always.
In addition, fractional distance was also covered. Fractional distance is used to find the portion of the distances between two points. You can find fractional distance by finding the slope or the rise and the run. If you were asked for 3/4 of the distance, you would find 3/4 of the rise and then 3/4 of the run. You take this product and add it to the endpoint which gives you the point.
Using the following handouts will help a lot and will give you step-by -step instructions on how to create and bisect segments.
The connection here is the beginning of constructing and bisecting constructions. This is the basis of a lot of work later in the year and utilizes the rays, points, lines, and planes we learned at the beginning.
Our learning experiences began on the very first day of school, diving right in.
Almost always the first thing you learn in a subject is a creator. In geometry, that person is Euclid.
We began at lesson 1 with a short study of Euclid and his contributions to the world of geometry. Euclid is known as the Father of Geometry. He is so synonymous with the study of geometry that the basic subject of geometry is known as 'Euclidean Geometry'. He originates geometry through three terms that are the basis of all geometric studies- point,line, and plane. These are known as undefined as the definitions are left to be understood by the student or user. Then as we moved past the origin of geometry, we learned about the fundamentals. Our first lesson included points,lines,planes,collinear points,coplanar points,segments,endpoints,rays,opposite rays, and intersections. These terms also brought along quite some postulates.
Postulates~
Through any two points there is exactly one line.
Through any 3 non-collinear points there is exactly one plane.
If two points lie in a plane, then the line containing those points lies in the plane.
If two unique lines intersect, then they intersect at exactly one point.If two unique planes intersect, they intersect at exactly one line.
This is connected from the origins and beginnings of geometry.Points, rays, lines, and planes are the very foundations of geometry.
Lesson 2-Inductive Reasoning and Use of Postulates and Diagrams
As we began to get into the groove of a new school year, we were introduced to a new aspect of geometry-logic. This warrants the use of inductive and deductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because of specific cases. This part of geometry also includes conjectures which is a statement believed to be true based on this inductive reasoning. Another key vocabulary term of inductive reasoning is a counterexample which is a specific example that shows why a certain conjecture or statement is false, and it basically gives a real world or general example of why it is wrong. For example, if I gave the conjecture that 'If you are in geometry, then you are in 10th grade', then all of us in this class would be an example of a counterexample since we are all in geometry but in 8th grade rather than 10th grade. In order to prove a conjecture false, only one counterexample is required. We can also use inductive reasoning to find the rule of a sequence of numbers or figures. The rule would therefore be the conjecture.
On the other hand, deductive reasoning is the process of using logic to draw conclusions based on facts, definitions, and properties. This utilizes the Law of Detachment and the Law of Syllogism. The first one states that 'If If p → q is a true statement and p is true, then q is true.' On the other hand, the law of syllogism states that 'If p → q and q → r are true statements, then p → r is a true statement.' These are both forms of deductive reasoning. In order to differentiate the two, you can go back to the definitions through you head. Inductive reasoning is logic based on specific examples. Using a counterexample to disprove a conjecture is an example of inductive reasoning. Deductive reasoning is logic based on given facts, definitions, and properties. Conclusions made based on known definitions, research, or data are examples of deductive reasoning.
Practice Problem: There is a myth that tapping the side of a carbonated beverage will prevent it from foaming over. However, if you were to shake it, tap it, and then open it, you would see that this myth is false. What type of reasoning is this?
In this example, the conclusion is that the myth is false. This conclusion is based on observation of a specific example, opening a carbonated beverage after it is shaken and tapped, which is a counterexample to the given conditional. Therefore, the type of reasoning used is inductive reasoning.
The connection to geometry here is the real world application. You can find conjectures,hypothesis and conclusions everywhere. This relates to the origins of geometry as everyone began to use logic when building on top of the basics.
Lesson 3-Analyze Conditional Statements
Euclid began to use a formal structure to write relationship-packed statements called theorems and postulates. These statements are sometimes called “IF-THEN statements”. Understanding how they are written will help you to understand how identifying relationships can save you time and effort when trying to solve a problem. The big idea here is to learn that knowing one relationship can be a shortcut for you discovering another new relationship. That can be very powerful in anything you do – especially if you see yourself in a science- or math-related profession. 4 types of IF-THEN statements – the conditional, the converse, the inverse, and the contrapositive. Pay close attention to the pattern that emerges regarding the truth value of these statements. That pattern will become very helpful to you in the future as you begin to study very complicated IF-THEN statements. The hypothesis of an If-Then Statement would be the part after the if but before the then, and the conclusion would be after the then.
Here's a practice problem to figure out hypotheses and conclusions.
If Ed has a test, then he will study.
Since the hypothesis is the part after the 'if', then the hypothesis would be 'Ed has a test'. On the other hand, the conclusion-the part after the 'then' would be 'Ed will study'.
The 4 types of conditional statements are built on negation and exchanges or in a simpler form 'nots' and 'flips'.
The types are
- conditional
- converse
- inverse
- contrapositive
Converse is the statement formed by exchanging the hypothesis and conclusion. In other words, this would be a 'flip'!
Inverse is the statement formed by negating the hypothesis and conclusion. In other words, this would be a 'not'!
Lastly, the contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. In other words, the contrapositive would be the 'flip' and the 'not'!
Here's another problem that was quite mind boggling and took some thought.
Consider the true statement, “If it’s snowing, then it’s cold outside.”
1. Suppose it’s snowing. Does that mean it’s cold outside?
2. Suppose it’s cold outside. Does that mean it’s snowing?
For number one, if it's snowing, it must b cold. The statement states that as the conclusion of the hypothesis that it is snowing. Since it is true it must be cold outside.
For number 2 on the other hand, just because it is cold doesn't mean it is snowing. It can be 40 degrees and be cold, but still not necessarily be snowing which makes the second conjecture false. The example I gave for the 40 degrees was a counterexample!
The last type of statement is a bi-conditional statement. This is in the form of "p if and only if q." This means "if p then q" and "if q then p."
This is connected through different conditional statements. This relates to the previous lesson but in-depth.
Lesson 4
The problem is 2(3x+5)=40. We are now going to take that problem and show every single step and justify the reason why we are allowed to take that step.
First, you should have distributed the 2 over the parenthesis to get 6x+10=40. You are allowed to do that step because of the Distributive Property.
The second step is to undo the addition of 10 on the left by adding a negative 10 to both sides. Some of you may have learned to subtract 10 from both sides. Both ways are correct and both can be used in this course as long as you specify the property that you are using. You should then have 6x=30. You are allowed to do that step because of the Addition Property of Equality (or the Subtraction Property of Equality). It’s called a property of equality because we are working with equations, which have equal signs.
Next you should now undo the multiplication of six by dividing both sides by six. You should get x=5. Again, you may have been taught to multiply by the reciprocal of six, which is 1/6th. You are allowed to do this step because of the Division Property of Equality (or the Multiplication Property of Equality). Again, the word equality is used because we are working with equations.
Overall, you’ve used 3 properties to solve this simple equation: Distributive, Addition (or Subtraction), and finally, Division (or Multiplication) to come up with an answer of 5.
It is important to note at this point that it is sometimes possible to solve an equation in more than one way. For instance, you could divide both sides of the equation by 2 using the Division Property on the first step. The result would be 3x + 5 = 20. Then, subtract 5 from both sides (Subtraction Property) to get 3x = 15 and finally, divide both sides by 3 (Division Property) to get the same answer of 5. One way is not more correct than the other. The focus this lesson will be on justifying the steps that you do with a property of mathematics.
The above is a way to use algebra to reason and solve equations. We have already learned algebra so it should be fairly easy and mostly a review to do the basics.
The 8 main properties of equality are
- Addition Property of Equality- If a=b, then a+c=b+c
- Subtraction Property of Equality- If a=b, then a-c=b-c
- Multiplication Property of Equality- If a=b, ac=bc
- Division Property of Equality- If a=b, and c doesn't equal 0, then a/c=b/c
- Reflexive Property of Equality- a=a
- Symmetric Property of Equality- If a=b, then b=a.
- Transitive Property of Equality- If a=b and b=c, then a=c.
- Substitution Property of Equality- If a=b, then b can be substituted for a in any expression.
- Distributive Property - a(b + c) = ab + ac and a(b – c) = ab – ac
Properties of Congruence- There are three Properties of Congruence: Reflexive, Symmetric, and Transitive.
The following is a practice problem for proofs.
-8=5n+2
The variable in this equation is n. In the original equation, 2 is added to 5n. So, use the Subtraction Property of Equality and subtract 2 from both sides of the equation. After the equation is simplified, n is multiplied by 5. So, use the Division Property of Equality and divide both sides of the equation by 5. After the equation is simplified again, the result is –2 = n. Since the variable n is isolated on one side of the equation, the equation is solved. However, the Symmetric Property of Equality can be applied to write the equation with n on the left side, as equations are more commonly written.
-8=5n+2 Given
-10=5n Subtraction POE
-10=5n Simplify
-2=n Division POE
Simplify
n=-2 Symmetric POE
This relates to last year's lessons of Algebra. This connected to the later lesson of geometric proofs later.
Lesson 5
This lesson was a review unit so I think this is a great time for a practice problem.
If I am in athletics, then I play basketball.
Using this conditional statement, show the 5 statements and their truth vlue. If it is false, give a counterexample.
Conditional- If I am in Athletics, then I play basketball. This is false as you can be in athletics and play either football or volleyball.
Converse- If I play basketball, then I am in athletics. This statement is also false, since you can play basketball for fun or play outside of school athletics.
Inverse- If I am not in Athletics, then I do not play basketball. This is a false conjecture, since I can still play basketball even though I am not in athletics.
Contrapositive- If I do not play basketball, then I am not in Athletics. This is false since I can be in Athletics even though I don't play basketball as I would be able to play another sport.
Bi-conditional- I am in athletics if and only if I play basketball. This is false, since you can be in basketball and not play basketball.
Since the conditional contains the hypothesis of being in athletics, this gives us multiple choices. You can be in numerous sports and be in athletics. Since the conclusion is playing basketball, the group of people in athletics is not selective to those in basketball. As a result all of these statements were false. In addition, every statement resulted in the possibility of their outcomes. If you weren't in athletics, you could still play basketball.
Unit 1B Notes
Lesson 6
Another area of geometry that is vitally important is the drawings that accompany the concepts, proofs, and problems that you will encounter. These drawings are called constructions. Constructions are drawn with a compass and straight edge – no rulers and no protractors. Using a ruler or protractor to measure creates greater human error than a compass and straight edge. Because the point is to be accurate, you will learn how to properly use these two tools as you navigate through the course.
Use a compass and a straight edge in such a way that your work will be draftsman quality. That means that your constructions will never be hand-drawn or sketched. You will always use a straight edge and you will always measure precisely with arc marks from your compass. We will start with the simplest of constructions today – copying a segment and bisecting a segment. As a big part of geometry, this is essential forever and always.
In addition, fractional distance was also covered. Fractional distance is used to find the portion of the distances between two points. You can find fractional distance by finding the slope or the rise and the run. If you were asked for 3/4 of the distance, you would find 3/4 of the rise and then 3/4 of the run. You take this product and add it to the endpoint which gives you the point.
Using the following handouts will help a lot and will give you step-by -step instructions on how to create and bisect segments.
The connection here is the beginning of constructing and bisecting constructions. This is the basis of a lot of work later in the year and utilizes the rays, points, lines, and planes we learned at the beginning.
Lesson 7
In this lesson, we learned about constructing and bisecting along with measuring angles. We also discussed different types of angles.
• Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points.
• A linear pair of angles is a pair of adjacent angles whose noncommon sides are opposite rays.
• Complementary angles are two angles whose measures have a sum of 90°.
• Supplementary angles are two angles whose measures have a sum of 180°.
• Vertical angles are two nonadjacent angles formed by two intersecting lines.
For example, two angles that have a common vertex and side, but do not overlap, are called adjacent angles. The figure to the left contains a pair of adjacent angles, ∠BAC and ∠CAD. Notice that ∠BAD and ∠BAC share a common vertex, point A, and a common side, ray AB, but those two angles are not adjacent angles because they overlap.
A linear pair is a pair of adjacent angles that combine to form
a straight angle. In a linear pair, the noncommon sides of the
two adjacent angles are opposite rays.
This connects angles into the original foundation. On top of the segments, this introduces constructions of angles.
In this lesson, we learned about constructing and bisecting along with measuring angles. We also discussed different types of angles.
- Acute angle-measures greater than 0° and less than 90°.
- Right Angle- measures 90°.
- Obtuse Angle- An obtuse angle measures greater than 90° and less than 180°.
- Straight angle- formed by two opposite rays and measures 180°.
- Angle bisector-is a ray that divides an angle into two congruent angles.
• Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points.
• A linear pair of angles is a pair of adjacent angles whose noncommon sides are opposite rays.
• Complementary angles are two angles whose measures have a sum of 90°.
• Supplementary angles are two angles whose measures have a sum of 180°.
• Vertical angles are two nonadjacent angles formed by two intersecting lines.
For example, two angles that have a common vertex and side, but do not overlap, are called adjacent angles. The figure to the left contains a pair of adjacent angles, ∠BAC and ∠CAD. Notice that ∠BAD and ∠BAC share a common vertex, point A, and a common side, ray AB, but those two angles are not adjacent angles because they overlap.
A linear pair is a pair of adjacent angles that combine to form
a straight angle. In a linear pair, the noncommon sides of the
two adjacent angles are opposite rays.
This connects angles into the original foundation. On top of the segments, this introduces constructions of angles.
Lesson 8
This lesson begins the crust of geometry – the formal geometrical proof. You will see today a kind of elegance and precision that is not seen in other areas of mathematics. You will learn that a very formal process used to show your understanding of a concept. Much like science uses the scientific method to prove that something is true, geometry uses formal proof to prove that things are true. Let's first learn a little about how real trials use proofs to find the truth.
At first, you will be exposed to one-step proofs that only require a careful look at a figure, then stating the relationship that you see between the items that you are hoping to prove alike. A few important theorems necessary for understanding follow.
•
- Linear Pair Theorem If two angles form a linear pair, then they are supplementary.
- Congruent Supplements Theorem- If two angles are supplementary to the same angle (or to two congruent angles), then
the two angles are congruent.
- Right Angle Congruence Theorem- All right angles are congruent.
- Congruent Complements Theorem- If two angles are complementary to the same angle (or to two congruent angles),
then the two angles are congruent.
A proof is an argument where logic, definitions, properties, and previously proven statements are used to demonstrate that a
conjecture is always true. There are several types of proofs in geometry, but a two-column proof is most commonly seen.
The process of writing a proof is the same for all types of proofs. A proof is based on a conjecture. The conjecture contains a
hypothesis (statements providing information that can be assumed to be true) and a conclusion (the statement that is to be
proven true). In a proof, deductive reasoning is used to write a list of connected steps beginning with the hypothesis and ending
with the conclusion.
Each step in a proof contains two parts: a statement and a justification, or reason. "Given" is the reason for statements in the
hypothesis. The reason for all other statements in the proof must be either a theorem, postulate, property, or definition.
The given statements are typically listed at the start of a proof. After the given statements are made in a proof, each subsequent
statement is first a conclusion based on a preceding step (or steps), and then that conclusion becomes a hypothesis for conclusions
made in later steps. In other words, once a statement is justified, that statement can be used in the hypothesis for a later
conclusion. The final statement in a proof is the original conjecture's conclusion. Once the conjecture's conclusion is stated
and justified using at least one previous statement, the conjecture has been proven.
If a figure is not given with the conjecture, it is often helpful to draw a figure and use markings, such as tick marks, to show
the given information. Mark only the given information on the figure; do not mark the information from the statement to be
proven.
Lesson 9
~Aniket Matharasi