Honors Geometry Unit 2 Notes

Definitions, Axioms, and Theorems

This is the beginning of our study of geometry. Geometry is one of the oldest branches of mathematics and is still studied for its historical significance as well as its applicability. Approximately 2000 years ago, a man named Euclid compiled the worlds knowledge of geometry and number theory into a series of thirteen books called the Elements that organized this knowledge into a new format. Each idea was either a definition, an word that we assigned a meaning to, a postulate (or axiom) or an idea that is considered simple enough that it can be accepted without proof and as a premise for building arguments, or theorems, statements that must be proven in order to be accepted.

Examples

• We can define a line as that which has length but no breadth, so that we all have the same understanding what we mean when we say "line."
• We can postulate that between any two points a line can be drawn.
• We can prove the Pythagorean Theorem.

Importance

This process provides formality to mathematics and allows us to show that the procedures we are following are true in all cases. It removed doubt of the validity of solutions.

Day 0

We will begin by looking at some algebraic properties to which you are probably more familiar.

• Reflexive Property
• Symmetric Property
• Transitive Property of Equality
• Associative Property of Addition/Multiplication
• Commutative Property of Addition/Multiplication
• Distributive Property
• Additive Property of Equality
• Multiplicative Property of Equality
• Additive Identity
• Multiplicative Identity
• Additive Inverse
• Multiplicative Inverse
  • What number does not have a multiplicative inverse?

These handful of properties allow us to solve a large number of algebraic problems and guarantee correctness.

$x+3=5$
$2x=8$
$4x-7=9$
$2(x+5)=14$

Proof

In order to prove a statement, for each action taken in the proof we must provide a justification. For example, when solving an equation like $x+4=9$ we know $x+4+(-4)=9+(-4)$ is an equivalent equation, but knowing is not sufficient and we must justify this action by referring to the Additive Property of Equality.

Day 1

To continue discussing our introduction to proofs, we need to establish a common idea of the types of numbers we can work with.

Counting

When you first learned to count, you initially began with 1, 2, 3, 4, and could continue indefinitely. However, what if you had no objects?

Gaps

This is sufficient in many situations, but what if you bought a pizza for dinner? With the numbers you currently know, you would either have to eat the entire pizza or none of it.

More Gaps

We know a lot of numbers now, but we still cannot represent all numbers. For instance, if you drew a square 1 meter by 1 meter, how long is the diagonal?

Gaps, again

We know have no more gaps in the numbers we are used to working with, but have we discovered every type of number we need?

Sets

These numbers will be divided into sets, or collections of objects, that have similar properties.

• Natural Numbers
• Whole Numbers
• Integers
• Rational Numbers
• Irrational Numbers
• Real Numbers
• Complex Numbers

Day 2

Now that we know more about types of numbers, lets link this with something we already know. How can we define an even number?

Once you have the idea that it is a multiple of 2, we can represent any even number by 2n, with what condition on n? What about for an odd integer?

• Prove: The sum of an even integer and an odd integer is odd.
• Prove: The sum of two odd integers is even.
• Prove: The sum of two even integers is even.
• Prove: The product of two odd integers is odd.
• Prove: The product of two even integers is even.
• Prove: The product of an even integer and an odd integer is even.
• Prove: The sum of a number and its opposite is zero.
• Prove: The product of a number and its reciprocal is one.
• Find the general form of the solution to equations in the family of functions $x+a=b$.
• Find the general form of the solution to equations in the family of functions $ax=b$.
• Find the general form of the solution to equations in the family of functions $ax+b=c$.
• Prove: The solution to any two step equation of the form $ax+b=c$ is $x=\frac{c-b}{a}$. Use the two previous proofs.

        Geometric

•  Prove: Vertical angles are congruent.

•  Prove: Alternate interior angles are congruent, given the Corresponding Angles Postulate.
•  Prove: Alternate exterior angles are congruent, given the Corresponding Angles Postulate.
•  Prove: Consecutive interior angles are supplementary, given the Corresponding Angles Postulate.
•  Prove: Consecutive exterior angles are supplementary, given the Corresponding Angles Postulate.
•  Prove: The sum of the measures of the interior angles of a triangle is $180^{\circ }$.

Day 5

All of these ideas apply regardless of the statement we are trying to prove. We have focused on the more familiar topic of algebra and related properties, introduced basic ideas of logic, and now are ready to apply these to some introductory geometric ideas.