• How can triangles be divided and what relationships develop from doing so

Isosceles triangles can be divided in half to make right triangles. once the triangle is divided we can find its area.
once we have the are of one triangle we can multiply it by two to get the full triangle. 

Triangle Relationships and How can they be used

 

Isosceles triangles are triangles that have two congruent sides and two congruent angles adjacent to the sides

Equilateral triangles all congruent angles and sides all of their angle have to be 60 degrees so that they add up to 180 degrees. the incenter orthocenter circumcenter and centroid are all at the same piont

Why are circles so amazing?

Great link for further learning http://windowseat.ca/circles/


Just the Facts

• The equation for finding the area of a circle is pie* radius squared
• A circle has an infinite amount of lines of symmetry
  
• They are the only one-sided shape with an area
• A straight line is a circle with an infinite area
  
• interior intersecting chords are proportional to each other

Also THIS!

Which is actually THIS! 

Circles can also be used for non mathematical things like music.

Triangle Congruence and Understanding Polygons

At A Glance:

• the area for a triangle is 1/2B*H
• Large polygons can be divided into triangles

within this square, if we know the hypotenuse of one, we an find the length of the diagonal


if you make triangles out of the diagonals, you can easily find the area. if we know the base length, we can use that to find the area of on triangle, than we would just have to multiply that by eight.

The properties of Right Triangles

At a Glance

A 30-60-90 Triangle has a hypotenuse that is twice as long as the short leg and the longer leg is always three times the square root of the length. with this knowledge we can find out all of the leg lengths by knowing just one.



in 45-45-90 the hypotenuse is equal to the leg length times the square root of two.you can also find all of the length with just one known length.  triangles the legs

The similarities and differences of quadrilaterals

At a Glance

• all quadrilaterals interior angles add up to 360 degrees 
• all quadrilaterals are coplanar
• all quadrilaterals have two diagonals
• all quadrilaterals have four sides

here are the most common quadrilaterals:

                               

A Rectangle                     A Parallelogram                               A Square        

                          

A Rhombus                 A Trapezoid                           and don't forget the Kite!

Here is a helpful table from math.com with the properties and differences of these quadrilaterals

Quadrilaterals show up in everyday life

quadrilaterals can be found in everyday life more than we think

They can be found in street signs


They can be found in foods like rice crispy treats


Or Toast!



Quadrilaterals are also found in architecture




quadrilaterals can even be found in kites!

 


Quadrilaterals can be found everywhere!

PART 2!

a quick citation note*

information and pictures have been accessed through Mr. Skafidas's amazingly wonderful power-points

dveloping and expanding concepts

we can develop and expand on concepts that we have learned already by using algebra and new conditional statements.

parallel lines and planes

lines within a coordenate plane

you can describe lines within a coordinate plane with line segments.

The Fundamentals of geometry

we describe our world geometrically using planes, angles, lines, segments, points and rays. here are there definitions.

Euclidean gometry and its reationship to paralell lines

Our definition of parallel lines greatly effects our understanding of euclidean geometry. Euclidean geometry allows us to define things using certain postulates and others as theorems. this allows us to better identify lines intersecting parallel lines.

formal logic

formal logic intersects with geometry in more ways than we would think. when making a proof in mathematics, formal logic and common knowledge are your two greatest tools. the idea that P->Q if Q-P is the best example of formal logic. this type of logic intersects with geometry in a big way.

Congruent Triangles

When you say triangles are congruent, you are saying that all 6 pairs of corresponding angles and sides are congruent.  here is an example:

inductive and deductive reasoning

Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties.

Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. 

These types of reasoning are appropriate when making and verifying conjectures. A conjecture is a statement that has not been proven true but also has not been proven false. Inductive reasoning helps us make conjectures. Deductive reasoning helps us verify conjectures

defining triangles!

when defining triangles there are a couple main types. here are some examples

Proofs

Proofs!!!!!

A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. proofs are important for finding a justifiable way to prove your argument

there are several was that we can develop and present effective arguments and proofs. 
  The most common type of proofs are table proofs. table proof are the most straightforward way to explain your process. here is an example.













another example of a proof is a paragraph proof. paragraph proofs give the same information as a table proof but in a paragraph instead of a chart. here is an example

the third type of proof is the flowchart proof. the flowchart proof still provides an effective argument but puts the information into a flowchart. here is an example