First off where did humans get the idea of 360 degrees in a circle ?
What else in our lives has approximately 360 things in it ? Give up ? Days in a year !!
Yep we got our 360 degrees concept from the 365 days in year. What did they do with those extra 5 days ? Party or fiesta, the same way we celebrate New Years and 4th of July. (by they way in the business world there are 30 days in EACH month and 360 days in a year)
So what is half of this 360 degrees circle ? Yep, a line and therefore 180 degree in a line. And what is a quarter of a circle ? (You know North-West or South-East angles). Yep, that is 90 degrees. (and we call those perpendicular lines)
So a straight line has 180 degrees. Makes sense right ?
Now lets say we have 2 parallel lines (see the post on Euclid and parallel lines)
And these two parallel lines are crossed by a third line. (check out picture on side bar)
If two lines are parallel, then the alternate interior angles are equal.
(if line one is parallel to line two, then angle d = angle e)
Ok now you will need your own paper and pencil. Draw a triangle. Then draw a line through the top point of your triangle parallel to the base of the triangle.
Label the angles of the line on top, angle 1, angle 2 (the top angle of the triangle) and angle 3.
Do you agree that angles 1 + 2 + 3 = 180 degrees a straight line ?
(you might need your child or grandchild to help you with this. They have done this several times in school, i hope.)
Now comes the tricky part. Label the two "base angles" inside the triangle on the bottom, angle 4 and 5.
Here comes the magic of Euclid and his proofs. (Don't be scare you already got a 100 for the effort to read all this and wanting to just learn. Remember when we stop learning, we stop growing and we collect moss like a rock.)
So since the lines are parallel, the angle 1 = 4 and the angle 3 = 5 (these are called alternate interior angles like the picture shows on the side bar d = e).
So now in the equation 1 + 2+ 3 = 180, we can replace the 1 with the 4 and the 3 with the 5.
And WHALA, we get angles 4 + 2 + 5 = 180 which are the three angles of the triangle.
(if your picture worked and you got this proof, you deserve an ice cream, well done)
I once saw a 3rd grade teacher take a straw and ask the class how many degrees are there in a straight line. The class yell out "180 !!" and then the teacher folded the staw up into a triangle and asked and how many degrees does this triangle have now ? and the class yell, "180 degrees !!"
Another proof is to cut a Big triangle out of a piece of paper. And label the angles 1,2,3. Then cut the triangle into three pieces. Line up the angles 1, 2, 3 into a straight line. WHALA there is your proof. Angles 1 + 2 + 3 = 180.
Tuesday, February 1, 2011
The Father of Geometry 300 BC
When ever i ask my students, "who is the father of geometry ?", they always yell back, "Mr. Suarez !!". (They know how to get a good grade on a test...lol)
Since we live in Tampa, we go driving south down Dale Mabry (in our heads)...we pass Kennedy Blvd, and then Bay to Bay...and El Prado...Gandy is too far.
Finally some students yells out EUCLID !! and sure enough most cities in America have a Euclid street or avenue (the books says MLK has surpassed Washington and Jefferson as the number one named street in America).
According to Trivial Pursuit, Euclid, Ohio had the first Stop Light in the USA.
So Euclid lived in Alexandria, Egypt. (At this point my students all say, why are these guys all GREEKS, but no one lives in Greece....good question...research that one)
As it turns out Alexander the Great conquered the known world (Western Civilization...Greece to India to Egypt)...and he established the University of Alexandria in Egypt. He brought ALL the known mathematics to Alexandria.
It just happened that Euclid was the math department chair. And he wrote the number TWO best seller of all time called "The Elements". It had 13 chapters and one of the chapters was on Pythagoras. This book today (2,500 years later)is till being taught in our high school geometry class.
So Euclid ranks in the Top 100 most influential men of western civilization.
(the sad part is that Archimedes who was the greatest mathematician before christ, is not mentioned in many lists)....and worse yet, many Geometry students do NOT even know who Euclid or Archimedes were...shame on you math teachers.
So Euclid was a textbook publisher, but more than that he organized formal proofs into a system that could be followed by other students of mathematics.
He also gave us the Parallel Postulate.
This postulate reads like a theorem, but Euclid could NOT proof it, so he had to accept it as postulate (something that we must accept without proof...or something so obvious we accept without proof).
("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
(you got to be kidding....this makes NO sense)...how about
parallel postulate ("through a point outside a line there is exactly one parallel")
that is much better !! (I think ?)
Just look around you, there are parallel lines every where. All the parallel definition states is that through a point NOT on the line, there exist only one line parallel to the original line. It is obvious, right ?
Interesting that even though it sounds like a theorem, Euclid could NOT prove it. (Even with his 13 chapters in his book The Elements.)
Many years later, Einstein would show that parallel lines do NOT exist...OUCH !!
but that is called non-Euclidean geometry and for another blog someday. Einstein was a little different....lol...probably one of the greatest minds of ALL time.
The problem in space is that the gravity of the sun bends the light coming from the stars...and this is an important concept if you are trying to land on the MOON !!
It would not have been "one step for mankind" if the space agency did not have Einstein to tell them that light bends in space....it would have been a BIG step for mankind...and they would have missed the Moon altogether !!! next stop MARS...lol
Since we live in Tampa, we go driving south down Dale Mabry (in our heads)...we pass Kennedy Blvd, and then Bay to Bay...and El Prado...Gandy is too far.
Finally some students yells out EUCLID !! and sure enough most cities in America have a Euclid street or avenue (the books says MLK has surpassed Washington and Jefferson as the number one named street in America).
According to Trivial Pursuit, Euclid, Ohio had the first Stop Light in the USA.
So Euclid lived in Alexandria, Egypt. (At this point my students all say, why are these guys all GREEKS, but no one lives in Greece....good question...research that one)
As it turns out Alexander the Great conquered the known world (Western Civilization...Greece to India to Egypt)...and he established the University of Alexandria in Egypt. He brought ALL the known mathematics to Alexandria.
It just happened that Euclid was the math department chair. And he wrote the number TWO best seller of all time called "The Elements". It had 13 chapters and one of the chapters was on Pythagoras. This book today (2,500 years later)is till being taught in our high school geometry class.
So Euclid ranks in the Top 100 most influential men of western civilization.
(the sad part is that Archimedes who was the greatest mathematician before christ, is not mentioned in many lists)....and worse yet, many Geometry students do NOT even know who Euclid or Archimedes were...shame on you math teachers.
So Euclid was a textbook publisher, but more than that he organized formal proofs into a system that could be followed by other students of mathematics.
He also gave us the Parallel Postulate.
This postulate reads like a theorem, but Euclid could NOT proof it, so he had to accept it as postulate (something that we must accept without proof...or something so obvious we accept without proof).
("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
(you got to be kidding....this makes NO sense)...how about
parallel postulate ("through a point outside a line there is exactly one parallel")
that is much better !! (I think ?)
Just look around you, there are parallel lines every where. All the parallel definition states is that through a point NOT on the line, there exist only one line parallel to the original line. It is obvious, right ?
Interesting that even though it sounds like a theorem, Euclid could NOT prove it. (Even with his 13 chapters in his book The Elements.)
Many years later, Einstein would show that parallel lines do NOT exist...OUCH !!
but that is called non-Euclidean geometry and for another blog someday. Einstein was a little different....lol...probably one of the greatest minds of ALL time.
The problem in space is that the gravity of the sun bends the light coming from the stars...and this is an important concept if you are trying to land on the MOON !!
It would not have been "one step for mankind" if the space agency did not have Einstein to tell them that light bends in space....it would have been a BIG step for mankind...and they would have missed the Moon altogether !!! next stop MARS...lol
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