Euclid was a Greek mathematician in Alexandria about three hundred years before Christ.
He wrote a book called “Elements” about the geometry of lines and angles on a plane.
Well, actually it was a textbook.
Possibly one of the “greatest mathematical textbooks of all time”, it has had a major
influence on education for a very, very long time.
For hundreds of years at a time, this textbook was used as the basis of Mathematics education—or
even all education.
Indeed, the basic ideas are still being taught in schools around the world to this day.
Euclidean geometry is as much about logic and problem solving as it is about geometry.
The Australian Curriculum calls it “geometric reasoning”.
You start with some basic principles about the relationship between lines and angles
in certain configurations, and you reason your way from there towards determining other
relationships.
For example, using the information in this diagram, you might try to work out the size
of this angle over here.
Again, it’s not just about the geometric content, although that is useful knowledge
in itself, especially in more advanced mathematics where these fundamental ideas are used to
solve more complex problems.
But it’s really about how to reason your way logically through a complex problem.
“If I know this and this are true, well, what else must be true?”
“If I need to determine that, how can I get there?
What could I work out that would then allow me to go that next step?”
I guarantee that if you practice solving problems with Euclidean geometry, you will get better
at problem solving in general.
Before we start, there are some mathematical terms you need to know.
We’ll use these words to describe relationships between different angles.
Congruent means exactly the same.
Complementary means angles that add up to ninety degrees in total.
And supplementary means they add up to one hundred and eighty degrees.
Congruent means exactly the same; complementary means adding up to ninety degrees; and supplementary
means adding to one hundred and eighty degrees.
Alright?
Let’s begin.
Each rule we learn, each geometric description, has a name and a symbol.
The coded symbol is used when we’re solving problems, so we can explain what we’re doing
without having to write lots of long words all the time.
The first two rules here are just describing the sizes of right angles and straight angles.
Angles in a right angle are complementary; they add up to ninety degrees.
And angles on a straight line are supplementary; they add to one hundred and eighty degrees.
Sounds obvious, but that’s the thing about Euclid’s “Elements”.
He starts with very simple, basic ideas and then builds them up carefully, logically,
step by step into a grand design that allows you to solve some quite complex situations
without having to go back to first principles all the time.
Angles at a point all add up to three hundred and sixty degrees.
That again is just a simple consequence of our definition of angle sizes in degrees.
A complete rotation makes three hundred and sixty degrees, so obviously that’s the sum
we’ll get by adding up the sizes of all these angles.
Vertically opposite angles are congruent.
Where two lines intersect at a point, the angles opposite each other at that point must
be the same size; they’re congruent.
Let’s try using those rules to solve some simple geometric problems.
Angle a and forty degrees are on a straight line, so they must add up to one hundred and
eighty degrees.
So a must be one hundred and forty degrees.
Notice the coded symbol to indicate how I arrived at my answer.
Now for b, I have a choice: a and b are supplementary, because they’re on a straight line, but
notice that b is also vertically opposite that forty degrees.
That’s actually going to be easier: vertically opposite angles are equal, so b must be forty
degrees as well.
You can calculate b either way, but it’s more reliable to use information in the question
if you can, rather than something you calculated yourself.
That’s why it’s important to give the coded reason each time you work something
out.
It’s analogous to showing your working in an algebraic problem.
It’s part of communicating your reasoning to other people.
That’s part of what Euclidean geometry is about: getting from what you know to a solution
with a logical, iron-clad explanation for why it’s true.
If you show your working and reasoning, someone else can double-check your work and you can
be sure you’ve got it right.
All of these angles are at a point, so they all add up to three hundred and sixty degrees.
These three add to two hundred and fifty-two, so c must be one hundred and eight degrees.
(Just subtract two hundred and fifty-two from three hundred and sixty.)
These two angles are inside a right angle, so they’re complementary: they add to ninety
degrees.
So d must be sixty-six degrees.
Last set.
e is on a straight line with these other two, so what would it have to be to make them all
add up to one hundred and eighty degrees?
e must be fifty degrees.
And then f can be solved either way, just like b earlier.
It’s got to be thirty-five degrees.