Euclidean Geometry
Logic and Problem Solving
Eucildean Logical Thinking Skills for Digital Content Creator Minimal knowledge for logical thinking 2
Hello this is Jonghwa Park
In this lecture we will learn about the minimal knowledge that Euclidean geometric requires
We often emphasize idea as more important than knowledge
You will also feel this as digital content creators
But can we come up with ideas without any knowledge?
We can't
There is no idea made with no knowledge at all
Basic knowledge is needed to create an idea
What we must be cautious of is
being tangled up in the former knowledge and doing only what was done before without anything new
In this lecture, we will learn about the minimal knowledge that Euclidean geometric requires
If you know this, you won't need anything more to make ideas
I hope you can create more ideas more confidently
with the knowledge from this lecture in the future
Euclid's axiom
Euclidean geometry starts from very basic things
and creates mathematical theorems with error-less deductive logic
The most basic starting point of Euclidean geometry is
corresponding angles are identical
This explained in diagram is as followed
If a straight line crosses two parallel lines
angle alpha and beta are corresponding angles
The size of angle alpha and beta are identical
These two angles facing each other are called vertical angles
Their sizes are also identical
This can be proved like following
With the red line, the dotted angle is 180-beta
and with the blue line, the dotted angle and angle alpha's sum would be 180
So 180-beta+alpha is 180
So we can conclude that alpha=beta
Angle alpha's verticle angle and angle beta are alternate interior angles
Angle alpha and beta are identical as corresponding angles
and angle alpha and alpha prime are identical as vertical angles
which makes alternate interior angles angle alpha prime and beta identical as well
Euclidean geometry starts from the fact that corresponding angles
vertical angles, and alternate interior angles are identical
Remembering this and applying it to many situations to solve problems
is Euclidean geometry
We know that the sum of the interior angles of a triangle is 180
Then how can we prove this
When triangle ABC is given as followed
when we draw a straight line L that is parallel to the base BC and crosses point A
the interior angles of the triangle
are identical as alternate interior angles
Like this, we can see that alpha, beta, and gamma are on top of a straight line
Therefore alpha+beta+gamma is 180
Like this, based on what we already know
logically calculating what we don't know
is what math is
From the fact that the sum of interior angles of a triangle is 180
we can think of the following relationship
This relationship appears very often in problem solving
which is also called the external angle theorem
It would be good to remember this and apply it in problem solving
One vertical angle's external angle is identical to the sum of other two vertical angles' internal angles
It would be good to understand this with problems
Problem 1, two straight lines are parallel
What is the size of angle X?
We can solve this with what we've just learned
Let's see
Just with the fact that corresponding angles and alternate interior angles are identical in parallel lines
we must understand the relationship between the straight lines and angles to solve this problem
Let's think of the given angles divided
For other angles as well, we can draw a parallel line
and divide the angles
We can calculate the divided angles
and eventually get the size of X
As you can see, the size of angle X is 44
Problem 2, there is quadrilateral ABCD between two parallel straight lines E and F
The angles of same color are of identical sizes
What is the size of angle D?
Let's say the angle between A and C is angle AB
A line parallel with straight lines E and F
and that passes point B is drawn
From the fact that two alternate interior angles are identical
we can conclude that a+b is 150
The sum of the internal angles of a quadrilateral is 360
Therefore a+b+150+angle D is 360
Therefore angle D is 60
The sum of internal angles of a quadrilateral is 360
Because all quadrilaterals can be divided into two triangles
and since the sum of internal angles of a triangle is 180
the sum of internal angles of a quadrilateral is 180X2, 360
Problem 3, what is the sum of all five colored angles?
You might try solving this problem yourself first and then look at the solution
The sum of internal angles of a triangle is 180
And the external angle of an angle
is identical to the sum of two other angles' internal angles
This was the external angle theorem
The given shape can be divided into a few triangles
As you can see
you can use the external angle theorem with two triangles
Like this, the five angles
are the same as on triangle's internal angle
Since the sum of internal angles of a triangle is 180
the sum of the five angles are also 180
How were these problems
Did you solve them well?
Or were they harder than you expected?
There are very basic necessary knowledge
and using them many things are solved
But without the basic knowledge, everything just feels difficult
Content creation is also the same
It is creation of new things but there are basic necessary things
I hope you look for what you need
like the Euclidean axioms, what the
basic knowledge needed are
Proportion
The beginning of math starts from Thales who lived 600 years BC
The reason why Thales is considered the first mathematician
is because he gathered mathematical knowledge and taught people
Thales' idea applied in math
is that shapes that are similar keep the same proportion
Even if the sizes are different, when there is a small triangle
if we expand that into a larger one keeping the proportions
the angles and the ratio of the length of the sides remain the same
Thales was who solved many problems with this simple idea
For example, if this rectangle was expanded keeping the proportion
the rectangle that had the width of 6 and height of 2
will become 12 wide and 4 tall
It increased in a certain ratio
This is applied to all shapes
and it goes the same for the simplest shape, triangle
Two similar triangles have maintain a certain ratio for the length of their sides
Remember what you see in the screen
It's very obvious in a way
Similar triangles maintain a certain ratio
This was Thales' idea
and this became the key in Euclidean geometric problem solving
There is a key idea in problem solving
It would also be good for you
digital content creators, to create and find
a key idea for your creation
When content consumers experience content
around that idea, they will understand and enjoy the content easier
We'll solve problems to see how useful Thales' idea is
Problem 4, find the area of the colored rectangle
At first sight you might be confused
and think we only have information about what we don't need while there is no information on what do need
How can I solve this problem?
You might think this way
But people born in 600 BC
learned math from Thales
and will have thought 'this is an easy one'
You can try as well
In the problem, you can see that for triangles A and B
sizes of all three angles are identical
because they are corresponding angles
We say that A and B are similar
Using the fact that A and B are similar
we can think of the rate of similar
7:a = b:12
Therefore by using the rate of similar
ab=7 X 12
84
The area of the rectangle we are looking for is a, b
so the answer is 84
To add on to the calculation
from a:b=c:d
to b x c = a x d
In the following proportionality, the product of the inner two are the same as the outer two
This is how it was done
This is a calculation of the following fraction expression of proportions
a / b = d / c
This is calculated as
b x c = a x d
Problem 5, what is the area of the colored square?
In this problem as well, finding similar triangles is needed
Let's find similar triangles
We can see that large triangle ABC and
small triangle DEC are similar
If we say the length of a side of the colored square is a
Triangle DEC's height is 10-a
and the base is a
Since triangles ABC and DEC are similar
we can make the following expression
10:15 = 10 - a:a
and we can calculate this
There for the length of a side of the colored square is 6
and the area of the square is 6 squared
36
One most basic thing about triangles is about the area
The area of a triangle is as follows
Base multiplied by height multiplied by 1/2
This is what we saw at lecture 1
This applies the same for not only acute triangles
but also for obtuse triangles with one angle larger than 90
It's good to remember that two triangles with the same base length and height have the same area
That is because we often need to remember this when solving problems
Let's take a look at some more problems
Problem 6, find the area of x
We can think of two similar triangles in the problem
First is saying that we draw a foot of perpendicular from point E to AD and name that H
triangle ADE's area is 1/2 x EH x 8
8
Therefore the length of EH is 2
The second triangle is, drawing foot of perpendicular from point C to AB and calling it J
Triangles AEH and ACJ are similar
Therefore 3:2=11:CJ
The area of triangle ABC is 1/2 x 12 x 22/3
44
Therefore x is 44-8
36 is the answer
Similarity and congruence
Thinking that A and B are similar is an abstract thought
People who lived 600 BC also thought abstractly
and used this to solve specific problems
In many problems we face, there are similarities
and there are also ones that are
of exact same sizes
This is called congruence
We can use the relationship of similarity and congruence very usefully in problem solving
Let's see in problems
Problem 7, find the area of the colored area
The key of this problem is that the two triangles facing each other are similar
and the rate of similar is 2:1 from the base length
Therefore the height is also 2:1
and it's four grids horizontally
so 2:1=4x2/3:4x1/3
Therefore the colored area's area can be calculated as followed
1/2x2x8/3+1/2x1x4/3
This comes out of 10/3
Problem 8, there is trapezoid ABCD
When the areas of the two parts of the four parts are given as followed
find the area of trapezoid ABCD
The key to solving this problem is again finding two similar triangles
Which two triangles are similar?
Triangles AED and BEC are
The two similar triangles' area's rate is given as 1:4
Therefore the ratio of the two triangles' base and height
is all 1:2
Since the area of triangle AED is 1
a x h is 2
The area of trapezoid ABCD can be calculated as followed
1/2 x 3a x 3h = 9
Also, areas of parallelograms and trapezoids can be calculated as shown on the screen
The proof for these formula
are applications of what we've learned
so you could think about it
Try thinking of the triangles inside the parallelogram and trapezoid
Euclidean geometry, an ancient Greek math, began
from understanding and using similarity and congruence
The first mathematician Thales learned a lot of
useful math knowledge going back and forth to Egypt during his trade business
In Egypt, there are giant Pyramids
To the people of that time, Pyramids were more than great architectures
but also a subject of respect
The height of a Pyramid is known to be about 150m
and the length of one side of the base plane is over 200m
To people of Thales' time, 2600 years ago
Pyramids would have been considered an incredibly huge and great architecture
It is said that Egyptians of that time also couldn't calculate the exact height of the Pyramids
They could measure the length of the base plane with a tape measure, but they couldn't for the height
But Thales accurately calculated the height of the Pyramids
using similarity and ratio
What's more important than what you can see is the invisible key idea
I hope you content creators
also find out the invisible key idea needed for you
and use that
Euclid's Axiom The fundamental starting point of Euclidean geometry is the principle that "corresponding angles are equal" In parallel lines, corresponding angles, vertically opposite angles, and alternate angles all have the same measure
Exterior Angle Theorem: An exterior angle equals the sum of the two opposite interior angles To solve problems, one must understand the basic knowledge and apply it to various situations
Proportion Thales' idea applied in math is that "similar objects maintain a certain proportion"
When a shape is expanded, if the angles are maintained the length of bases are also maintained Using similarity and proportion, the length and area can be calculated
Similarity and Congruence Euclidean geometry starts from understanding and applying similarity and congruence
Similarity is two shapes with same shape and when they are shrunk or expanded they become congruent Two identical figures in shape and size, fitting perfectly when overlaid, are called congruent