MATHS SYMBOLS - TRIANGLES - FIRST PROPERTIES - POLYGONAL CHAINS and POLYGONS - DEFINITION - MEASURES of the SIDES - TYPES of TRIANGLES - EQUILATERAL ACUTE - ISOSCELES ACUTE - ISOSCELES RIGHT - ISOSCELES OBTUSE - SCALENE ACUTE - SCALENE RIGHT - SCALENE OBTUSE - SUM of the INTERIOR ANGLES - EXTERIOR ANGLES THEOREM - SUM of the EXTERIOR ANGLES - PROOFS STEP by STEP

1. ENZO EXPOSYTO
MATHS
SYMBOLS
TRIANGLES - FIRST PROPERTIES
Enzo Exposyto 1

2. TRIANGLES
-
FIRST PROPERTIES
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3.
Enzo Exposyto 3

4. 1 - Definition 5
2 - Measures of the Sides 13
3 - Types of Triangles 23
4 - Sum of the Interior Angles 33
5 - Exterior Angles Theorem 41
6 - Sum of the Exterior Angles 53
7 - SitoGraphy 57
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5. DEFINITION
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6. Polygonal Chains - Examples
a simple open polygonal chain
a simple closed polygonal chain
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7. Polygonal Chains - 2
polygonal it is a finite sequence
chain of connected line-segments,
called sides.
This sides are connected
by consecutive points,
called vertices.
For example:
an angle
has
a simple open polygonal chain;
a triangle,
a square, …
have
a simple closed polygonal chain
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8. Polygonal Chains - 3
polygonal More precisely,
chain a closed polygonal chain
is one in which
the first vertex
coincides
with the last one,
or, alternatively,
the first and the last vertices
are connected
by a line segment.
A simple closed polygonal chain
in the plane
is the boundary
of a simple polygon.
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9. Polygons - 1
polygon is a planar figure
that is bounded
by a finite sequence
of straight line-segments
which form
a closed polygonal chain.
Often the term "polygon"
is used in the meaning
of "closed polygonal chain",
but, in some cases,
it's important
to do a clear distinction
between
a polygonal area
and a polygonal chain.
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10. Polygons - 2
polygon Since two line-segments
(triangle) of an angle
always form
a simple open polygonal chain,
are needed, at least,
three line-segments
to have a simple closed polygonal chain
and, then, a polygon.
This type of polygon,
with 3 line-segments,
is called triangle.
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11. Definition - 1
A triangle is a polygon
with THREE SIDES (a, b, c)
and THREE VERTICES (A, B, C).
It is one of the basic shapes
in Geometry.
The symbol ΔABC
represents a triangle
with vertices A, B, C.
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12. Definition - 2
Euclid was a great greek scholar,
author of “The elements”,
written 3 centuries b. C..
In Euclidean Geometry,
any three points,
when non-collinear,
determine an unique triangle
and an unique plane,
i. e.
an Euclidean Plane.
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13. MEASURES
of the
SIDES
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14. Measures of the Sides - 1
a, b, c We can form a triangle
if, and only if,
every sum
of the measures of two sides
is greater than
the measure of the third side.
In other words,
for every
three line-segments
which form a triangle
and whose measures are a, b, c,
it's
a + b > c
a + c > b
b + c > a
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15. Measures of the Sides - Example 1
a, b, c Since a = 4.31
b = 4.55
c = 10.15
and a + b = 4.31 + 4.55 = 8.86
then a + b < c because 8.86 < 10.15
It's impossible
to form a triangle
by line-segments
with a, b, c lengths
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b a
c

16. Measures of the Sides - Example 2
a, b, c Since a = 5.17
b = 4.98
c = 10.15
and a + b = 5.17 + 4.98 = 10.15
then a + b = c because 10.15 = 10.15
It's impossible
to form a triangle
by line-segments
with a, b, c lengths
Enzo Exposyto 16
b a
c

17. Measures of the Sides - 3
a, b, c Besides,
we can form a triangle
if, and only if,
every side has a measure
greater than
the absolute value of the difference
of the measures of the other two sides.
In other words, for every
three line-segments
which form a triangle
and whose measures are a, b, c,
it's
a > |b - c| or a > |c - b|
b > |a - c| or b > |c - a|
c > |a - b| or c > |b - a|
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18. Measures of the Sides - 4
a > |b - c| Let's start from (page 14)
(proof) a + c > b
We can subtract c
from both sides:
a + c - c > b - c
and we get
a > b - c
If b < c
then
b - c will be a negative number,
which hasn't geometric meaning.
So, we must write
a > |b - c|
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19. Measures of the Sides - 5
a > |c - b| If we consider (page 14)
(proof) a + b > c
we can subtract b
from both sides:
a + b - b > c - b
and we have
a > c - b
If c < b
then c - b will be a negative number,
which hasn't geometric meaning.
Thus, we must write
a > |c - b|
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20. Measures of the Sides - 6
a > |b - c| Therefore,
a > |c - b| a > |b - c|
and
a > |c - b|
are both true.
In a similar way,
it's possible to proof that
b > |a - c| or b > |c - a|
c > |a - b| or c > |b - a|
Enzo Exposyto 20

21. Measures of the Sides - Example 3
a, b, c Since a = 4.31
b = 4.55
c = 10.15
and b - c = 4.55 - 10.15 = - 5.60
then a < |b - c| because 4.31 < |-5.60|
It's impossible
to form a triangle
by line-segments
with a, b, c lengths
Enzo Exposyto 21
b a
c

22. Measures of the Sides - Example 4
a, b, c Since a = 5.17
b = 4.98
c = 10.15
and b - c = 4.98 - 10.15 = -5.17
then a = |b - c| because 5.17 = |-5.17|
It's impossible
to form a triangle
by line-segments
with a, b, c lengths
Enzo Exposyto 22
c
ab

23. TYPES
of
TRIANGLES
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24. Types of Angles and Degrees
Right Angle (90°)
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25. Types of Angles and Degrees
Right Angle (90°) and Perpendicularity
perpendicularity sign
... is at 90° (90 degrees) to ...
... forms a right angle with ...
AB CD
a line segment (AB) drawn so that
it forms 2 right angles (90°)
with a line (CD)
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26. Types of Angles and Degrees
α - Acute Angle (less than 90°)
β - Obtuse Angle (greater than 90° and less than 180°)
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27. Identifying Triangles
NAME PROPERTY
EQUILATERAL 3 sides have equal length
ISOSCELES 2 sides have equal length
SCALENE 3 sides have different lengths
ACUTE 3 angles are acute
RIGHT 1 angle is right
OBTUSE 1 angle is obtuse
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28. Grouping Triangles by …
(SIDES) (ANGLES)
Equilateral Acute
Isosceles Right
Scalene Obtuse
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29. Grouping Triangles by
(SIDES)
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30. Grouping Triangles by
(ANGLES)
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31. Grouping Triangles by …
(ACUTE) (RIGHT) (OBTUSE)
Equilateral
Isosceles Isosceles Isosceles
Scalene Scalene Scalene
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32. Grouping Triangles by …
Enzo Exposyto 32
equilateral
acute
isosceles
right

33. SUM
of the INTERIOR
ANGLES
Enzo Exposyto 33

34. Sum of the Interior Angles
In an Euclidean Plane,
the sum of the measures
of the interior angles
of a triangle
is ALWAYS 180°
(180 DEGREES).
Enzo Exposyto 34

35. Sum of the Interior Angles
Proof 1 - Step 1
Let’s see the image.
Let's draw the red line on C
which is parallel to side AB.
Now, we can see that
the angles
with the same colours
are congruent:
this means that
they have
the same measure
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36. Sum of the Interior Angles
Proof 1 - Step 2
Let’s see the image:
- the green colour represents
the measure of the angle C,
- the red colour represents
the measure of the angle A,
- the black colour represents
themeasureoftheangleB.
The three angles together
form, clearly,
a straight angle
and, then, the sum
of their measures
is 180°
Enzo Exposyto 36

37. Sum of the Interior Angles
Proof 2
Let’s see the image.
Let's draw the green line segment on B
which is parallel to side AC.
Now, we can see that
the angle α ‘in’ B
and the angle α which's in A
are congruent:
they have the same measure;
besides,
the angle γ ‘in’ B
and the angle γ that's in C
are congruent:
they have the same measure.
The 3 angles together in B form
a straight angle
and the sum of their measures is
α + β + γ = 180°
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38. Sum of the Interior Angles
If we know the measures
of two angles of a triangle,
we can determine
the measure of the third angle,
subtracting the known measures
from 180°.
For example:
a triangle ha 2 angles
with 2 known measures:
70° and 80°.
The measure of the third angle is:
180° - (70° + 80°) = 180° - 150° = 30°
Enzo Exposyto 38

39. Sum of the Interior Angles
Examples
(image from https://www.ck12.org/geometry/triangle-angle-sum-theorem/)
a. EQUILATERAL ACUTE: 60° + 60° + 60° = 180°
b. ISOSCELES RIGHT : 90° + 45° + 45° = 180°
c. SCALENE ACUTE : 70° + 30° + 80° = 180°
d. SCALENE OBTUSE : 25° + 120° + 35° = 180°
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40. Sum of the Interior Angles
In Euclidean Geometry,
the sum of the interior angles
of a triangle
is ALWAYS 180°.
This is equivalent to
the Euclid's Parallel Postulate.
In Hyperbolic Geometry,
the sum of the interior angles of a hyperbolic triangle
is less than 180°.
In Elliptic Geometry,
the sum of the interior angles of an elliptic triangle
is greater than 180°.
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41. EXTERIOR
ANGLES
THEOREM
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42. INTERIOR and EXTERIOR ANGLES
with TWO SETS of EXTERIOR ANGLES
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43. 1st SET of EXTERIOR ANGLES
1st set
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44. 2nd SET of EXTERIOR ANGLES
exterior angles - 2nd set
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45. SUM of an INTERIOR ANGLE and ITS EXTERIOR
From the last figure, we can see that
α + α exterior = 180°
β + β exterior = 180°
γ + γ exterior = 180°
We get the same result from the 2nd figure at page 43
The SUM
of an INTERIOR ANGLE
and ITS EXTERIOR ANGLE
IS ALWAYS 180°
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46. EXTERIOR ANGLES THEOREM - 1
Besides, we can state the following theorem:
EVERY EXTERIOR ANGLE of a TRIANGLE
ALWAYS EQUALS
the SUM of the OTHER TWO FAR AWAY INTERIOR ANGLES
α exterior = β + γ
β exterior = α + γ
γ exterior = α + β
Enzo Exposyto 46

47. EXTERIOR ANGLES THEOREM - 2
EXAMPLE
γ exterior = α + β
Enzo Exposyto 47
γ = 180° - (α + β)
= 180° - (70° + 50°)
= 180° - 120°
= 60°
γ exterior = α + β
= 70° + 50°
= 120°

48. EXTERIOR ANGLES THEOREM - 3
PROOF - Premise 1
Let's see the images:
if we call a, b, c
respectively
the interior angles α, β, γ
and A, B, C
the respective exterior angles,
we can write:
γ exterior = C
Therefore,
the thesis
which we must prove
γ exterior = α + β
becomes
C = a + b
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49. EXTERIOR ANGLES THEOREM - 4
PROOF - Premise 2
Besides,
how we can see,
the sum
of the interior angles
α + β + γ = 180°
becomes
a + b + c = 180°
and, now,
the sum
γ exterior + γ = 180°
becomes
C + c = 180°
Enzo Exposyto 49

50. EXTERIOR ANGLES THEOREM - 5
PROOF - 1
We must prove that
C = a + b
Since (let's see the image)
C + c = 180°
and
a + b + c = 180°
we get
C + c = a + b + c
If we simplify,
it gives us
C = a + b
Q.E.D.
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51. EXTERIOR ANGLES THEOREM - 6
PROOF - 2
We must prove that
C = a + b
Since this equality is true (let's see the image):
C + c = 180°
we obtain
C = 180° - c
From the sum of the interior angles,
a + b + c = 180°
we get the c value:
c = 180° - (a + b)
If we substitute the c value
in the equality with C, we get
C = 180° - [180° - (a + b)]
Simplified, it becomes
C = a + b
Q.E.D.
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52. EXTERIOR ANGLES THEOREM - 7
PROOF - 3
Since
C = a + b
is equivalent to
γ exterior = α + β
we also proved this last thesis.
In similar ways,
we can prove that
A = b +c
namely
α exterior = β + γ
and
B = a + c
namely
β exterior = α + γ
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53. SUM
of the EXTERIOR
ANGLES
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54. the SUM
of the EXTERIOR ANGLES of a TRIANGLE
IS EQUAL to 360°
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55. the SUM
of the EXTERIOR ANGLES of a TRIANGLE
IS EQUAL to 360°
PROOF - STEP 1
If
the EXTERIOR ANGLES of a TRIANGLE ARE CALLED A, B, C
and
the SUM of these EXTERIOR ANGLES is CALLED S
we get
S = A + B + C
Now (page 52),
A = b + c
B = a + c
C = a + b
where a, b and c
are the corresponding
interior angles
of the exterior angles A, B, C.
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56. the SUM
of the EXTERIOR ANGLES of a TRIANGLE
IS EQUAL to 360°
PROOF - STEP 2
If we substitute the values of
A, B and C
in the first equality, we get:
S = (b + c) + (a + c) + (a + b)
= (a + a) + (b + b) + (c + c)
= 2 * a + 2 * b + 2 * c
= 2 * (a + b + c)
= 2 * 180°
= 360°
Q.E.D.
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57. SitoGraphy
Enzo Exposyto 57

58. https://en.m.wikipedia.org/wiki/Angle
https://en.m.wikipedia.org/wiki/Triangle
http://www.gogeometry.com/problem/p040_geometry_help_theorem.htm
http://www.math-salamanders.com/printable-shapes.html
https://www.ck12.org/geometry/triangle-angle-sum-theorem/
…
Enzo Exposyto 58