1. ENZO EXPOSYTO
MATHS
SYMBOLS
TRIANGLES - FIRST PROPERTIES
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2. TRIANGLES
-
FIRST PROPERTIES
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3.
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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
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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|
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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
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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
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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 …
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equilateral
acute
isosceles
right

33. SUM
of the INTERIOR
ANGLES
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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).
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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°
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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°
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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
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47. EXTERIOR ANGLES THEOREM - 2
EXAMPLE
γ exterior = α + β
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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°
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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
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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/
…
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