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
8. Polygonal Chains - 3
polygonal More precisely,
chain a closed polygonal chain
is one in which
the ļ¬rst vertex
coincides
with the last one,
or, alternatively,
the ļ¬rst 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 ļ¬gure
that is bounded
by a ļ¬nite 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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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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24. Types of Angles and Degrees
Right Angle (90Ā°)
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25. Types of Angles and Degrees
Right Angle (90Ā°) and Perpendicularity
perpendicularity sign
... is at 90Ā° (90 degrees) to ...
... forms a right angle with ...
AB CD
a line segment (AB) drawn so that
it forms 2 right angles (90Ā°)
with a line (CD)
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26. Types of Angles and Degrees
Ī± - Acute Angle (less than 90Ā°)
Ī² - Obtuse Angle (greater than 90Ā° and less than 180Ā°)
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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 ā¦
(SIDES) (ANGLES)
Equilateral Acute
Isosceles Right
Scalene Obtuse
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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 180Ā°
(180 DEGREES).
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35. Sum of the Interior Angles
Proof 1 - Step 1
Letās 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
Letās 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 180Ā°
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37. Sum of the Interior Angles
Proof 2
Letās 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 Ī± āinā B
and the angle Ī± which's in A
are congruent:
they have the same measure;
besides,
the angle Ī³ āinā B
and the angle Ī³ 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
Ī± + Ī² + Ī³ = 180Ā°
Enzo Exposyto 37
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 180Ā°.
For example:
a triangle ha 2 angles
with 2 known measures:
70Ā° and 80Ā°.
The measure of the third angle is:
180Ā° - (70Ā° + 80Ā°) = 180Ā° - 150Ā° = 30Ā°
Enzo Exposyto 38
39. Sum of the Interior Angles
Examples
(image from https://www.ck12.org/geometry/triangle-angle-sum-theorem/)
a. EQUILATERAL ACUTE: 60Ā° + 60Ā° + 60Ā° = 180Ā°
b. ISOSCELES RIGHT : 90Ā° + 45Ā° + 45Ā° = 180Ā°
c. SCALENE ACUTE : 70Ā° + 30Ā° + 80Ā° = 180Ā°
d. SCALENE OBTUSE : 25Ā° + 120Ā° + 35Ā° = 180Ā°
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40. Sum of the Interior Angles
In Euclidean Geometry,
the sum of the interior angles
of a triangle
is ALWAYS 180Ā°.
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 180Ā°.
In Elliptic Geometry,
the sum of the interior angles of an elliptic triangle
is greater than 180Ā°.
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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 ļ¬gure, we can see that
Ī± + Ī± exterior = 180Ā°
Ī² + Ī² exterior = 180Ā°
Ī³ + Ī³ exterior = 180Ā°
We get the same result from the 2nd ļ¬gure at page 43
The SUM
of an INTERIOR ANGLE
and ITS EXTERIOR ANGLE
IS ALWAYS 180Ā°
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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
Ī± exterior = Ī² + Ī³
Ī² exterior = Ī± + Ī³
Ī³ exterior = Ī± + Ī²
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48. EXTERIOR ANGLES THEOREM - 3
PROOF - Premise 1
Let's see the images:
if we call a, b, c
respectively
the interior angles Ī±, Ī², Ī³
and A, B, C
the respective exterior angles,
we can write:
Ī³ exterior = C
Therefore,
the thesis
which we must prove
Ī³ exterior = Ī± + Ī²
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
Ī± + Ī² + Ī³ = 180Ā°
becomes
a + b + c = 180Ā°
and, now,
the sum
Ī³ exterior + Ī³ = 180Ā°
becomes
C + c = 180Ā°
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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 = 180Ā°
and
a + b + c = 180Ā°
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 = 180Ā°
we obtain
C = 180Ā° - c
From the sum of the interior angles,
a + b + c = 180Ā°
we get the c value:
c = 180Ā° - (a + b)
If we substitute the c value
in the equality with C, we get
C = 180Ā° - [180Ā° - (a + b)]
Simpliļ¬ed, 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
Ī³ exterior = Ī± + Ī²
we also proved this last thesis.
In similar ways,
we can prove that
A = b +c
namely
Ī± exterior = Ī² + Ī³
and
B = a + c
namely
Ī² exterior = Ī± + Ī³
Enzo Exposyto 52
56. the SUM
of the EXTERIOR ANGLES of a TRIANGLE
IS EQUAL to 360Ā°
PROOF - STEP 2
If we substitute the values of
A, B and C
in the ļ¬rst 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 * 180Ā°
= 360Ā°
Q.E.D.
Enzo Exposyto 56