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Plane Geometry - Plane Figures (Geometric
Figures) - Triangles |
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Triangles |
Types of triangles
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Main
properties of triangles
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Congruence of triangles
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Theorems about congruence
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Similarity
of triangles, division of a line segment in a given ratio |
Similarity criteria of triangles
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Area of a triangle
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Types of triangles
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Types of triangles categorized by their sides
are; a scalene triangle, isosceles triangle and
equilateral triangle. |
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The types of triangles categorized by their angles
are; an acute triangle, obtuse triangle and
right triangle. |
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Main
properties of triangles
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1) A sum of triangle angles
a +
b
+ g
= 180°. |
2)
Angles lying
opposite the equal sides are also equal, and inversely. |
All angles in
an equilateral triangle are also equal. It follows, that each angle in an
equilateral triangle is equal to 60 degrees. |
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3) In any triangle, if one
side is extended, the exterior angle is equal to a sum of interior
angles, not supplementary.
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The
sum of exterior angles is 360°. |
4)
Any side of a triangle is less than a sum of two other sides and
greater than their difference. |
5)
An angle lying opposite the greatest side, is also the greatest
angle, and inversely. |
Proof:
By turning the side BC,
of the scalene triangle ABC
below, around the vertex C
by the angle g
into
direction of the side AC,
obtained is the isosceles triangle BCD
with equal angles on the base BD.
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Its
lateral side CD
< AC.
Angle a <
b', as b'
is the exterior angle of the triangle ABD,
and b'
< b,
therefore
a <
b. Thus, proved is the above
statement.
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Congruence of triangles
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Two
figures are called congruent if they have identical size and shape, i.e., if their corresponding angles and sides are equal. |
The
two congruent figures fit on top of each other exactly. We prove the
congruence of two figures by rotation and translation. |
Theorems about congruence of triangles are; Two triangles are
congruent: |
1) If a pair of corresponding sides and the included angle
are equal SAS (Side-Angle-Side). |
2) If their corresponding sides
are equal SSS. |
3) If a pair of corresponding angles and the included
side are equal ASA. |
The congruence of two triangles we denote as
D ABC
@
D A'B'C'. |
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Similarity of triangles,
division of a line segment in a given ratio
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Two plane figures are similar if differ in scale not in shape. |
Two
polygons are similar, if their angles are equal and sides are proportional. |
Similarity criteria of triangles are; Two triangles are
similar: |
1) If all their corresponding angles are
equal. |
2) If all their
sides are proportional. |
3) If one angle of a triangle is congruent to
one angle of another triangle and the sides that include those angles are proportional. |
Similarity of
two triangles is denoted as D
ABC ~
D A'B'C'. |
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Division of a line segment
AB to
equal parts: Example:
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Division of the line segment
AB
in a given ratio: Example:
AC
: BC
= 1 : 2 |
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Area of a triangle
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The area
A
of any triangle is equal to one-half the product of any base and corresponding height
h. |
A height or
altitude of a triangle is a straight line through a vertex and perpendicular to the opposite
side. |
This opposite side is called the
base of the altitude, and the point where the altitude intersects the base (or its
extension) is called
the foot of the altitude. |
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