
Plane Geometry  Plane Figures (Geometric
Figures)  Triangles 



Triangles 
Types of triangles

Main
properties of triangles

Congruence of triangles

Theorems about congruence

Similarity
of triangles, division of a line segment in a given ratio 
Similarity criteria of triangles

Area of a triangle








Types of triangles

Types of triangles categorized by their sides
are; a scalene triangle, isosceles triangle and
equilateral triangle. 


The types of triangles categorized by their angles
are; an acute triangle, obtuse triangle and
right triangle. 


Main
properties of triangles

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. 



3) In any triangle, if one
side is extended, the exterior angle is equal to a sum of interior
angles, not supplementary.

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.

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.




Congruence of triangles

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 (SideAngleSide). 
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'. 

Similarity of triangles,
division of a line segment in a given ratio

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'. 



Division of a line segment
AB to
equal parts: Example:



Division of the line segment
AB
in a given ratio: Example:
AC
: BC
= 1 : 2 


Area of a triangle

The area
A
of any triangle is equal to onehalf 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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