
Plane Geometry, Plane Figures (Geometric
Figures)  Triangles 



Similarity and congruence of
triangles use

Congruence 
The
sine law (or the sine rule) and the cosine law 
The
cosine law 
The
cosine law examples 
Calculating angles
of an oblique triangle 
Solving
the oblique triangle  use of the sine law and the cosine law 





Congruence

Two triangles are congruent if they have identical size and shape
so that they can be exactly superimposed. 
Thus, two triangles are
congruent: 
a) if a pair of corresponding sides and the included
angle are equal, 
b) if their corresponding sides are
equal, 
c) if a
pair of corresponding angles and the included side are equal. 

The
sine law
(or the sine rule) and the cosine law 
From the congruence of triangles follows that an oblique triangle is determined by three of its parts, as are 
 two sides and the included angle
(SAS), 
 two angles and the included side
(ASA), 
 three sides
(SSS) and 
 two sides and the angle opposite one of them
(SSA), which does not always determine a unique triangle. 
By using definitions of
trigonometric functions of an acute angle and Pythagoras’ theorem, we can
examine 
mutual relationships of sides and corresponding angles of an oblique triangle. 

The cosine law 
From
the right triangles,
BCD and
ABD
in the figure 




Expressing the same way the squares of the altitudes
as common legs of another two pairs of the right

triangles, obtained are 


As
in every above formula included are all three sides and the angle that correspond to the
side to be

calculated, of the given oblique triangle, it follows that the
cosine law can be used in the cases if known are, 
 two sides and the included angle
(SAS) and 
 three sides
(SSS).


Example:
Given is a triangle with sides, a
= 64 cm, c = 29 cm and
angle b
= 147°,
find the side
b, and 
angles
a and
g. 
Solution: Given
a
= 64 cm, c
= 29 cm and b
= 147°.
b,
a and
g
= ? Applying
the cosine law 


Then,
using the sine law 

a
+ b + g
= 180°,
g
= 180°

(a
+ b)=10°8′20″ 



Calculating angles
of an oblique triangle 
The formulas to calculate angles
of an oblique triangle are
derived from the
cosine law, thus 


Example:
In a triangle
side, a
= 17 cm, b
= 7 cm and c
= 13 cm, find angles
a ,
b and
g. 
Solution: Given
a
= 17 cm, b
= 7 cm and c
= 13 cm. a ,
b and
g
= ? Plug the given values into formulas 

As
a
+ b + g =
180°
then g
= 180°

(a
+ b) =
180°

135°14′30″
= 44°45′30″,
g
=
44°45′30″. 

Solving
the oblique triangle  use of the sine law and the cosine law 











Geometry
and use of trigonometry contents  A 



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