Plane Geometry, Plane Figures (Geometric Figures) - Triangles

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