2. Plane triangle formulae
A plane triangle, Figure C2, has six parts: three angles and three sides.

Figure C2 
Individual angles may be acute (<90°), right (=90°), or obtuse (>90°) with their sum exactly 180°.
To geometrically define a triangle requires three parts, including at least one side, be fixed. Why a side? Because fixing angles alone does not constrain the size of the triangle. The two triangles in Figure C3 are different sizes even though they have identical angles.
Figure C3 
Equations for triangle trigonometry are:
Angle condition: 

Equation C1 
Law of Sines: 

Equation C2 
Law of Cosines: 

Equation C3 
A right triangle is a special case where one angle is exactly 90°. Applying the Law of Cosines, Equation C3, to a right triangle, Figure C4:
Figure C4 
Because cos(90°) = 0 The Law of Cosines becomes the Pythagorean Theorem. 
Depending on which parts of it are known the area of a triangle can be determined using one of two equations:
Using two sides and an included angle: 
Equation C4 

Using three sides: 
Equation C5 

where:  Equation C6 