To solve a triangle means to find all the angles and all the side lengths.
This section summarizes information related to solving triangles,
and presents ‘safe’ solution approaches for all triangle types.
Complete discussions are offered in the sections listed below; consult them as needed.
NotationCapital letters are used to denote angles/vertices.Corresponding lowercase letters are used to denote lengths of opposite sides. Thus:


If you just need to solve a triangle reliably, here's an easytouse
triangle solver.
It can also be used to check your work.
GOOD HABIT:
When solving triangles, select tools that use exact values, whenever possible. For example, suppose you're solving the right triangle shown at right. Green numbers are given (exact) values ($\,\color{green}{2}\,$ and $\,\color{green}{39^\circ}\,$). Black numbers are computed exact values ($\,51^\circ\,$). Red numbers are computed approximate values ($\,\color{red}{2.574}\,$). To compute $\,b\,,$ you should use $$ \cssId{s97}{\tan 39^\circ = \frac{b}{2}}\qquad \cssId{s98}{\text{ or }} \qquad \cssId{s99}{\tan 51^\circ = \frac{2}{b}} $$ since they use only exact values. You should not use $$ \cssId{s102}{\cos 51^\circ = \frac{b}{\color{red}{2.574}}}\qquad \cssId{s103}{\text{ or }}\qquad \cssId{s104}{\sin 39^\circ = \frac{b}{\color{red}{2.574}}}\qquad \cssId{s105}{\text{ or }}\qquad \cssId{s106}{b^2 + 2^2 = (\color{red}{2.574})^2} $$ since they use computed approximate values, which introduce more computational error. GOOD HABIT: Use the full accuracy of your calculator, whenever possible. For example, if you've computed $\,2.574...\,$ and must use it later on, then use your calculator features (e.g., memory, recent values) to use all the calculator digits. Don't just type in the four digits $\,2.574\,$! 

Be on the lookout for these special triangles.
When you recognize them, you should be able to fill in values from memory (no computations).
$\,30^\circ\,$$\,60^\circ\,$$\,90^\circ\,$ triangle

$\,45^\circ\,$$\,45^\circ\,$$\,90^\circ\,$ triangle hypotenuse is $\,\sqrt{2}\,$ times the length of the leg 
The following example illustrates what can go wrong when solving triangles, if good habits are not followed.
Consider a triangle with sides of length $\,3\,,$ $\,4\,$ and $\,6\,.$ Solve the triangle. Faulty Solution:

An application of just one of the two recommended ‘good habits’ would
have prevented the impossible triangle.
Here, both good habits are applied: Correct Solution:
Note: At left, the corrections would be: $$\begin{gather} \cssId{sb56}{F = 180^\circ  \arcsin(0.8886) \approx 117.3^\circ}\cr\cr \cssId{sb57}{E = 180^\circ  26.38^\circ  117.3^\circ = 36.32^\circ} \end{gather} $$ 
CONGRUENCE THEOREMS A unique triangle is determined by each of these configurations: 

SAS sideangleside (two sides and an included angle) 
SSS sidesideside (three sides) 
AAS/SAA angleangleside (or, sideangleangle) (two angles and a nonincluded side) 
ASA anglesideangle (two angles and an included side) 
If $$\begin{gather} \cssId{sb65}{e > 0}\cr \cssId{sb66}{f > 0}\cr \cssId{sb67}{0^\circ < D < 180^\circ} \end{gather} $$ then there exists a unique triangle having the two sides and included angle. 
Name the sides so that: $$\cssId{sb73}{0 < f \le e \le d}$$ Then, if $$\cssId{sb75}{d < e + f}$$ the triangle exists and is unique. That is: the longest side (if there is one) must be strictly less than the sum of the two other sides. 
If $$ \begin{gather} \cssId{sb84}{0^\circ < D < 180^\circ}\cr \cssId{sb85}{0^\circ < E < 180^\circ}\cr \cssId{sb86}{D + E < 180^\circ}\cr \cssId{sb87}{d > 0} \end{gather} $$ then there exists a unique triangle having the two angles and nonincluded side. Note: Two angles uniquely determine the third. Three angles uniquely define the shape. One side uniquely defines the size. 
If $$ \begin{gather} \cssId{sb97}{0^\circ < D < 180^\circ}\cr \cssId{sb98}{0^\circ < E < 180^\circ}\cr \cssId{sb99}{D + E < 180^\circ}\cr \cssId{sb100}{f > 0} \end{gather} $$ then there exists a unique triangle having the two angles and included side. Note: Two angles uniquely determine the third. Three angles uniquely define the shape. One side uniquely defines the size. 
‘SAFE’ TRIANGLESOLVING STRATEGIES By using the guidelines here, you will avoid a mistake leading to an ‘impossible’ triangle. When it is safe to do so, the Law of Sines is preferred over the Law of Cosines, since it requires fewer computations. Once two angles are known, find the third angle using the fact that the angles sum to $\,180^\circ\,.$ 



There is no danger in using the Law of Sines when the unknown is a side. 
There is no danger in using the Law of Sines when the unknown is a side. 
OTHER TRIANGLE CONFIGURATIONS  


Each step is illustrated with this example: Suppose a triangle has a $\,50^\circ\,$ angle opposite a side of length $\,6\,.$ A second side has length $\,7\,.$ Solve the triangle. Start by defining notation: Let $\,E\,$ be the angle opposite the side of length $\,7\,.$ Let $\,F,\,f\,$ be the remaining unknown angle and opposite side. 

JUST BE CAREFUL 
SOLVE A QUADRATIC EQUATION 
Compare the results.
Different approaches may yield slightly different results. Greater accuracy at intermediate steps will help to minimize differences. 

On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. Although only a few decimal places are displayed in problem solutions, additional accuracy is used in intermediate calculations. 
PROBLEM TYPES:
