In your solving toolbox (along with your pen, paper and calculator) you have these 3 equations:
1. The angles always add to 180°:
A + B + C = 180°
When you know two angles you can find the third.
2. Law of Sines (the Sine Rule):
Law of Sines
When there is an angle opposite a side, this equation comes to the rescue.
Note: angle A is opposite side a, B is opposite b, and C is opposite c.
3. Law of Cosines (the Cosine Rule):
Law of Cosines
This is the hardest to use (and remember) but it is sometimes needed
to get you out of difficult situations.
It is an enhanced version of the Pythagoras Theorem that works
on any triangle.
With those three equations you can solve any triangle (if it can be solved at all).
Six Different Types (More Detail)
There are SIX different types of puzzles you may need to solve. Get familiar with them:
1. AAA:
This means we are given all three angles of a triangle, but no sides.
AAA Triangle
AAA triangles are impossible to solve further since there are is nothing to show us size ... we know the shape but not how big it is.
We need to know at least one side to go further. See Solving "AAA" Triangles .
2. AAS
This mean we are given two angles of a triangle and one side, which is not the side adjacent to the two given angles.
AAS Triangle
Such a triangle can be solved by using Angles of a Triangle to find the other angle, and The Law of Sines to find each of the other two sides. See Solving "AAS" Triangles.
3. ASA
This means we are given two angles of a triangle and one side, which is the side adjacent to the two given angles.
ASA Triangle
In this case we find the third angle by using Angles of a Triangle, then use The Law of Sines to find each of the other two sides. See Solving "ASA" Triangles .
4. SAS
This means we are given two sides and the included angle.
SAS Triangle
For this type of triangle, we must use The Law of Cosines first to calculate the third side of the triangle; then we can use The Law of Sines to find one of the other two angles, and finally use Angles of a Triangle to find the last angle. See Solving "SAS" Triangles .
5. SSA
This means we are given two sides and one angle that is not the included angle.
SSA Triangle
In this case, use The Law of Sines first to find either one of the other two angles, then use Angles of a Triangle to find the third angle, then The Law of Sines again to find the final side. See Solving "SSA" Triangles .
6. SSS
This means we are given all three sides of a triangle, but no angles.
SSS Triangle
In this case, we have no choice. We must use The Law of Cosines first to find any one of the three angles, then we can use The Law of Sines (or use The Law of Cosines again) to find a second angle, and finally Angles of a Triangle to find the third angle.
where s is the side.