Angle rules
In a nutshell
There are four rules you have to know about angles. You can apply these rules to solve problems involving missing angles.
The four angle rules
Here are the four rules you need to know.
RULE  DIAGRAM 
Angles in a triangle add up to $180\degree$  $\alpha+\beta+\gamma=180\degree$

Angles on a straight line add up to $180\degree$  
Angles in a quadrilateral add up to $360\degree$  $\alpha+\beta+\gamma+\delta=360\degree$ 
Angles around a point add up to $360\degree$  $\alpha+\beta+\gamma+\delta=360\degree$

Note: A quadrilateral is another name for a shape with 4 sides.
Missing angle problems
To solve problems involving missing angles, look at the question and see which rule applies to it. You may have to label unknown angles with $x$ and use algebra to solve for them.
Example
In the triangle below, the value of $\beta$ is $60\degree$. The angle $\gamma$ is $20\degree$ more than the angle $\alpha$. What is the size of the angle $\alpha$?
This question concerns angles in a triangle, so use the fact that the angles in a triangle add up to $180\degree$.
$\alpha+\beta+\gamma=180$
$\alpha+60+\gamma=180$
$\alpha+\gamma=120$
The question also says that the angle $\gamma$ is $20\degree$ more than the angle $\alpha$. Algebraically, this means that:
$\gamma=20+\alpha$
Use this new information together with the first equation to solve for $\alpha$:
$\alpha+\gamma=120$
$\alpha+(20+\alpha)=120$
$2\alpha+20=120$
$2\alpha=100$
$\alpha=50$
The size of the angle $\alpha$ is $\underline{50\degree}$.