Two rays that share an endpoint make an angle. The shared endpoint is the vertex. What does the size of an angle measure? Not the length of its sides: the amount of turn between them.
Key idea
Imagine one ray turning around the vertex until it lands on the other ray. A full turn is $360^\circ$, so one degree is $\dfrac{1}{360}$ of a full turn. A quarter turn is $90^\circ$, a half turn is $180^\circ$, and an angle that turns through 50 one-degree angles measures $50^\circ$.
Why it works
To measure, you need a unit, just like inches for length. The unit for turning is the one-degree angle: split a full circle into 360 equal slices, and each slice is 1 degree. Measuring an angle means counting how many of those slices fit inside it. Making the rays longer does not add any slices.
Worked example: A fraction of a turn
How many degrees is $\dfrac{1}{3}$ of a full turn?
- A full turn is $360^\circ$.
- Split it into 3 equal parts: $360 \div 3 = 120$.
Answer: $120^\circ$
Worked example: A clock hand
The minute hand turns from the 12 to the 3. How many degrees does it turn?
- The 12 numbers split a full turn into 12 parts: $360 \div 12 = 30^\circ$ each.
- From 12 to 3 is 3 parts: $3 \times 30 = 90$.
Answer: $90^\circ$ (a quarter turn)
Common mistake
Thinking longer sides make a bigger angle. The size of an angle is how far it opens, not how long its rays are drawn. You could extend both rays forever and the turn would stay the same.