You can sort angles as acute, right or obtuse. A protractor measures them exactly, in degrees. It has two scales, and choosing the right one is the whole trick.
Key idea
Put the protractor's center point on the vertex. Line up the 0 line with one ray. Find the scale that reads 0 on that ray, and follow it to the other ray. Then check: an acute angle must be less than $90^\circ$, an obtuse angle more.
Why it works
Each scale counts degrees from one side. The outer scale starts at 0 on the right; the inner scale starts at 0 on the left. Both are correct, but only one of them starts where your angle starts. The two readings always add to $180^\circ$, so the wrong one is the other part of a straight line.
Worked example: Outer scale
An angle's first ray lies along the 0 on the right. The second ray crosses the outer scale at 70. What is the angle?
- The angle starts at 0 on the outer scale, so read the outer scale.
- It is less than a right angle, so 70 (not 110) makes sense.
Answer: $70^\circ$
Worked example: Inner scale
An angle opens from the left. Its second ray crosses the scales at 40 (inner) and 140 (outer). What is the angle?
- The angle starts at 0 on the left, which is the inner scale.
- The angle looks acute, so $40^\circ$ makes sense.
Answer: $40^\circ$
Common mistake
Reading the wrong scale, so an acute angle is reported as $130^\circ$. Always decide first whether the angle is acute or obtuse, then pick the reading that agrees.