You can measure an angle with a protractor. When a ray splits an angle into two parts, the measures behave like lengths: the parts add up to the whole.
Key idea
If ray BC is inside $\angle ABD$, then $\angle ABC + \angle CBD = \angle ABD$. To find a missing part, subtract the known part from the whole. Remember the wholes you know: a right angle is $90^\circ$ and a straight angle is $180^\circ$.
Why it works
Angle measures count one-degree turns. Turning $35^\circ$ and then $50^\circ$ more in the same direction is $85^\circ$ of turning in all, as long as the two angles sit side by side with no gap and no overlap.
Worked example: Add the parts
$\angle ABC = 35^\circ$ and $\angle CBD = 50^\circ$. Find $\angle ABD$.
- The parts sit side by side, so add.
- $35^\circ + 50^\circ = 85^\circ$.
Answer: $85^\circ$
Worked example: A missing part of a right angle
A right angle is split into two parts. One part is $28^\circ$. What is the other?
- The whole is $90^\circ$.
- $90 - 28 = 62$. Check: $28 + 62 = 90$.
Answer: $62^\circ$
Worked example: On a straight line
Two angles make a straight line. One is $115^\circ$. What is the other?
- A straight angle is $180^\circ$.
- $180 - 115 = 65$.
Answer: $65^\circ$
Common mistake
Adding when you should subtract (or the other way around). Ask which angle is the whole: the whole is the biggest one, made of the others. Missing part = whole minus known part.