Sometimes the angles in a diagram are labeled with expressions like $(2x + 10)^\circ$. The angle facts give you an equation, and you know how to solve it.
Key idea
Use the angle fact to write an equation: the parts of a right angle add to 90, angles on a line add to 180, angles around a point add to 360, and vertical angles are equal. Solve for $x$, then substitute to find each angle.
Why it works
The labels describe the sizes of real angles. If two angles make a straight line, their sizes add to $180^\circ$ no matter how they are written, so the expressions must add to 180. That sentence is the equation.
Worked examples
Worked example: On a straight line
Two angles on a line measure $(2x + 10)^\circ$ and $(3x + 20)^\circ$. Find $x$ and both angles.
- They add to $180$: $2x + 10 + 3x + 20 = 180$.
- Combine like terms: $5x + 30 = 180$. Subtract 30: $5x = 150$. Divide: $x = 30$.
- Angles: $2(30) + 10 = 70^\circ$ and $3(30) + 20 = 110^\circ$. Check: $70 + 110 = 180$ ✓.
Answer: $x = 30$; the angles are $70^\circ$ and $110^\circ$
Worked example: In a right angle
A right angle is split into $x^\circ$ and $(2x + 15)^\circ$. Find $x$.
- The parts add to $90$: $x + 2x + 15 = 90$.
- $3x + 15 = 90$, so $3x = 75$ and $x = 25$.
- Check: $25 + (2 \cdot 25 + 15) = 25 + 65 = 90$ ✓.
Answer: $x = 25$
Worked example: Vertical angles
Vertical angles measure $(4x - 10)^\circ$ and $70^\circ$. Find $x$.
- Vertical angles are equal: $4x - 10 = 70$.
- Add 10: $4x = 80$. Divide: $x = 20$.
Answer: $x = 20$
Common mistake
Stopping at $x$ when the question asks for an angle. In the first example, $x = 30$ but the angles are $70^\circ$ and $110^\circ$. Reread the question, and check that your angles fit the fact you used.