Skill: Apply vertical angle relationships

When two lines intersect, they form two pairs of vertical angles.

• Vertical angles are opposite each other, so they have equal measures.
• Adjacent angles share a side. If they form a straight line, they make a linear pair and add to $180^\circ$.

Suppose two lines intersect at point $O$. The opposite angles are labeled:

• $\angle AOC = 3x+10$
• $\angle BOD = 5x-30$

Because $\angle AOC$ and $\angle BOD$ are vertical angles, set their measures equal:

$$3x+10=5x-30$$

Solve for $x$:

$$40=2x$$ $$x=20$$

Now substitute $20$ into either angle expression:

$$\angle AOC=3(20)+10=70^\circ$$

So the vertical angle $\angle BOD$ is also:

$$\angle BOD=70^\circ$$

If $\angle AOD$ is next to $\angle AOC$, they form a linear pair. Therefore, they add to $180^\circ$:

$$\angle AOD+70^\circ=180^\circ$$ $$\angle AOD=110^\circ$$

Remember: use equal measures for vertical angles and use a sum of $180^\circ$ for adjacent angles that form a straight line.