Skill: Apply angle relationships formed by intersecting lines

When two lines intersect, use the positions of the angles:

• Vertical angles are opposite each other, so they have equal measures.
• Adjacent angles share a side. If their other sides form a straight line, they are a linear pair, so their measures add to $180^\circ$.
• If the lines are perpendicular, each angle is a right angle, or $90^\circ$.

Example

Two lines intersect at point $O$. One angle measures $70^\circ$. The angle next to it is labeled $x+20^\circ$. Find $x$ and the measures of all four angles.

The $70^\circ$ angle and the $(x+20)^\circ$ angle are adjacent and form a linear pair. Therefore,

$$70+(x+20)=180$$

Combine like terms:

$$x+90=180$$

Subtract $90$ from both sides:

$$x=90$$

Now find the angle labeled $x+20$:

$$x+20=90+20=110^\circ$$

The two angles are $70^\circ$ and $110^\circ$. Their vertical angles have the same measures:

• The angle opposite $70^\circ$ is also $70^\circ$.
• The angle opposite $110^\circ$ is also $110^\circ$.

So the four angles are

$$70^\circ,\ 110^\circ,\ 70^\circ,\ 110^\circ.$$

Remember: opposite angles are equal, but adjacent angles are usually not equal; adjacent angles in a linear pair add to $180^\circ$.