Skill: Apply angle relationships formed by intersecting lines

When two lines intersect, they form four angles.

• Vertical angles are opposite each other, so they are equal:

$$m\angle 1=m\angle 3$$

• Adjacent angles share a side. If their other sides form a straight line, they make a linear pair and add to $180^\circ$:

$$m\angle 1+m\angle 2=180^\circ$$

Example

Two lines intersect. The upper-left angle measures $2x+20^\circ$, and the upper-right angle measures $4x+10^\circ$.

These two angles are adjacent and form a linear pair, so their measures add to $180^\circ$:

$$(2x+20)+(4x+10)=180$$

Combine like terms:

$$6x+30=180$$

Subtract $30$ from both sides:

$$6x=150$$

Divide by $6$:

$$x=25$$

Now substitute $25$ into each angle expression:

Upper-left angle:

$$2(25)+20=70^\circ$$

Upper-right angle:

$$4(25)+10=110^\circ$$

The angle opposite the $70^\circ$ angle is a vertical angle, so it also measures $70^\circ$. The angle opposite the $110^\circ$ angle also measures $110^\circ$.

So the four angles are:

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

Remember: use equal when angles are vertical, and use sum to $180^\circ$ when adjacent angles form a linear pair.