Skill: Determine unknown angles formed by intersecting lines

When two lines intersect:

• Vertical angles are opposite each other, and they have equal measures.
• Adjacent angles are next to each other, and they form a linear pair, so their measures add to $180^\circ$.

Example: Two lines intersect at point $O$. One angle measures $3x+20$ degrees, and an angle next to it measures $5x$ degrees. Find $x$ and the angle measures.

Because the two given angles are adjacent, they form a linear pair:

$$(3x+20)+5x=180$$

Combine like terms:

$$8x+20=180$$

Subtract $20$ from both sides:

$$8x=160$$

Divide by $8$:

$$x=20$$

Now substitute $x=20$ into each angle expression:

$$3x+20=3(20)+20=80^\circ$$ $$5x=5(20)=100^\circ$$

So the two adjacent angles measure $80^\circ$ and $100^\circ$. The vertical angle opposite the $80^\circ$ angle is also $80^\circ$, and the vertical angle opposite the $100^\circ$ angle is also $100^\circ$.

Always decide first whether the unknown angle is opposite a given angle or next to it. Use equal measures for vertical angles and a sum of $180^\circ$ for adjacent angles in a linear pair.