Skill: Apply angle relationships formed by parallel lines and a transversal

When two parallel lines are crossed by a transversal, their angle relationships help you find unknown measures.

• Corresponding angles are congruent: they have equal measures.
• Alternate interior angles are congruent.
• Alternate exterior angles are congruent.
• Same-side interior angles are supplementary, so their measures add to $180^\circ$.
• Adjacent angles in a linear pair are also supplementary.

These relationships work because the lines are parallel.

Example: Two parallel lines are cut by a transversal. One interior angle measures $70^\circ$. The alternate interior angle is labeled $(3x+10)^\circ$. Find $x$.

1. The two angles are alternate interior angles, so they are congruent.

$$3x+10=70$$

2. Subtract $10$ from both sides:

$$3x=60$$

3. Divide both sides by $3$:

$$x=20$$

4. Check the angle measure:

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

So, $\boxed{x=20}$. The unknown angle measures $\boxed{70^\circ}$ because alternate interior angles formed by parallel lines are equal.