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

When a transversal crosses two parallel lines, use the angles’ positions, not how the picture looks.

• Corresponding angles are congruent: they have the same relative position at each intersection.
• Alternate interior or exterior angles are congruent: they lie on opposite sides of the transversal.
• Same-side interior angles are supplementary: they add to $180^\circ$.
• Vertical angles are congruent: they are opposite angles at the same intersection.

Worked example

Two parallel lines are crossed by a transversal. At the top intersection, the lower-right interior angle measures

$$3x+15.$$

At the bottom intersection, the upper-right interior angle measures

$$5x+5.$$

Find $x$ and both angle measures.

Step 1: Identify the relationship.

Both angles are between the parallel lines, so they are interior. They are also on the same side of the transversal. Therefore, they are same-side interior angles and must be supplementary:

$$(3x+15)+(5x+5)=180.$$

Step 2: Solve the equation.

Combine like terms:

$$8x+20=180$$

Subtract $20$:

$$8x=160$$

Divide by $8$:

$$x=20.$$

Step 3: Find each angle measure.

Substitute $x=20$ into each expression:

$$3(20)+15=75^\circ$$

and

$$5(20)+5=105^\circ.$$

Step 4: Check your answer.

Same-side interior angles should add to $180^\circ$:

$$75^\circ+105^\circ=180^\circ.$$

The answer is consistent with the lines being parallel. Remember: angle relationships come from the angles’ positions and the parallel-line markings, not from the diagram’s appearance.