Use angle relationships (vertical angles; angles formed by parallel lines and a transversal). | Lines, angles, and triangles | SAT Math

Vertical, corresponding, alternate interior, and same-side interior angles are the key line-angle relationships.

Core Idea

When two lines cross, opposite (vertical) angles are equal. When a transversal cuts parallel lines, alternate interior angles are equal, corresponding angles are equal, and co-interior (same-side interior) angles are supplementary.

Understanding

Rule: Vertical angles are equal, and parallel lines create two more equal-or-supplementary angle families.

Vertical angles are equal.
Alternate interior and corresponding angles are equal.
Same-side interior angles add to $180 \circ$ .

Match the angle pair first, then write the equation.

Step by Step

Identify whether the lines are parallel (the problem will state this or mark it).
Find the angle pair: are they vertical, corresponding, alternate interior, or co-interior?
Apply the rule: equal for vertical/corresponding/alternate interior, supplementary for co-interior.
Solve the resulting equation.

Misconceptions

Treating co-interior (same-side interior) angles as equal instead of supplementary.
Assuming lines are parallel when the problem doesn't state or mark it.
Confusing alternate interior angles with alternate exterior angles — they follow the same equal rule, but you must identify the correct pair.

Question

Worked Example

Two parallel lines are cut by a transversal. One of the alternate interior angles measures $(3 𝑥 + 15) °$ and the other measures $(5 𝑥 - 1) °$ . What is the value of $𝑥$ ?

Select an answer to see the explanation