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Geometry & Trig

SAT angles: parallel lines, transversals, and triangle rules

When a line crosses two parallel lines, eight angles appear, but they take only two values. Explore an interactive diagram, then use the triangle rules that solve most angle questions.

SPSAT Practice Editorial7 min read

The short version

What you'll take away

  • When a transversal crosses parallel lines, every angle equals either x or 180° − x.
  • Corresponding, alternate interior, and vertical angles are equal; same-side interior angles add to 180°.
  • A triangle’s angles add to 180°, and an exterior angle equals the sum of the two remote interior angles.
In this article

Eight angles, two values

When a line, called a transversal, crosses two parallel lines, it creates eight angles. Because the lines are parallel, the angles take only two values: an angle x and its supplement, 180° − x. Change the angle and highlight different pairs to see how they match.

Try it yourself · interactive

Parallel lines cut by a transversal

Corresponding angles equal: 60° = 60°

Blue values equal the transversal angle and amber values equal its supplement. Same-side interior angles always add to 180°.

The angle-pair rules

Two of these rules hold for any pair of intersecting lines: vertical angles are equal, and angles that form a straight line add to 180°. The others depend on the lines being parallel.

See the skill

Which pairs are equal?

Vertical angles

Equal

Across from each other at one intersection.

Linear pair

Add to 180°

Side by side on a straight line.

Corresponding / alternate interior

Equal

Same position at each intersection, or opposite sides between the lines.

Same-side interior

Add to 180°

Between the parallel lines, on the same side of the transversal.

Use the visual as a reasoning aid: tie each answer choice to the rule, relationship, or evidence it depends on.

Triangle rules

The three angles of a triangle add to 180°. An exterior angle, formed by extending one side, equals the sum of the two interior angles that are not next to it. In an isosceles triangle, the angles opposite the equal sides are equal.
Similar triangles have equal corresponding angles, and two pairs of equal angles are enough to prove similarity. That is why parallel lines inside a triangle often create a smaller, similar triangle.

Your turn

Quick practice

Two parallel lines are cut by a transversal. Two alternate interior angles measure (3x + 10)° and (5x − 30)°. What is the value of x?

Hint: Are alternate interior angles equal or supplementary?

Choose one answer

Practice: try these before checking the answers

Name the angle relationship before calculating.
  • 1. One angle of a linear pair measures 115°. What is the other?
  • 2. Two angles of a triangle measure 50° and 60°. What is the third?
  • 3. The two remote interior angles of an exterior angle measure 40° and 75°. What is the exterior angle?
  • 4. An isosceles triangle has a vertex angle of 40°. What are its base angles?
  • 5. Same-side interior angles measure 2x° and 3x°. Find x.

Sources & further reading

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