All articles / SAT Math / Geometry & Trig
Lines, Angles & Triangles for the SAT: Complete Study Guide + Free Practice Problems
Everything the SAT tests about lines, angles, and triangles in one place: parallel lines and transversals, angle pairs, the triangle angle sum, the exterior angle theorem, triangle side relationships, and similar triangles — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupBasic angle relationships
Before tackling parallel lines and triangles, three fundamental angle pair relationships come up constantly on the SAT.
| Angle pair | Relationship |
|---|---|
| Complementary angles | Sum to 90° |
| Supplementary angles | Sum to 180° |
| Vertical angles | Angles across from each other at an intersection — always equal |
A straight line always measures 180°. Any time you see angles along a straight line, their measures must sum to 180° — this single fact unlocks most basic angle problems.
Parallel lines & transversals
When a transversal crosses two parallel lines, it creates eight angles that fall into predictable equal and supplementary groups.
| Angle pair type | Relationship |
|---|---|
| Corresponding angles | Equal — same position at each intersection |
| Alternate interior angles | Equal — between the parallel lines, on opposite sides of the transversal |
| Alternate exterior angles | Equal — outside the parallel lines, on opposite sides of the transversal |
| Co-interior (same-side interior) angles | Supplementary — between the parallel lines, same side of the transversal |
Two parallel lines are cut by a transversal. One angle measures 65°. Find the measure of its co-interior angle.
These angle relationships only hold when the two lines are actually parallel. If a figure doesn't state or mark the lines as parallel, don't assume corresponding or alternate angles are equal.
The triangle angle sum
A triangle has angles of 42° and 79°. Find the third angle.
In an isosceles triangle, the two base angles opposite the equal sides are also equal to each other — combine this fact with the 180° angle sum to solve for all three angles from limited information.
The exterior angle theorem
An exterior angle of a triangle is formed by extending one side. The exterior angle theorem gives a fast shortcut without needing the full 180° sum.
A triangle has two interior angles of 50° and 65°. Find the exterior angle at the third vertex.
Triangle side relationships
Triangle inequality
The sum of any two sides must be greater than the third side, or the triangle can't exist.
Larger side ↔ larger angle
The longest side is always opposite the largest angle; the shortest side is opposite the smallest angle.
Isosceles & equilateral
Equal sides mean equal opposite angles. An equilateral triangle has all sides and all angles equal (60° each).
A triangle has sides of length 5 and 9. Which of the following could NOT be the length of the third side: 4, 6, 13?
The triangle inequality must hold strictly — the sum of two sides must be greater than, not equal to, the third side. Equal to the third side collapses the triangle into a straight line, which isn't a valid triangle.
Similar triangles
Similar triangles have the same angle measures and proportional side lengths. If you can show two triangles are similar, every pair of corresponding sides shares the same ratio.
Triangle ABC is similar to triangle DEF. AB = 6, DE = 9, and BC = 8. Find EF.
Correctly matching corresponding vertices is the most important step in a similar triangles problem — the order of the letters in triangle names (ABC ~ DEF) tells you exactly which vertices and sides correspond.
Common SAT traps
- Assuming lines are parallel without confirmation: transversal angle relationships only apply when the lines are actually parallel.
- Mismatching corresponding sides in similar triangles: always match vertices in the order given by the triangle names.
- Forgetting the strict inequality in the triangle inequality theorem: the sum of two sides must be strictly greater than the third.
- Assuming a figure is drawn to scale: SAT figures are not always drawn to scale unless stated — rely on given angle and side values, not visual appearance.
Test day strategy for lines, angles & triangles
| Question signal | Fastest approach |
|---|---|
| Parallel lines with a transversal | Identify the angle pair type first (corresponding, alternate, co-interior) |
| Two angles of a triangle given | Subtract their sum from 180° for the third |
| Exterior angle of a triangle | Add the two non-adjacent interior angles directly |
| Missing side length, triangle given | Check the triangle inequality range before finalizing an answer |
| "Similar to" language between two triangles | Set up a proportion using correctly matched corresponding sides |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Lines, Angles & Triangles 1
Parallel lines, transversals, and angle pairs.
Start quiz → Quiz 2Lines, Angles & Triangles 2
Triangle angle sum, side relationships, and similarity.
Start quiz →