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Lines, Angles & Triangles for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT 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 a free practice quiz.
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Free · No signupLines, Angles & Triangles
Angle pairs, transversals, triangle angle sum, and similar triangles.
Basic angle relationships
| 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°.
Parallel lines & transversals
| Angle pair type | Relationship |
|---|---|
| Corresponding angles | Equal — same position at each intersection |
| Alternate interior angles | Equal — between the parallel lines, 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 72°. Find the measure of its co-interior angle.
These angle relationships only hold when the two lines are actually parallel. Don't assume angles are equal or supplementary unless the figure states or marks the lines as parallel.
The triangle angle sum
A triangle has angles of 38° and 85°. Find the third angle.
The exterior angle theorem
A triangle has two interior angles of 45° and 70°. 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.
A triangle has sides of length 6 and 10. What is the range of possible values for the third side?
The triangle inequality must hold strictly — the third side can't equal the sum or the difference of the other two sides, only fall strictly between them.
Similar triangles
Similar triangles have the same angle measures and proportional side lengths.
Triangle ABC is similar to triangle DEF. AB = 4, DE = 10, and BC = 6. Find EF.
Correctly matching corresponding vertices is the most important step — the order of the letters in the triangle names (ABC ~ DEF) tells you exactly which sides correspond.
Common PSAT 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 third side must fall strictly between the difference and sum of the other two.
- Assuming a figure is drawn to scale: PSAT figures are not always drawn to scale unless stated.
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
One comprehensive quiz covering the full topic — angle pairs, transversals, the triangle angle sum, and similar triangles.
Lines, Angles & Triangles
Angle pairs, transversals, triangle angle sum, and similar triangles.