PSAT Geometry — angles, congruence and similarity

College Board's Geometry and Trigonometry domain for the PSAT/NMSQT includes lines, angles and triangles, and its description is specific: determine congruence, similarity and sufficiency using concepts and theorems about vertical angles, triangles and parallel lines cut by a transversal. This material practices those theorems one at a time, with every figure described in words so that the question can be answered without a drawing, and with the numbers set so that the algebra is quick and the reasoning is what counts.

The twelve quiz questions start where two lines cross: vertical angles set equal to find x, and then the adjacent angle, which is the step students skip. Two questions use parallel lines cut by a transversal, one with alternate interior angles, which are equal, and one with same-side interior angles, which add to 180 degrees; in each, the option built from the other rule is waiting. Triangle facts follow: angles in the ratio one to two to three, the base angles of an isosceles triangle, an equilateral triangle whose sides are written as expressions, and the exterior angle theorem.

Then sufficiency. One question lists four sets of three facts about two triangles and asks which does NOT guarantee congruence; the answer is two sides with an angle that is not between them. Another gives two angles of each of two triangles and asks whether they are similar, which needs the third angle before the match is visible. A segment parallel to one side of a triangle splits the other two sides in proportion, a shadow problem sets up similar triangles from the sun's angle, and a thirty-sixty-ninety triangle closes the set using the side ratios printed on the test's reference sheet.

The flashcards hold each angle relationship, the triangle theorems, the congruence rules and why side-side-angle fails, angle-angle similarity, scale factor, the side-splitter theorem and both special right triangles.

The printable written work has eight open problems: all four angles at an intersection, three angles named and measured around parallel lines, an isosceles triangle with algebraic base angles, an exterior angle with algebra, a written explanation of why side-side-angle fails with a worked pair of different triangles, similarity and two lengths from a parallel segment, a flagpole's height, and a square's diagonal and an equilateral triangle's height in exact form.

The oral exam asks one question at a time and wants each step named: which angle relationship, which rule, which scale factor.

Every figure is invented for practice, and nothing is taken from any published test.

• Use vertical angles and linear pairs to find unknown angles, including with algebra
• Apply corresponding, alternate interior and same-side interior angle relationships for parallel lines
• Use the triangle angle sum, isosceles and equilateral triangle facts and the exterior angle theorem
• Decide whether given information is sufficient for congruence, and explain why side-side-angle is not
• Establish similarity by angle-angle and use the scale factor to find lengths
• Use a segment parallel to a side of a triangle to set up proportions
• Solve shadow problems with similar triangles
• Use the 30-60-90 and 45-45-90 side ratios from the reference sheet

Practice material written by Zestly, based on the PSAT/NMSQT Math skill "Lines, angles, and triangles" as described in College Board's Fall 2026 PSAT/NMSQT Student Guide and the Assessment Framework for the Digital SAT Suite (version 3.01, August 2024). All figures are invented. Zestly is not affiliated with College Board.