Lines, angles, and triangles: use vertical, complementary and supplementary angles, parallel lines cut by a transversal, the triangle angle sum, and similar and congruent triangles to find unknowns.

What this skill is asking

This skill is the angle and triangle geometry of the Digital SAT (Geometry and Trigonometry domain): angle relationships at points and along lines, the angles formed when a transversal crosses parallel lines, the triangle angle sum, and similar and congruent triangles. These questions are about setting up an equation from a geometric fact and solving it.

The angle facts

A handful of relationships generate almost every answer.

A worked parallel-lines question

Parallel-line questions chain a couple of facts.

Similar triangles and proportion

Similar triangles have the same angle measures but possibly different sizes, so their corresponding sides are proportional. This is one of the most useful SAT geometry ideas because it converts a geometry question into a proportion. If two triangles are similar with a pair of corresponding sides $6$ and $9$ (a scale factor of $\frac{9}{6} = \frac{3}{2}$), then every side of the larger triangle is $\frac{3}{2}$ times the matching side of the smaller. Look for similarity when a figure has a smaller triangle nested in a larger one (sharing an angle) or when parallel lines create equal corresponding angles; then write the side ratio and solve. The angle-angle criterion (two equal angles force similarity) is the usual trigger.

Isosceles and special triangles

The SAT also leans on the isosceles triangle fact: a triangle with two equal sides has two equal base angles, and vice versa. Combined with the angle sum, this fills in unknown angles quickly: an isosceles triangle with a $40^\circ$ apex has base angles $\frac{180 - 40}{2} = 70^\circ$ each. An equilateral triangle has three $60^\circ$ angles. These facts pair naturally with the angle relationships above, so a typical question gives a figure with parallel lines or an isosceles triangle and asks for one missing angle, which you reach by applying two or three facts in sequence.