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SAT Geometry: Lines, Angles, and Triangles

Angle relationships, Pythagorean triples worth memorizing, special right triangles (45-45-90 and 30-60-90), and the similar-triangle setup that solves most SAT geometry word problems.

9 phút TEKS GEO SAT Math

Triangles are the most-tested geometry topic on the Digital SAT — usually 2–3 questions on angle sums, similar triangles, and special right triangles. Memorize a handful of patterns and these are nearly free points.

Angle relationships

Five facts that solve 90% of angle problems
  • Straight line = 180°
  • Triangle sum = 180°
  • Quadrilateral sum = 360°
  • Vertical angles are equal (the X-pattern)
  • Parallel lines + transversal: corresponding and alternate-interior angles are equal
a b c d a = c (corresponding) b = c (alternate interior)
When a transversal cuts two parallel lines, eight angles form just two distinct values (sum to 180°).

Triangle types

  • Equilateral: all sides equal, all angles 60°
  • Isosceles: two equal sides; base angles equal
  • Right: one 90° angle; Pythagorean applies
  • Scalene: all sides different

Pythagorean theorem

a² + b² = c²    (c = hypotenuse, opposite the right angle) Only works for right triangles. The hypotenuse is always the longest side.

Pythagorean triples — memorize

These show up constantly. If you see two of the numbers, the third is free.

3 — 4 — 5 (and scalars: 6-8-10, 9-12-15, 12-16-20) 5 — 12 — 13 8 — 15 — 17 7 — 24 — 25 Skipping the Pythagorean computation when you spot a triple saves 30+ seconds.

Special right triangles

1 1 √2 45-45-90 √3 1 2 30-60-90
Two right-triangle shapes the SAT uses repeatedly. The reference sheet has both — but recognition is faster.
45-45-90: sides in ratio 1 : 1 : √2 30-60-90: sides in ratio 1 : √3 : 2 (short : long-leg : hypotenuse) The side opposite 30° is the short side; opposite 60° is √3 times that; opposite 90° is 2 times the short.

Similar triangles

Two triangles are similar when their angles match — sides are then in proportion. SAT signals similarity with parallel lines, shared angles, or the magic phrase "AA similarity."

Triangle ABC ~ Triangle DEF (similar) AB/DE = BC/EF = AC/DF (sides proportional) Find a side: set up the proportion and cross-multiply Always match corresponding vertices in the same order on both sides.

Area formulas

Triangle: A = ½ · base · height Rectangle: A = length · width Parallelogram: A = base · height (perpendicular) Trapezoid: A = ½ · (b₁ + b₂) · h All on the Bluebook reference sheet — but knowing them saves a tab-switch.

Common mistakes

  • Using Pythagorean on a non-right triangle
  • Confusing the legs and the hypotenuse in 30-60-90
  • Matching non-corresponding vertices in similar triangles
  • Using slant height instead of perpendicular height for area

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