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Math · 30 · Geometry and Trigonometry

Triangles

Notes

Key points

  • Angles sum to 180∘180^\circ — on the reference sheet.
  • Exterior angle = sum of the two remote interior angles.
  • Isosceles: equal sides face equal angles. Equilateral: all angles 60°.
  • Largest angle faces the longest side; smallest faces the shortest.
  • Triangle inequality: each side < sum of the other two, > their difference.
  • Area 12bh\tfrac{1}{2}bh: height is ⟂ to the base, can fall outside.

Formulas

Angle sumA+B+C=180∘A+B+C=180^\circ
Exterior angled=a+bd=a+ba, b: the two non-adjacent interior angles
AreaA=12bhA=\dfrac{1}{2}bhOn the reference sheet
Triangle inequality∣a−b∣<c<a+b|a-b|<c<a+b
Equilateral areaA=34s2A=\dfrac{\sqrt{3}}{4}s^2Not on the sheet; from 30-60-90

Common traps

  • Using a slanted side as the height in 12bh\tfrac{1}{2}bh.
  • Assuming isosceles from the picture — it needs tick marks, equal angles or a statement.
  • Allowing c=a+bc=a+b — that is a flat line, not a triangle.

SAT tip

Third-side questions: the length lies strictly between the difference and the sum of the other two.

Test yourself

10 official SAT questions on Triangles, easiest first.

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