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Math · 32 · Geometry and Trigonometry

Similarity and Congruence

Notes

Key points

  • Similar: equal angles, proportional sides. Same shape, any size.
  • AA is enough to prove similarity.
  • Congruent: same shape and size — SSS, SAS, ASA, AAS. Never SSA or AAA.
  • Scale factor kk: lengths ×k\times k, areas ×k2\times k^2, volumes ×k3\times k^3.
  • Match vertices in order: △ABC ~ △DEF means A↔D, B↔E, C↔F.
  • A line parallel to one side cuts off a smaller similar triangle.
  • Similar right triangles share trig ratios — the bridge to SOH-CAH-TOA.

Formulas

Proportional sidesABDE=BCEF=ACDF=k\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{AC}{DF}=k
Area ratioA2A1=k2\dfrac{A_2}{A_1}=k^2
Volume ratioV2V1=k3\dfrac{V_2}{V_1}=k^3
Parallel to a sideDE∥BC ⇒ ADAB=AEAC=DEBCDE\parallel BC\ \Rightarrow\ \dfrac{AD}{AB}=\dfrac{AE}{AC}=\dfrac{DE}{BC}D on AB, E on AC

Common traps

  • Scaling area by kk: double the sides and the area is ×4\times 4.
  • Pairing sides by position on the page instead of by the equal angles.
  • In the parallel-line setup, using a segment piece where the whole side belongs.

SAT tip

Redraw the two triangles separately in the same orientation, then write one proportion of matching sides.

Test yourself

10 official SAT questions on Similarity and Congruence, easiest first.

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