Congruent triangles and corresponding parts, the SSS, SAS, ASA, AAS and HL congruence shortcuts, why SSA and AAA don't work, and CPCTC in proofs.
Congruent triangles
Two triangles are congruent if one can be moved (slid, turned or flipped) to fit exactly on the other. Then all six pairs of corresponding parts are congruent: three pairs of sides and three pairs of angles.
The order of the letters matters. \(\triangle ABC \cong \triangle DEF\) means \(A\) matches \(D\),\(B\) matches \(E\) and \(C\) matches \(F\). So \(\overline{AB} \cong \overline{DE}\), \(\angle C \cong \angle F\), and so on.
Five shortcuts
You don't need to check all six pairs. These combinations of three are enough.
SSS explains why triangles are used in bridges and roof trusses: a triangle with fixed side lengths can't change shape. A rectangle made of four rods, by contrast, can be pushed into a slanted parallelogram.
CPCTC
Once you've shown two triangles are congruent, every pair of corresponding parts is congruent. In proofs this reason is abbreviated CPCTC: "corresponding parts of congruent triangles are congruent". It's how congruent triangles help you prove that two segments or two angles are equal.
Practice
Quick check
Which of these is not a valid way to prove two triangles congruent?
SSA can lead to two different triangles. HL is the special case that works for right triangles.
Quick check
\(\triangle JKL \cong \triangle MNO\). Which segment is congruent to \(\overline{KL}\)?
\(K\) matches \(N\) and \(L\) matches \(O\), so \(\overline{KL}\) corresponds to \(\overline{NO}\).
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