Congruence & geometric proof
The idea
Two figures are congruent when a sequence of rigid motions — translations, rotations, reflections — carries one exactly onto the other, forcing all matching sides and angles to be equal. For triangles you never need all six matches: the shortcut criteria SSS, SAS, ASA, and AAS each guarantee congruence from just three well-chosen pieces. A geometric proof is a chain of justified statements — every claim cites a given fact, a definition, or a previously proven result — so the conclusion inherits certainty from its premises.
Build proofs from both ends at once: ask which criterion could finish the job (SAS needs two sides and the included angle), then hunt the diagram for free information — vertical angles, shared sides, midpoints, angles made by parallel lines. Once congruence is established, the principle that corresponding parts of congruent triangles are congruent (CPCTC) releases the remaining equalities. The misconception to bury is SSA: two sides and a non-included angle do NOT force congruence, because the free angle lets the third side swing into two different triangles. The included angle is what locks the shape.
Worked example
Segments AB and CD bisect each other at point M. Prove that triangle AMC is congruent to triangle BMD, and conclude that AC = BD.
- Plan before writing: both triangles meet at M, so aim for two sides and the included angle at M — an SAS setup.
- Since M bisects AB, the two halves are equal: AM = MB. Since M bisects CD, the same definition gives CM = MD.
- Angles AMC and BMD are vertical angles formed by the two crossing segments, so angle AMC = angle BMD — and in each triangle this angle sits between the two sides already matched.
- Two sides and the included angle of triangle AMC equal the corresponding parts of triangle BMD, so triangle AMC ≅ triangle BMD by SAS.
- By CPCTC, the corresponding sides AC and BD are equal. The rigid motion behind the proof is concrete: rotating triangle AMC by 180° about M lands it exactly on triangle BMD.
Answer. Triangle AMC ≅ triangle BMD by SAS, and therefore AC = BD by CPCTC.
Check your understanding
- Why does SAS guarantee congruence while SSA fails, and what picture exposes the failure?
- How does the rigid-motion definition of congruence explain why corresponding parts must match once congruence is proven?
- What free pieces of information do geometric diagrams routinely hide, and how do you train yourself to notice them?
- How do you choose which congruence criterion to aim for before writing the first statement of a proof?
Build the foundations first
Congruence & geometric proof builds on these concepts. If any feel shaky, start there.