Proving two triangles congruent
Congruence proofIn triangle ABC, M is the midpoint of BC and AM ⊥ BC. Prove that △ABM ≅ △ACM.
- BM ≅ CM. M is the midpoint of BC, and a midpoint divides a segment into two congruent segments (definition of midpoint).
- ∠AMB ≅ ∠AMC. AM ⊥ BC, so both angles are right angles, and all right angles are congruent.
- AM ≅ AM. A segment is congruent to itself (reflexive property).
- △ABM ≅ △ACM by SAS. Two sides and the included angle of one triangle are congruent to the corresponding parts of the other (steps 1–3).
△ABM ≅ △ACM by the SAS congruence postulate — and by CPCTC it follows that AB ≅ AC, so △ABC is isosceles.
Common mistake the solver flags: citing SSA, which is not a valid congruence criterion — the right angle here is included between the two known sides, which is why SAS applies.