Proving triangles congruent
The Given hands you a midpoint, a bisector or two parallel lines. How do you turn that into a proof that two triangles, and then their parts, are congruent?
Where you are 3 inside
What this area is about
A congruence proof is a two-column proof that ends at a line such as \triangle ABC \cong \triangle DEF, or passes through one to reach a side or an angle. The tools are fixed: the SSS, SAS and ASA postulates, the AAS theorem, and HL for right triangles, each needing three matching pairs. The Given almost never lists three pairs, and this area is the craft of finding them.
How these fit together
The core procedure. Planning a congruence proof: start from the Prove, unpack each given into a pair of parts, find the pairs nobody states (a shared side, vertical angles), and choose the criterion they fit. It also separates overlapping triangles sharing a side, an angle or a piece of one, using the reflexive property and segment or angle addition.
Going past the triangles. Proving parts congruent after the triangles: prove the triangles congruent, then claim a matching side or angle, because corresponding parts of congruent triangles are congruent. Its two stage proofs use one congruent pair to supply the missing part for another.
A classic, worked in full. Proving the base angles theorem adds one line from the vertex to split an isosceles triangle into two congruent halves.
A path through it
Planning, then parts after the triangles, then the base angles proof, which uses them.
You need the congruence criteria first. After this, congruent triangles serve proofs about other figures, such as why opposite sides of a parallelogram are congruent.