We need a second triangular part with exactly the same shape and size. Must we measure all three sides and all three angles? No: some sets of three measurements are enough. But “some” is the crucial word.
Two triangles are congruent if they can be made to coincide completely without resizing. The three main criteria are two sides and their included angle, a side and its two adjacent angles, or three sides. The prerequisite is triangle sides and angles.
Match the vertices first
We may translate, rotate and reflect a figure. Equal area or perimeter alone is insufficient. ΔABC≅ΔDEF specifies A↔D, B↔E, C↔F. AB therefore corresponds to DE, BC to EF, and AC to DF. These are the pairs we must compare.
Once congruence is established, all corresponding sides and angles are equal, not just the three measurements checked.
Three sets that fix a triangle
SAS: two sides and their included angle. If AB = DE, AC = DF and ∠A = ∠D, the triangles are congruent. Imagine matching A with D and side AB with DE. The angle fixes the second side’s direction and its length fixes the third vertex. It cannot slide freely anymore. “Included” cannot be omitted. Euclid I.4.
ASA: a side and its adjacent angles. Match the given side; the two angles define rays from its endpoints. Their intersection fixes the third vertex. The angles must be positive and sum to less than 180°. Given two other corresponding angles and a side, first find the third angle using the 180° sum and reduce to this case. Euclid I.26.
SSS: three sides. If all three corresponding length pairs are equal, the triangles are congruent. For a fixed base the other lengths allow two reflected vertex positions; these produce congruent triangles, not different shapes. The strict side inequalities from the previous lesson must hold. Euclid I.8.
SAS · two sides and their included angle
ASA · a side and adjacent angles
SSS · three sides
SSA · two different positions of C
This explains the geometric rigidity of a triangular frame with fixed side lengths: its angles cannot change while all three lengths stay fixed. The model does not calculate the strength of actual materials or loaded joints.
Why arbitrary three data do not work
Given AB=8, BC=5 and ∠A=30°, with lengths in one chosen unit. Angle A is not included between the given sides: they meet at B. The diagram has two possible positions of C on the same ray. Both are 5 units from B, but the AC lengths differ. Therefore “two sides and any angle” is not a general criterion. This is the ambiguous SSA case. Commentary on I.26.
Three angles do not fix size either: double the whole triangle and its angles stay unchanged while its lengths double. These figures are not congruent.
What a right angle adds
The two sides adjoining a right angle are legs; the opposite side is the hypotenuse. It is the longest because it faces the largest angle.
Two right triangles are congruent if their corresponding legs are equal: the included 90° is already given, so SAS applies. A leg and a corresponding acute angle also suffice, as do a hypotenuse and a corresponding acute angle: the other acute angle completes it to 90°, giving two angles and a side.
A special criterion says that equal hypotenuses and one pair of equal corresponding legs also guarantee congruence. Both triangles must be right triangles; removing this condition brings back the ambiguity of two sides and an angle. The special case in the commentary on Euclid I.26.
For a copy of a part, write down the vertex correspondence and name the criterion being used. That is what justifies determining the remaining measurements.