When are two triangles said to be congruent?

Answer 1

Congruent means the same size and shape. Two triangles are congruent if the 3 sides and 3 angles of one are equal to the respective sides and angles, in order, of the other. Thus the triangles ABC and DEF are congruent if the lengths of AB and DE are equal, as well as BC and EF, and CA and FD, and the angle at A equals the angle at D, likewise that at B and at E, and of course if those two are true, the angle at C must equal the one at F since the 3 angles in a triangle always add up to 180 degrees. Two triangles are congruent if you can rigidly move one to exactly coincide with the other. It might be necessary to rotate it through 3-dimensional space, if the triangles are mirror images of each other.

There are some theorems that give criteria that guarantee triangles to be congruent. One is side-side-side, abbreviated SSS, meaning that if the sides of two triangles, in order, are equal, so are the angles. Another is SAS, meaning two sides of one triangle and the angle included between them are equal to the corresponding parts of the other. If 2 of the angles of two triangles are the same (AA), so is the third, and the triangles are similar (same shape, but not necessarily the same size). Then all you need is that one side and the corresponding side in the other triangle are equal to prove congruence. There is one ambiguous case: SSA. Depending on the length of the side opposite the given angle, there may be 0, 1, or 2 different (non-congruent) triangles having the given part measures: 0 if the side is too short, 1 if it is the length of the perpendicular to the other side, and 2 if it is longer than that.

Answer 1

=======

When they both have the same 3 interior angles and the same length of sides