The simplest congruency statement pertains to lines. A line can be drawn between any two points on a plane. If the length of the line between points A and B, AB is equal to CD, the length of the line between points C and D, then these lines are congruent.
Angles are formed when two lines meet. Given angles A and D, if their measurements in degrees are the same, they are considered to be congruent. The length the lines forming the angles or their position on a plane have no bearing with their congruency. As with the line, the congruence statement for angles is the same as a statement of equality.
Triangles are formed of three points which are connected to each other by lines. Given triangles ABC and DEF, if each of the three sides of these triangles and each of the angles are equal, or congruent, then the two triangles are congruent. You may not always have all this information, however, and there are other congruence statements that may be applied.
Given triangles ABC and DEF, if side AB = side DE, side BC = side EF and side AC = side DF, these triangles are congruent. This is called the side-side-side or SSS congruence statement. It provides that whenever two triangle have all three sides of the same length they are congruent and it is the simplest of the congruence statements pertaining to triangles. Note that two triangles having all three angles with the same measurements are not necessarily congruent as the lengths of their sides could be different.
Given triangles ABC and DEF, if angle C which is enclosed by sides AC and CB has the same measurement as angle F which is enclosed by sides DF and FE, the length of AC equals the length of DF, and the length of CB is equal to the length of FE, these two triangles are congruent. This is called Side-Angle-Side, or SSS, Congruence.
Given triangles ABC and DEF, if angle A equals angle D, angle B equals E, and the length of side AB equals the length of side DE, these triangles are congruent. This congruence statement applies to any two angles in the compared triangles and the sides joining them. It is known as the ASA or Angle-Side-Angle Congruence.