Two shapes are congruent (identical) (1) The angles in one shape are equal to the corresponding angles of the other (2) All pairs of corresponding sides are the same size A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Congruent shapes are fundamental in geometry as they are identical in both shape and size. This means that every angle and side in one shape matches exactly with the corresponding angle and side in the other shape. When two shapes are congruent, they can be superimposed perfectly on each other. This concept is essential for solving many geometric problems, including those in the GMAT. By understanding congruence, you can determine if two shapes are identical or just similar. This leads to deeper insights about spatial relationships in mathematics. For example, consider two triangles with the same angles and sides, making them congruent. Such shapes are a key aspect of GMAT geometry questions and can prepare you to identify congruence in various problems.

Geometric conditions refer to the specific properties that must hold true for shapes to meet certain criteria, such as being congruent. In the context of congruent shapes, there are two primary conditions to consider: equal angles and equal sides. Firstly, if the angles in one shape match the corresponding angles in another shape, it indicates that the shapes have the same shape, but not necessarily the same size. Secondly, if the sides of one shape are equal to the corresponding sides of another shape, it suggests that the shapes are the same size, but doesn't guarantee the same shape. Only when both these conditions are satisfied can we confidently say that the shapes are congruent. Understanding these conditions helps in breaking down complex geometric problems into manageable parts.

Angle equality is a crucial component in the study of congruent shapes. When the angles in one shape are identical to the corresponding angles in another shape, we know that they share the same shape. To illustrate, consider two triangles where all three interior angles are equal. Even if their sides are of different lengths, these triangles will have the same angular relationships. However, having equal angles alone is not enough to conclude that the shapes are congruent. We need to also consider the sides to ensure the complete congruence criteria is met. This principle is vital in many geometric proofs and GMAT questions, which often require not just visual inspection but rigorous verification of both angles and sides.

Corresponding sides are pairs of sides that are located in the same relative position in different shapes. For instance, in congruent rectangles, each side of one rectangle will have a corresponding side in the other rectangle that is exactly equal in length. For shapes to be congruent, it is essential that all pairs of corresponding sides are of the same length. This ensures that the shapes are of the same size. However, having corresponding sides alone, without considering the angles, does not confirm congruence. This is because different shapes can sometimes have sides of the same lengths but arranged differently, leading to different overall shapes. Considering corresponding sides is thus a part of the comprehensive check for congruence in geometry.