Answer :
To determine why shape 1 and shape 2 are not congruent, we must understand what congruence means in geometry.
Congruence in Geometry:
Two shapes are said to be congruent if they are exactly the same size and shape. This implies that one shape can be transformed into the other using a series of rigid transformations. Rigid transformations include translations (sliding), rotations (turning), and reflections (flipping). These transformations do not alter the size or shape of the figure.
To determine if two shapes are congruent:
- Side Correspondence: All corresponding sides must be the same length.
- Angle Correspondence: All corresponding angles must be the same measure.
- Sequence of Transformations: It must be possible to map one shape onto the other using a sequence of rigid transformations.
Now, let's analyze the given options:
- A. There is no sequence of rigid transformations that will map shape 1 onto shape 2.
This option suggests that no combination of slides, turns, or flips can make shape 1 and shape 2 overlap perfectly, meaning they are not congruent. This is the essence of what it means for two shapes to be congruent.
- B. Not all corresponding pairs of sides on the two shapes are parallel.
Parallelism is not a prerequisite for congruence; it's more relevant in similarity or in special shape properties (e.g., parallelograms).
- C. Shape 1 cannot be mapped onto shape 2 using a reflection.
Though reflection is a rigid transformation, congruence requires that any combination of rigid transformations (not just reflections) must map one shape to the other.
- D. Shape 1 cannot be mapped onto shape 2 using a dilation.
Dilation is not a rigid transformation as it involves resizing, which affects congruence. Hence, this option is irrelevant to congruence.
- E. Not all corresponding pairs of sides on the two shapes are perpendicular.
Perpendicularity doesn't determine congruence either.
Conclusion:
The best explanation for why shape 1 and shape 2 are not congruent is that "A. There is no sequence of rigid transformations that will map shape 1 onto shape 2." This addresses the core requirement of congruence that one shape can be perfectly transformed into the other using only moves that do not change size or shape.
