E1.3 Identify congruent lengths, angles, and faces of three-dimensional objects by mentally and physically matching them, and determine if the objects are congruent.
Skill: Identifying Congruent Faces
Congruence is a special relationship between two-dimensional shapes whose measurements of all corresponding elements (sides and angles) are equal. In Grades 3, students could explore this idea of with ”regular” two-dimensional shapes, which have congruent sides and angles. At this point, students discover that these ”look alike in every way“ by overlapping them to see how well they match.
Congruent:
- said of two angles or two line segments whose measures are equal;
- said of two two-dimensional shapes or two three-dimensional objects whose measures of all corresponding elements (sides and angles; faces and edges) are equal.
Elementary students will often encounter concepts related to congruence, such as:
- the faces of a three-dimensional object can be congruent (for example, the faces of a cube are congruent squares; the two bases of a pentagon-based prism are congruent pentagons);
- translations (sliding a shape) and reflections (reflecting a shape as in a mirror) give an image congruent to the initial shape.
Source: translated from Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, Géométrie et sens de l'espace, p. 18.
Skill: Determining if Three-Dimensional Objects are Congruent
All faces of two congruent three-dimensional objects match perfectly when they are superimposed.
Two non-congruent three-dimensional objects can still have congruent elements. For example, two three-dimensional objects can have one face that is congruent (namely, the same size and shape), but if the other faces are different in any way (for example, if the magnitude or size of their angles or the length of their sides differ), they are not congruent. Similarly, when all the faces of two three-dimensional objects are congruent, but they are arranged differently in the three-dimensional objects, these two three-dimensional objects are not congruent because they do not have the same geometric shape.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.