A Venn diagram like the one shown in figure 6.14 might be used to summarize observations that a square is a special case of a rhombus and rectangle, each of which is a special case of a parallelogram. The teacher might also ask students to draw the diagonals of multiple examples of each shape, as shown in figure 6.15, and then measure the lengths of the diagonals and the angles they form. The results can be summarized in a table like that in figure 6.16. Students should observe that the diagonals of these parallelograms bisect each other, which they might propose as a defining characteristic of a parallelogram. Using dynamic geometry software, students could explore the adequacy of this definition by trying to generate a counterexample.
Students can draw the diagonals of parallelograms to make further observations.
A table of students' observations about the properties of the diagonals of special types of quadrilaterals Middle-grades students also need experience in working with congruent and similar shapes. Students can investigate congruence and similarity in many settings, including art, architecture, and everyday life. For example, observe the overlapping pairs of triangles in the design of the kite in figure 6.17. The overlapping triangles, which have been disassembled in the figure, can be shown to be similar. Students can measure the angles of the triangles in the kite and see that their corresponding angles are congruent.