Why did we spend so much time learning about when triangles are congruent? Because we can decompose other shapes into triangles. By looking for triangles that must be congruent, we can prove other shapes have many properties. For example, we could learn more about these types of quadrilaterals:

• A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
• A rectangle is a quadrilateral with 4 right angles.
• A rhombus is a quadrilateral with 4 congruent sides.
• A square is a quadrilateral with 4 right angles and 4 congruent sides.
• A kite is a quadrilateral which has 2 sides next to each other that are congruent and where the other 2 sides are also congruent.

Knowing how to decompose quadrilaterals into triangles using their diagonals lets us prove how the different quadrilaterals’ definitions lead to their diagonals having different properties. We can also look at whether arranging the diagonals to have certain properties gives us enough information to prove which type of quadrilateral must be formed. For example, we might conjecture that if one diagonal is the perpendicular bisector of the other, the figure is a kite. But how do we turn that into a statement that we can prove?

Here is a specific statement that shows what we mean by “One diagonal is the perpendicular bisector of the other” and “The figure is a kite.” In quadrilateral with diagonals and , the diagonals intersect at . Segment is congruent to segment , and is perpendicular to . Prove that segment is congruent to segment  and that segment is congruent to segment . This specific statement lets us draw and label a diagram, which might give us some ideas about how to prove the statement is true.