184 Part 3: The Shape of the World
Rectangles .....................................................................................................................................
A rectangle is a parallelogram in which adjacent sides are perpendicular, and so it has four right
angles. Because the rectangle is a parallelogram, it has all the properties of a parallelogram, but also
the special property that all four of the angles are the same size. It’s an equiangular parallelogram.A rectangle is a parallelogram with four right angles.You’ve probably worked with rectangles before, because there are so many rectangles all around
us. You’re reading this on a rectangular page, and you could be sitting in a room built from
rectangles.
Every rectangle is a parallelogram, so its opposite sides are parallel and congruent, its consecu-
tive angles are supplementary (90r + 90r = 180r), and its opposite angles are congruent. Those
are properties of every parallelogram. The rectangle also has consecutive congruent angles,
because all angles are congruent.
In any parallelogram, a diagonal makes two congruent triangles. That’s still true in a rectangle,
but those congruent triangles will both be right triangles. In any parallelogram, the diagonals
bisect each other, and because the rectangle is a parallelogram, that’s still true. But the rectangle
has another special property. The diagonals of a rectangle are congruent. In most parallelograms,
which sort of slant to one side, there’s a long diagonal and a short one, but in a rectangle, both
diagonals are the same length.Suppose that in rectangle ABCD, the diagonals intersect at E. If BE = 8, how long is AE? Because
the diagonals are congruent and bisect each other, AE = EC = BE = ED. So if BE = 8, AE = 8 as
well.