Chapter 14: Quadrilaterals 185
Rhombuses and Squares.......................................................................................................
A rhombus is a parallelogram with four sides of the same length, an equilateral parallelogram.
Because the rhombus is a parallelogram, it has all the properties of a parallelogram, and it still
may have that lean to one side, if its angles are different sizes. It’s an equilateral parallelogram,
but not always an equiangular one.
A square is a parallelogram that is both a rhombus and a rectangle. Squares have four right angles
and four equal sides. They are equilateral and equiangular.
A rhombus is a parallelogram in which all sides are congruent. A square is a parallelogram with
four congruent sides and four right angles.
In a rhombus or a square, drawing one diagonal will make two congruent triangles, just as it does
in any parallelogram. In a rhombus, those triangles will be isosceles, and in a square, they will be
isosceles right triangles. You know that in 45-45-90 right triangles the length of the hypotenuse
is the length of a leg times the square root of two. That means that the length of the diagonal of
a square will be the length of a side times radical two. Because the square is a rectangle, both
diagonals are the same length.
When you draw both diagonals in a rhombus or a square, a bunch of things happen. The diago-
nals bisect each other, just as they do in any parallelogram. Because the sides of the rhombus (or
square) are all the same length, you can figure out that the diagonals of any rhombus, including
a square, are perpendicular to one another.
In RSTU, RU = UT, RV = VT, and UV equals itself, so 'RVU and 'TVU match each other. That
means RYU is congruent to TVU, but they’re also supplementary, so they must both be 90r.
If the diagonals cross at right angles, they’re perpendicular.
U T
RS
V
