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Theorems for Quadrilaterals

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Rhombi

The diagonals of a rhombus have three special
properties.

- They are perpendicular to each other.
- They bisect each other.
- They bisect the interior angles of the rhombus.

Thus, when you draw a rhombus, you can also draw this:

Figure %: The diagonals of a rhombus have useful properties

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Rectangles

Recall that a rectangle is a
parallelogram. It therefore has all the properties
of a parallelogram. One more useful fact is true of rectangles, though. The
diagonals of a rectangle are equal. This is true because rectangles are
equiangular.

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Squares

Now recall that a square is both a rhombus and a rectangle. Its sides and
angles are all congruent. From this fact, it follows:

- The diagonals of a square are perpendicular to each other.
- They bisect each other.
- They bisect the angles of the square.
- They are equal.

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Isosceles Trapezoids

An isosceles trapezoid is the name given to a trapezoid with equal legs. The
angles whose vertices are the vertices of the longer base are called the lower
base angles, and the other two angles are called the upper base angles.

Figure %: An isosceles trapezoid

For every isosceles trapezoid, the following is true:

- The legs are equal.
- The lower base angles are equal.
- The upper base angles are equal.
- The diagonals are equal.

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Theorem for Regular Polygons

One additional theorem applicable to all regular
polygons must be mentioned. You have probably
already assumed as much from drawings, but to make it official, we'll state it
as a theorem: the radii of a regular polygon bisect the interior angles.

Figure %: The radii of a regular polygon bisect its internal angles

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Theorem for Perpendicular Bisectors

A final handy theorem with polygons has to do with perpendicular
bisectors. The points on a perpendicular
bisector are equidistant from the endpoints of the segment that they bisect.

Figure %: A perpendicular bisector