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Geometry: Theorems

Theorems for Other Polygons

Problems

Problems

Theorems for Quadrilaterals

Rhombi

The diagonals of a rhombus have three special properties.

  1. They are perpendicular to each other.
  2. They bisect each other.
  3. 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

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.

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:

  1. The diagonals of a square are perpendicular to each other.
  2. They bisect each other.
  3. They bisect the angles of the square.
  4. They are equal.

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:
  1. The legs are equal.
  2. The lower base angles are equal.
  3. The upper base angles are equal.
  4. The diagonals are equal.

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

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

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