Geometry: Quadrilaterals

On this page we hope to clear up problems that you might have with quadrilaterals. Quadrilaterals are the most used shape (they cover everything from squares to trapezoids) in geometry except for the triangle. Read on to start understanding quadrilaterals better!

Special Things to Remember

As noted above, a quadrilateral is any shape that has four sides. Outlined below are four definitions that are good to always keep in mind when working with quadrilaterals.

Consecutive angles are any two angles whose vertices are the endpoints of the same side. (In the accompanying figure, angle B and angle C are consecutive angles.)
Consecutive sides are any two sides that intersect. (In the accompanying figure, AB and BC are consecutive sides.)
Opposite angles are any two angles that are not consecutive. (In the accompanying figure, angle B and angle D are opposite angles.)
Opposite sides are any two sides that are not consecutive. (In the accompanying figure, AB and DC are opposite sides.)

Sum of Angles Alwyas Equals 360 Degrees

There is also a theorem in geometry that tells us that in all quadrilaterals, the sum of the measures of the angles is 360 degrees. This can be used in many situations, including problems where you need to find the measures of angles.

Example:

1.  Problem: Find the measure of angle x in the accompanying figure.
   Solution: Use the given information to help you find any new
             information.

             Angle y = 80° because vertical angles are
             congruent.
             Angle z = 120° because it is supplementary to the
             60° angle shown in the figure.

             You now know the measures of three of the four angles
             in the quadrilateral.  The other, angle w can be
             found by using the theorem that tells us all
             quadrilaterals have a sum of angles that equals 
             360°.  Set up an equation to do this.

             360 = w + 75 + 80 + 120

             Solve for w.  w equals 85°.

             Angle x can now easily be found because it is
             supplementary to angle w, which you found above.

             Angle x = 95°.

Angles of an N-gon

Although you won't encounter many odd shapes, such as shapes with twelve sides, it can happen. On most instances of this, you will need to find the sum of the measures of the angles. There is a special theorem that says, if n is the number of sides of any polygon, the sum (S) of the measure of its angles is given by the formula — S = (n - 2)180°

The example figure and table below will help this theorem make more sense.

     -------------------------------------------------------------------
     | Polygon  | No. Sides | Total No. of | No. Triangles |  Sum of   |
     |          |           |   Diagonals  |     Formed    |   Angle   |
     |          |           | fr. 1 vertex |               | Measures  |
     -------------------------------------------------------------------
     | Triangle |     3     |      0       |       1       |   180°    |
     | Quad.    |     4     |      1       |       2       |   360°    |
     | Pentagon |     5     |      2       |       3       |   540°    |
     | Hexagon  |     6     |      3       |       4       |   720°    |
     |    .     |     .     |      .       |       .       |      .    |
     |    .     |     .     |      .       |       .       |      .    |
     |    .     |     .     |      .       |       .       |      .    |
     |  n-gon   |     n     |    n - 3     |     n - 2     |(n-2)(180°)|
     -------------------------------------------------------------------

Take the quiz on quadrilaterals. The quiz is very useful for either review or to see if you've really got the topic down.

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