Properties of Quadrilaterals
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    This lesson page will inform you about the properties of quadrilaterals. Here are the sections within this lesson page:

    A quadrilateral is a polygon that has four sides. Quadrilaterals are sometimes called tetragons (4-sided polygon, like hexagons have 6-sides) or quadrangles (having 4-angles, like triangles have 3-angles).

    Besides having four sides and four angles, there is a property concerning its angles. Look at this quadrilateral ABCD.

quadrilateral ABCD

    If we draw diagonal (a segment that connects two non-adjacent vertices) BD, we can see the quadrilateral is divided into two triangles.

quadrilateral ABCD with diagonal

    One property of a triangle (in a plane) is that the sum of its internal angles is 180 degrees. Each triangle therefore has a sum of 180 degrees.

    If we add all of these angles, we get...

    Let's rearrange these angles, like so...

    We can see from the diagram that angle-ABC and angle-ADC have been broken into two smaller angles.

    So, we can make a substitution within our rearranged 360 degree equation. We can take pairs of angles and replace them.

    Removing the diagonal, we see our original quadrilateral.

quadrilateral ABCD

    We therefore no longer need to use three letters to describe our angles. This makes the equation a lot simpler.

quadrilateral interior angle sum 360 degrees

    This means...

The Sum of the Measures of a Quadrilateral's
Interior Angles is Always Equal to 360 Degrees

    Below is quadrilateral ABCD again.


    Here is the same quadrilateral with exterior angles shown.


    The angles need to be marked so that another property can be found.


    Notice the supplementary angles.

supplementary angles interior exterior quadrilateral

    If we add all of these equations, we get this new equation.

    We can rearrange the angles, like so.

    From the last section (Interior Angles) we discovered that the sum of the internal angles is always 360 degrees. So the sum of angles 1, 2, 3, and 4 has to be 360 degrees. The equation now becomes...

    We can subtract 360 degrees from both sides of the equation to get this equation.


    This means...

The Sum of the Measures of a Quadrilateral's
Exterior Angles is Always Equal to 360 Degrees

    Review this related lesson.

    esson: Geometry Proofs
    esson: Properties of Parallelograms
    esson: Properties of Rectangles