Recall from
Introduction to Polygons
that a polygon is a closed figure in a plane made by joining line segments,
where each line segment intersects exactly two others.
In this section, we continue our exploration of foursided polygons:
Sides, angles, and vertices that are next to each other in a polygon are called
consecutive.
For example, in the quadrilateral below:
When a quadrilateral is named, recall that the vertices are always listed in a consecutive order.
Therefore, the quadrilateral above could be named
[beautiful math coming... please be patient]
$\,ABCD\,$ or
$\,BCDA\,$,
but not
$\,ACBD\,$.
In a polygon, objects that are not consecutive are said to be
nonconsecutive.
In a quadrilateral, nonconsecutive objects are given a special name—they are said to be
opposite.
For example, in the quadrilateral above:
Every quadrilateral has two diagonals.
A diagonal of a quadrilateral separates the quadrilateral into two triangles, as shown below:
By using the fact that the angles in a triangle sum to 180°, we thus have the following:
It follows immediately that if a quadrilateral has four equal angles, then they must each be 90°.
There are several important types of quadrilaterals, which are classified according to extra properties that their angles or sides possess:
It is easy to prove (later on) that every square, rectangle, and rhombus must be a parallelogram.
Consider two oddlyshaped fields, shown below:
If you pick any two points in the first field,
and walk in a straight line from one point to the other,
then you'll remain inside the field.
Geometric figures with this property are said to be convex.
However, there are points in the second field with the property that the straightline path
between them takes you outside of the field.
Geometric figures with this property are said to be concave.
(Memory device: it is caved in)
Recall that, strictly speaking, a polygon does not include its interior—the polygon is only the "boundary".
So, strictly speaking, line segments connecting points on the polygon go outside the polygon (first picture below).
However, when we talk about polygons being convex or concave, it is conventional to include the interior.
Squares, rectangles, parallelograms, trapezoids and rhombuses are all convex polygons.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
