Download Quadrilaterals

Your study of geometry has included much information about triangles. Now
let's move on to the four-sided polygons - quadrilaterals.
Four-sided figures are called quadrilaterals.
QUAD = 4
LATERAL = Side
A quadrilateral is a four-sided figure that has four interior angles.
The word degree has a new meaning that applies to this lesson.
When a point is connected to other
points by segments (or curved edges
for that matter), the point has a degree
that is equal to the number of segments
that are connected to that point.
In this diagram, each connecting point is labeled with its degree.
In quadrilaterals, all vertices have a degree of 2.
A quadrilateral is a four-sided figure that has four interior angles. Each vertex
has a degree of two.
Two sides of a quadrilateral that share
a common vertex are called
consecutive sides.
Two angles that are connected by a
common side are called consecutive
angles.
Two sides of a quadrilateral that do not share a common vertex are called
opposite sides.
Can you identify the opposite sides in quadrilateral MNOP?
OPPOSITE SIDES
Pairs of opposite angles of a quadrilateral are NOT connected by a common
side.
Which angle pairs are opposite each other in quadrilateral MNOP?
DIAGONALS
A diagonal of a geometric figure is a
line segment that joins two
nonconsecutive vertices.
In quadrilaterals, diagonals connect the
vertices of opposite angles.
DIAGONALS
In quadrilateral DEFG identify the
following:
1. Opposite sides
2. Consecutive angles
3. Diagonals
4. Consecutive sides
5. Opposite angles
6. Consecutive vertices
7. Opposite vertices
If necessary, go back and review the previous
screens.
Let's prove a new theorem.
Look at this quadrilateral with its
diagonal. How many triangles do you
see?
There are two triangles. Each triangle has a total angle measure
of
The total of the angles in any quadrilateral is
Look at the difference in these two quadrilaterals. Now, think about their diagonals.
Now, look at the diagonals of each figure.
A convex quadrilateral has diagonals that intersect in the interior of the figure.
Notice that the interior angles of the
quadrilateral look like normal angles, with
measures between
(That won't be true for concave
quadrilaterals.)
Both of these quadrilaterals are convex . The angles are less than 180 degrees, and the
diagonals intersect inside the figures.
A concave quadrilateral has diagonals that do not intersect in the interior of the figure.
One interior angle will have a measure of more than
STUV is concave because the
measure of
if measured in the
interior of the quadrilateral, is greater
than
One of the diagonals of the
quadrilateral would lie outside the
figure.
Here's another test for concavity. If a segment can be drawn between any two
points in the interior of the quadrilateral so that the segment intersects the sides of
the quadrilateral, the figure is concave.
The word quadrilateral means four sides. There are other names for specific types
of quadrilaterals.
TYPESOFQUADRILATERALS
Special quadrilaterals can be made by applying rules to the sides or the angles.
Opposite sides might be parallel or congruent. Opposite angles might be
congruent. All sides or angles might be congruent. Different combinations make
different shapes.
Let's first talk about parallel sides.
This figure is simply a quadrilateral. It
has no special name. It has no parallel
sides.
These special figures are called trapezoids. They have exactly one pair of
parallel sides.
PARALLELOGRAM
In a parallelogram, BOTH pairs of opposite sides are parallel.
Some parallelograms have special names.
A rectangle
has four
right angles.
A rhombus
has four
congruent sides.
If a rectangle and a rhombus are combined, what is the result?
A rectangle
has four
right angles.
A rhombus
has four
congruent sides.
A square
has four right angles
AND four congruent sides.
The trapezoid is the only quadrilateral listed on the previous screen that is NOT a
parallelogram.
The rectangle, rhombus, and square are special types of parallelograms.
Here is a final thought with regard to parallelograms:
In a parallelogram, ANY two consecutive angles
are same-side interior angles. They will be
supplementary.
The opposite sides are on parallel lines.
Either of the other two sides could be used as
a transversal.
Later lessons will focus on the special properties of the rectangle, rhombus, and
square.
You have completed the lesson on Quadrilaterals.
You may go on to take the practice test or review the study guide if you are
unsure about some of the material.