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As we walk through the geometric jungle, we see all types of different mathematical creatures. The sun shines rays down on the landscape. A circle rolls across our path. Angles are chirping like the beaks of birds. And the colorful leaves at the top of cylindrical trees are shaped like polygons.
We need to be able to distinguish the quadrilaterals from the rest of the jungle's inhabitants. Simply put, a quadrilateral is just a four-sided polygon. We can remember this easily because "quad" means four and "lateral" means side. Nothing too shocking there.
There's one more part to our definition. The quadrilateral is a member of the family of polygons. They have only straight lines (no curvy hips), and all of the lines are connected at their endpoints to form a closed shape. Of course, quadrilaterals aren't like most polygons. They have some pretty special properties that set them apart from the rest of the polygonal world. Then again, that's what all polygons like to think.
People travel from all over the world to view da Vinci's masterpiece, the Mona Lisa. When people peep that famous painting, different things come to everyone's mind. Some people think she's too small to be famous. Others wonder if she's daydreaming about a peanut butter and fluff sandwich. And there are even those who think that a touch of makeup could really bring out her features and distract from that no-eyebrow thing she's got going on. No offense, Mona.
As mathematicians, we should analyze quadrilaterals the same way. Only more CSI and less Extreme Makeover.
When we lay our eyes on a quadrilateral for the first time, we should look at it from different angles. We'll see the many facets it has to offer us and maybe appreciate its inner beauty just as much as we appreciate its outer beauty. (And you have to admit, some of those quadrilaterals are pretty fabulous.) Just take a look at Carl, here.
Carl's got four sides and four angles, which makes him a quadrilateral. Unlike triangles in which any two sides are adjacent, quadrilaterals have specific sides that are adjacent or opposite. Adjacent sides are those that share a vertex (corner), while opposite sides do not. In the same vein (ouch—be careful with that needle), adjacent vertices share a side, while opposite vertices do not. Not exactly earth-shatteringly new information, is it?
What are all of CARL's adjacent and opposite sides?
We can see that CA and AR share the vertex A, so they must be adjacent. Same with AR and RL, RL and LC, and LC and CA. What about the opposite sides? Well, CA and RL don't share any vertices, so they're opposite sides. Same goes for AR and LC.
Identify all pairs of adjacent vertices in CARL's quadrilateral friend, ALIX.
This time, we need to find the vertices that are connected to the same side. So if two vertices make a side, they're adjacent. Side AL exists, so we know that vertices A and L are the vertices at either end of the side. This makes vertices A and L adjacent. Using this same logic, we know that L and I are adjacent, as are I and X, and X and A. Of course, this means A is opposite I and L is opposite X. In case you were wondering.
We can also find opposite angles of any quadrilateral. We don't really have to be more explicit about this, do we?
Identify all pairs of opposite angles in quadrilateral ALIX.
There are only two pairs of opposite angles in any quadrilateral. We can pick them out by noticing that opposite angles don't share any sides in common. For instance, ∠XAL is opposite to angle ∠LIX, and ∠ALI is opposite to ∠IXA.
Good thing we know where opposite vertices are because all quadrilaterals have two diagonals that connect opposite vertices. How can we draw all over CARL and ALIX? They aren't our personal doodle pads, nor are they Ross and Rachel on their way to Las Vegas. The diagonals of the quadrilateral connect the two pairs of opposite vertices. For example, C is connected to R by a diagonal (the dotted lines).
What about ALIX? We don't want to leave her hanging. What would be the names of her diagonals?
The diagonal connecting vertices L and X is pretty easy to pick out. The other diagonal lies outside the quadrilateral, connecting A and I. It's still a diagonal because it connects a pair of opposite vertices. It doesn't matter whether it's outside the quadrilateral or not.
One last thing we should know about any and all quadrilaterals: the sum of all the internal angles of a quadrilateral is 360°. Always, always, always. Don't believe us? Let's split CARL into two pieces by drawing only one diagonal. Don't worry, CARL. This won't hurt a bit.
If we take a close look with our spectacles, we can see that CARL is just a quadrilateral made up of two triangles. We already know all triangles have three internal angles that add up to 180°. The internal angles of CARL the quadrilateral are the same as the six internal angles of the two triangles. Since each triangle contributes 180° of angle measurement and there are two of them, CARL has a total of 180° × 2 = 360°. This is true for any and all quadrilaterals and their internal angles. They'll always add up to 360°.
Gee willikers! What happened to Pacman?
He looks so rigid and angular, nothing like his usual well-rounded self. Could it be that this isn't Pacman, but his evil twin—Quadman? Well, it only makes sense. After all, he's a quadrilateral, and look! Even his food dots are quadrilaterals. Clearly, we're in some major quadrilateral territory.
There's an important difference between Quadman and his food. (We aren't talking about the obvious polygonal predator-versus-prey situation going on here.) This Quadman is a perfect example of a concave quadrilateral, while his food is made of convex quadrilaterals.
Concave quadrilaterals are those that have a cavity, or a cave. In the game, Quadman's mouth is the "cavity" we're talking about. Convex quadrilaterals, like the more familiar squares and rectangles, don't have a cavity. Of course, as mathematicians, we might want to come up with a better definition than "has a cavity" or "doesn't have a cavity." It sounds too much like a painful trip to the dentist.
Let's take a closer look at Quadman and his food.
When we draw the diagonals on Quadman, we see that one of his diagonals (the horizontal one) lies inside his body, but the other one doesn't. If we do the same for his food, we see that both diagonals are completely contained within the shape itself. That's the difference between concave and convex.
So let's define them again, take two. A convex quadrilateral has both diagonals completely contained within the figure, while a concave one has at least one diagonal that lies partly or entirely outside of the figure. Those are good definitions, but the concave shapes having a cavity or cave is probably an easier way to remember it.