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?
Sample Problem
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.
Sample Problem
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?
Sample Problem
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).
Sample Problem
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°.