You're sitting in Geometry class staring at a monstrosity of lines when the following question comes out of your teacher's mouth in a language closely resembling Greek:
Line segment AB is parallel to line segment CD. Line segment EF is perpendicular to line segment CD. Line segment GH intersects both line segments AB and CD at points M and N, respectively. Solve for angles x, y, and z.
"Gesundheit," you reply. Your teacher looks you in the eye. Before you can go on to suggest some excellent throat lozenges, your teacher says, "Please come to the board and solve this problem for the class."
You take a deep breath and walk toward the board. Everything you just learned about angles rushes through your mind. "Complimentary angles add to ninety. Supplementary angles add to one-eighty..." your thoughts race as you grab the blue marker. The marker remains hovering above the board. "I... I don't know where to start," you stammer.
"What do you remember about opposite angles?" your teacher hints. You stare back at her and realize how dry your mouth has gotten when you reply, "Angles across intersecting lines are equal?"
"That's right!" she says and draws the following picture on the board:
"There's a relationship you missed," your teacher replies while picking up the green marker, "Let's redraw what we were given using the relationships described in the problem."
"Now what do you see?" she asks.
"Angle y equals forty," you say, starting to feel less flustered.
"Exactly!" your teacher explains, "We knew that line segments AB and CD were parallel. That means if they are intersected by the same line you can relate the angles. Now, let me show you something else using Angle Addition." She grabs a purple marker and writes quantities in some of the blank spaces.
You immediately see it. "Angle z equals twenty!" you exclaim.
"Can you explain it to the class?" your teacher asks.
"Well, angle y is opposite of angle z and the given angle. So you could write it like this," you say while you write the following equation: 40 = z + 20. "Solving for z, we find that z equals twenty degrees."
"Perfect!" your teacher smiles. "Just one unknown left."
On a roll you say, "Since line segment AB is a straight line we know that these top angles have to equal one-eighty. They are supplementary angles." You write on the board:
"So angle x is sixty degrees!" you say triumphantly.
"Well done!" says your teacher. "Class, do you see how redrawing the problem helps prevent yourself getting overwhelmed by a flurry of lines?"
"Does this only work for opposite angles?" a classmate asks.
"No, you can use this on any question. Check this out."
The class is buzzing. You smile because looking at all of those lines made you dizzy.
"Looking at the whole problem can sometimes distract you from the simple relationships," your teacher explains. "Try working through tonight's homework problems using this method. You may have to draw a few different simplifications before you find one that works, but don't give up! After enough practice you won't need to draw the simplifications, but they start to jump out at you."