High School: Geometry
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Congruence is the overarching theme of the next dozen standards in geometry, so let's tackle that one before running down the rest of the line. Pun intended.
In its most basic sense, "congruency" means being in a state of agreement. No, not with your partner, children, or even your students. In the world of geometry, two figures are called congruent if one can be "carried onto" another.
What does that mean? In brief, it means that all corresponding pairs of sides and angles are equal. They're kind of like "identical twins" without the hassles of sibling rivalry or having to decide how long they'll be dressing alike.
More than just congruence, students should know the precise definitions of an angle, circle, perpendicular line, parallel line, and line segment. After all, they can't work with something if they don't know what it is. That's like trying to kick a field goal without knowing what those big yellow posts are at each end of the field.
Speaking of which—those huge yellow posts at either end? Yeah, the goal posts. Each clearly has two massive poles standing straight up in the air. They are parallel. The one at the bottom of each, which is parallel to the ground but connects them? It's perpendicular to the first two and to the one in the center that reaches into the grass below and holds up the whole thing.
There are several angles here, too. For example, one at each end of the bottom of that square "U" formed by the two parallel posts and their perpendicular bar.
Okay, so what about those undefined notions? Don't worry. It's a lot simpler than it sounds. It all boils down to the fact that we have to start somewhere.
In football, for example, the kicker has to trust that when he sends the ball to the other team, it's going to move in a certain direction based on where and with how much force his foot hits it. He may not realize it, but he trusts the laws of physics.
In geometry, when your students set out to prove or construct something, they have to accept a few concepts that seem perfectly understandable, but are not actually formally defined. They can think of them as the laws of geometry.
In this case, these "notions" are a point, a line, the distance along a line, and the distance around a circular arc. A point is simply a location, represented by a dot. A point has no size, no length, no nothing. Only a location.
A line is simply is straight, infinite length. It's named simply by any two points on the line. It can also be named by a single letter, normally shown in the lower case. The distance along a line is a decided or arbitrary length along the line.
The distance around a circular arc is like the distance along a line, but it's not straight. It's a length equidistant from a point rather than equidistant from a line.
You may also see these called "fundamental" or "primitive" notions. They're the same thing, but try to be consistent. Your students might get confused if you refer to the same thing in three different ways.
Using these undefined notions, we can define an angle as a two lines or line segments or rays that share a common endpoint and a circle as a set of points equidistant from a given point (better known as "the center").
What about all this business with lines? Well, since lines in the real world can't extend forever, we have line segments, which are pieces of the line that run between two endpoints, but don't extend beyond them.
Parallel lines are lines that never intersect. Ever. No matter how far you extend them. Perpendicular lines not only intersect, they meet each other head-on, at a 90° angle.
Students should be able to not only define but also recognize and draw a proper angle, circle, perpendicular line, parallel line, and line segment, and more. These are concepts that most students shouldn't have a problem with. If some of your students are struggling, give lots of definitions and examples so that they master these concepts before moving on.