Points, Lines, and Angles
Imagine staring through a microscope at a red blood cell for an hour. It would be pretty boring. All alone, blood cells are pretty ho-hum. But you can’t really talk about the human body without mentioning blood, can you?
Points are the red blood cells of geometry. By themselves, they’re pretty dull, but you can’t really talk about larger geometric figures without using points. Lines, triangles, polygons, and angles are all described using the points that comprise them.
For the record, a point technically has no width and length. It’s just a place marker. In the figure below, point F is outside the square denoted by the points MNOP. Point G is inside the square.
The next step up on the geometric food chain is the line. A line is an infinite set of points assembled in a straight formation. Lines have no width, but they do have length. Lines go on infinitely, which we indicate visually by placing a double-arrowed hat over two points that define the line, like this: .
Lines can be parallel, meaning they never meet. How sad. Parallel lines are denoted like this: . If a line crosses another line at a 90-degree right angle, the two lines are perpendicular. In symbol speak: .
When two parallel lines are cut by a third straight line, that line, known as a transversal, will intersect with each of the parallel lines.
Line goes on forever, but a line segment (denoted AB or ) does not. Line segments don’t get to wear the cool, double-arrowed hat, but they do have a finite length: line segment AB = 4. Most geometric figures are made up of line segments.
If a line and a line segment had a child, it would be a ray.
Rays extend infinitely in one direction only, and they wear the not-as-cool, single-arrowed hats. and are both rays. Notice that you have to put the arrow point over the letter that is heading off into infinity. Point D—called the endpoint of the ray—isn’t going anywhere. It doesn’t get an arrow.
An angle consists of two lines, rays, or line segments sharing a common endpoint. This is very useful to know, since determining the value of angles is a game you will play over and over again on the SAT.
Angles are denoted by the symbol . There are many different ways to name an angle, and the SAT will use every one of them. You can use points, but the center letter has to be the vertex (the place where the rays start) of the angle. In the previous example, : they are both the same thing.
Angles are measured in degrees, which have nothing to do with the temperature. Geometric degrees are indicated by this little guy: ˚.
Types of Angles
Angle a is a right angle, which equals 90˚. You’ll see the sign to show that it’s a right angle. (This sign is also on the perpendicular lines on page 18.) Combined, angles b and c also form a right angle. Angles that sum up to 90˚ are called complementary angles.
Lines 2 and 3 cross to make angles b and g. These are vertical angles, and vertical angles are always equal. Angles c and h are another pair of vertical angles. Shifting over to the other diagram, angle e is an acute angle (less than 90˚) while angle d is obtuse (more than 90˚). There are 180˚ in a line, so angle f = 180˚. Angles d and e add up to make a line, which earns them the moniker supplementary angles. Supplementary angles always add up to 180˚.
An angle with a measure of 0˚ is called a zero angle. Lines 4 and 6 overlap each other, so they create a 0˚ angle.
You’ll use these basic facts to bop around multiple figures, determining angle values wherever you can. These geometry terms are also used extensively to accurately describe geometric figures.
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