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
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.
goes on forever,
but a line segment
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
Rays extend infinitely in one direction only, and they
wear the not-as-cool, single-arrowed hats.
both rays. Notice that you have to put the arrow point over the
letter that is heading off into infinity. Point D
of the ray—isn’t going anywhere. It doesn’t get
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
is a right angle,
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
form a right angle. Angles that sum up to 90˚ are called complementary
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.