Angles and Lines
An angle is a geometric figure consisting of two lines,
rays, or line segments that share a common endpoint called a vertex:
In the angle above, the vertex is point A
The angle can be called either angle CAB
or angle BAC
The only rule for naming an angle is that the vertex must be the
middle “initial” of the angle. The SAT
may also refer
to angles using symbols:
Angles are measured in degrees, which have nothing to
do with Nelly or temperature. Geometric degrees are sometimes denoted
by this little guy: º. There are 360º in
a complete rotation around a point (that’s why a circle has 360º).
Two Lines Meet in a Bar...
When two lines meet, they produce angles. And when two
lines meet, they form four angles! That must be exhausting.
These aren’t just any old four angles, either. Together,
the angles encompass one full revolution around the point of intersection
of the two lines. So, the four angles produced by two intersection
lines total 360º: angle a + b + c + d = 360º.
If you know the value of three of the four angles formed
by intersecting lines, you can always find the value of the fourth.
Types of Angles
The different types of angles are named and categorized
according to their number of degrees.
A zero angle has, you guessed it, 0º. To
visualize a zero angle, first picture two lines that form some angle
greater than 0º. Then imagine one of the lines rotating
toward the other until they both fall on the same line. The angle
they create has shrunk from its original measure to 0º,
forming a zero angle:
For some reason, an angle with a measure of 90º is
called a right angle. For some other reason, right angles are symbolized
with a square drawn in the corner of the angle. Whenever you see
that reliable little square, you know you’re dealing with a right
Right angles are extremely important on the SAT. They
appear in math questions all the time. Knowing their special properties
will help you solve right angle questions. We give you a detailed
look at those properties a little later in this chapter. For now, just
remember: Always be on the lookout for right angles
on the SAT.
An angle with a measure of 180º is called
a straight angle. It looks just like a line. Don’t confuse straight
angles with zero angles, which look like a single ray.
Acute and Obtuse Angles
An angle can also be classified according to whether its
measure is greater or less than 90º
. If an angle measures
less than 90º
, it’s called an acute angle. If it measures more
, it’s called an obtuse angle. Right angles
are neither acute nor obtuse. They’re just right. In the picture
Complementary and Supplementary Angles
Special names are given to pairs of angles whose sums
equal either 90º or 180º. Angles whose
sum is 90º are called complementary angles, while angles
whose sum is 180º are called supplementary angles.
In the picture above,
since together they make up a right angle. Angles
are supplementary, since they make up
a straight line.
On the SAT, you’ll have to use the rules of complementary
and supplementary angles to figure out the degree measure of an
the diagram below, AC is a line. What
is x in degrees?
The picture tells you that
, but how many
Well, since you
know that AC
is a line,
must be a straight angle (meaning
it equals 180º
are supplementary angles that add
up to 180º
. To find out
the value of
you can simply
and subtract 113º
. = 67º
When two lines (or line segments) intersect, the angles
that lie opposite each other, called vertical angles, are always equal.
are vertical angles and are therefore
are also vertical (and equal) angles.
We promise that the SAT will ask you at least one question involving
vertical angles. Promise.
Parallel and Perpendicular Lines
Pairs of lines that never intersect are parallel. Parallel
lines appear to line up right next to each other because they never
meet in space. However, on the SAT, you can’t assume two lines are
parallel just because they look parallel. The SAT will tell you
if two lines are parallel.
Lines (or segments) are perpendicular if their intersection
forms a right angle. And if one of the angles formed by the intersection
of two lines or segments is a right angle, then all four angles
created will also be right angles. By the way, this also shows that the
degree measurement of four angles formed by two intersecting lines
will add up to 360º, since 90º + 90º + 90º +
90º = 360º.
As with parallel lines, don’t assume that lines on the
SAT are perpendicular unless the SAT tells you they are. The SAT
will tell you either in words (“lines a and b are parallel”)
or by using the little reliable box to show that the angles are 90º.
Parallel Lines Cut by a Transversal
A transversal is a line that cuts through two parallel
lines. The SAT loves to cut parallel lines with transversals. Who
knows why? Not us. But we know how to get those questions right,
and you will too.
A transversal creates eight angles when it intersects
with two parallel lines. The eight angles created by these two intersections
have special relationships to each other.
You now have a choice to make: (1) spend all day figuring
out these relationships, or (2) use our list.
- Angles 1, 4, 5, and 8 are
equal to each other because they’re vertical angles.
- Angles 2, 3, 6, and 7 are equal
to each other because they’re vertical angles.
- The sum of any two adjacent angles, such as 1 and 2 or 7 and 8,
equals 180º, because these are supplementary angles.
By using these three rules, you can figure out the degrees
of angles that may seem unrelated. For example, since angles 1 and 2 sum
to 180º, and since angles 2 and 7 are
equal, the sum of angles 1 and 7 also
equals 180º. The SAT will almost definitely include
a question that asks you to solve for an angle whose measurement
at first glance seems impossible to determine.