Lines and Angles
A line is a collection of points that extends
without limit in a straight formation. A line can be named by a
single letter, like line l, or it can be named
according to two points that it contains, like line AB.
The second way of naming a line indicates an important property common
to all lines: any two points in space determine a line. For example,
given two points, J and K:
a line is determined:
This line is called JK.
A line segment is a section of a line. It
is named and determined by its endpoints. Unlike a line, whose length
is infinite, a line segment has finite length. Line segment AB is
Distance and Midpoint of a Line Segment
The midpoint of a line segment is the point
on the segment that is equidistant (the same distance)
from each endpoint. Because a midpoint splits a line segment into
two equal halves, the midpoint is said to bisect the
Because a midpoint cuts a line segment in half, knowing
the distance between the midpoint and one endpoint of a line segment
allows you to calculate the length of the entire line segment. For
example, if the distance from one endpoint to the midpoint of a
line segment is 5, the length of the whole line segment is 10.
The Math IC test often asks questions that focus on this
property of midpoints. The Math IC writers usually make their questions
a little trickier though, by including multiple midpoints. Take
the midpoint of WZ and Y is the
midpoint of XZ. If M is the midpoint
of XY and MY = 3, what is the length
All the midpoints flying around in this question can get
quite confusing. Instead of trying to visualize what is being described
in your head, draw a sketch of what the question describes.
Once you’ve drawn a sketch, you can see how the three
midpoints, and the new line segments that the midpoints create,
all relate to each other.
- Since X is the midpoint
of WZ, you know that WX = XZ and
that both WX and XY are equal
- Since Y is the midpoint of XZ,
you know that XY = YZ and that both XY and YZ are
equal to 1⁄2XZ and 1⁄4WZ.
- Since M is the midpoint of XY,
you know that XM = MY and that both XM and MY are
equal to 1⁄2XY and 1⁄8WZ.
Please note that you don’t have to write out these relationships
when answering this sort of question. If you draw a good sketch,
it’s possible to see the relationships.
Once you know the relationships, you can solve the problem.
For this question, you know that MY is equal to 1⁄8WZ.
Since, as the question tells you, MY = 3, you can
calculate that WZ = 24. The question asks for the
length of WX, which is equal to 1⁄2WZ,
so WX = 12.
Technically speaking, an angle is the union
of two rays (lines that extend infinitely in just one direction)
that share an endpoint (called the vertex of the angle). The measure
of an angle is how far you must rotate one of the rays such that
it coincides with the other.
In this guide and for the Math IC, you don’t really need
to bother with such a technical definition. Suffice it to say, angles
are used to measure rotation. One full revolution around a point
creates an angle of 360 degrees, or 360
. A half-revolution, also known as
a straight angle, is 180
degrees. A quarter
revolution, or right angle, is 90
In text, angles can also be indicated by the symbol
When two lines or line segments intersect, two pairs of
congruent (equal) angles are created. The angles in each pair of
congruent angles created by the intersection of two lines are called
In this figure,
are vertical angles (and
therefore congruent), as are
Supplementary and Complementary Angles
Supplementary angles are two angles that
together add up to 180º. Complementary angles are two
angles that add up to 90º.
Whenever you have vertical angles, you also have supplementary
angles. In the diagram of vertical angles above,
are all pairs of supplementary angles.
Parallel Lines Cut by a Transversal
Lines that will never intersect are called parallel
lines, which are given by the symbol ||. The intersection
of one line with two parallel lines creates many interesting angle
relationships. This situation is often referred to as “parallel
lines cut by a transversal,” where the transversal is the nonparallel
line. As you can see in the diagram below of parallel lines AB and CD and
transversal EF, two parallel lines cut by a transversal
will form eight angles.
Among the eight angles formed, three special angle relationships
Alternate exterior angles are pairs of
congruent angles on opposite sides of the transversal, outside of
the space between the parallel lines. In the figure above, there
are two pairs of alternate exterior angles: and , and and .
interior angles are pairs of congruent angles on opposite
sides of the transversal in the region between the parallel lines.
In the figure above, there are two pairs of alternate interior angles: and , and and .
angles are congruent angles on the same side of the transversal. Of
two corresponding angles, one will always be between the parallel
lines, while the other will be outside the parallel lines. In the
figure above, there are four pairs of corresponding angles: and , and , and , and and .
In addition to these special relationships between angles,
all adjacent angles formed when two parallel lines are cut by a
transversal are supplementary. In the previous figure, for example,
Math IC questions covering parallel lines cut by a transversal
are usually straightforward. For example:
the figure below, if lines m and n are
parallel and = 110Âº,
then f – g =
If you know the relationships of the angles formed by
two parallel lines cut by a transversal, answering this question
are alternate exterior angles, so
is adjacent to
, so it must be equal to 180º – 110º
= 70º. From here, it’s easy to calculate that f – g
110º – 70º = 40º.
Two lines that intersect to form a right (90º) angle are
called perpendicular lines. Line segments AB and CD are
A line or line segment is called a perpendicular bisector
when it intersects a line segment at the midpoint, forming vertical
angles of 90º in the process. For example, in the figure above,
since AD = DB, CD is the perpendicular
bisector of AB.
Keep in mind that if a single line or line segment is
perpendicular to two different lines or line segments, then those
two lines or line segments are parallel. This is actually just another
example of parallel lines being cut by a transversal (in this case,
the transversal is perpendicular to the parallel lines), but it
is a common situation when dealing with polygons. We’ll examine
this type of case later.