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
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 line segment.
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 IIC, you don’t 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 360º. A half revolution, also
known as a straight angle, is 180º. A quarter revolution, or right
angle, is 90º.
In text, angles can also be indicated through 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, angles 1 and 4 are vertical angles (and
therefore congruent), as are angles 2 and 3.
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,
4 are all pairs of supplementary
Lines that don’t intersect are called parallel lines.
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 8 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: 1 and 8, and 2and 7.
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: 3 and 6, and 4 and 5.
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: 1 and 5, 2 and 6, 3 and 7, and 4 and 8.
In addition to these special relationships between angles
formed by two parallel lines cut by a transversal, all adjacent
angles are supplementary.
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 above figure,
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.