A
line is a collection of points that extends without limitin a straight formation. A line can be named by a single letter, likeline
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 commonto 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.
Line Segments
A
line segment is a section of aline. It is named and determined by its endpoints. Unlike a line, whoselength is infinite, a line segment has finite length. Line segment
AB is pictured below.
M.^*Pus"SYy)x4W+b2Y
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 aline segment into two equal halves, the midpoint is said to
bisect the line segment.
Because a midpoint cuts a line segment in half,knowing the distance between the midpoint and one endpoint of a linesegment allows you to calculate the length of the entire line segment.For example, if the distance from one endpoint to the midpoint of aline segment is 5, the length of the whole line segment is 10.
The Math IC test often asks questions that focuson this property of midpoints. The Math IC writers usually make theirquestions a little trickier though, by including multiple midpoints.Take a look:
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| X is 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 of WX? |
|
All the midpoints flying around in thisquestion can get quite confusing. Instead of trying to visualize whatis being described in your head, draw a sketch of what the questiondescribes.
Once you’ve drawn a sketch, you can see howthe three midpoints, and the new line segments that the midpointscreate, 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 to 1⁄2WZ.
- 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 outthese relationships when answering this sort of question. If you draw agood 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 1⁄8
WZ. 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 1⁄2
WZ, so
WX = 12.
Angles
Technically speaking, an
angle is theunion of two rays (lines that extend infinitely in just one direction)that share an endpoint (called the vertex of the angle). The measure ofan angle is how far you must rotate one of the rays such that itcoincides with the other.
In this guide and for the Math IC, you don’treally need to bother with such a technical definition. Suffice it tosay, angles are used to measure rotation. One full revolution around apoint creates an angle of 360 de
grees, or 360
. A half-revolution, also known as a straight angle, is 180
degrees. A quarter revolution, or right angle, is 90
.
u o.U"ol)A In text, angles can also be indicated by the symbol
.
Vertical Angles
When two lines or line segments intersect,two pairs of congruent (equal) angles are created. The angles in eachpair of congruent angles created by the intersection of two lines arecalled vertical angles:
In this figure,
and
are vertical angles (and therefore congruent), as are
and
.
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,
and
,
and
,
and
, and
and
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 twoparallel lines creates many interesting angle relationships. Thissituation is often referred to as “parallel lines cut by atransversal,” where the transversal is the nonparallel line. As you cansee 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 exist:
- Alternate exterior angles are pairs of congruent angles onopposite sides of the transversal, outside of the space between theparallel lines. In the figure above, there are two pairs of alternateexterior angles:
and 
, and 
and 
. - Alternate interior angles are pairs of congruent angleson opposite sides of the transversal in the region between the parallellines. In the figure above, there are two pairs of alternate interiorangles:
and 
, and 
and 
. - Corresponding angles are congruent angles on the sameside of the transversal. Of two corresponding angles, one will alwaysbe between the parallel lines, while the other will be outside theparallel lines. In the figure above, there are four pairs ofcorresponding angles:
and 
, 
and 
, 
and 
, and 
and 
.
In addition to these special relationshipsbetween angles, all adjacent angles formed when two parallel lines arecut by a transversal are supplementary. In the previous figure, forexample,
and
are supplementary.
Math IC questions covering parallel lines cut by a transversal are usually straightforward. For example:
|
|
| In the figure below, if lines m and n are parallel and  = 110º, then f – g = |
| 
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If you know the relationships of the anglesformed by two parallel lines cut by a transversal, answering thisquestion is easy.
and
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.
Perpendicular Lines
Two lines that intersect to form a right (90º) angle are called
perpendicular lines. Line segments
AB and
CD are perpendicular.
A line or line segment is called aperpendicular bisector when it intersects a line segment at themidpoint, 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 linesegment is perpendicular to two different lines or line segments, thenthose two lines or line segments are parallel. This is actually justanother example of parallel lines being cut by a transversal (in thiscase, the transversal is perpendicular to the parallel lines), but itis a common situation when dealing with polygons. We’ll examine thistype of case later.
