# More Terminology for Segments and Angles

You may want to review: Segments, Rays, Angles

Harold R. Jacobs has written a great Geometry book, entitled
Geometry: Seeing, Doing, Understanding, third edition.
It has many excellent exercises.
The notation used here for *betweenness of points* is the same notation that is used by Jacobs.

Any line can be ‘transformed’
into a *number line* by choosing locations
for zero ($\,0\,$) and one
($\,1\,$).
With these choices, the positions of
all other real numbers are uniquely determined.

A number line is a conceptually perfect picture of the real numbers: each point on the line corresponds to a unique real number, and each real number corresponds to a unique point on the line.

When points are located on a number line, it is convenient to use the following convention:

The coordinate of point
$\,A\,$
is denoted by
$\,a\,.$

The coordinate of point $\,B\,$
is denoted by $\,b\,.$

Thus, uppercase letters are used to denote points, and the corresponding lowercase letter is used to denote the coordinate of the point.

In Geometry,
you want to be careful to distinguish
between a *point* (an exact location),
and a *number*
(that might be used to specify the
location of the point,
relative to a chosen coordinate system).

Recall that, given any two real numbers $\,x\,$ and $\,y\,,$ we can talk about the distance between them:

- If $\,x\gt y\,,$ then the distance between them is $\,x-y\,.$
- If $\,x\lt y\,,$ then the distance between them is $\,y-x\,.$
- A general formula for the distance between $\,x\,$ and $\,y\,$ is $\,|y-x|\,$ (or $\,|x-y|\,$). That is, to find the distance between any two real numbers, subtract them (in any order) and then take the absolute value.

The distance between points
$\,A\,$ and $\,B\,$ is the length of
the segment $\,\overline{AB}\,,$
and will be denoted by
$\,AB\,$ (no overline).
Thus, $\,\overline{AB}\,$ is a
geometric figure (a line segment),
but $\,AB\,$ is a *number*
that specifies the length
of the line segment $\,\overline{AB}\,.$

Given three points on a line,
it is often convenient to
talk about one of the points being
*between* the other two:

Let $\,A\,,$ $\,B\,,$ and $\,C\,$ be three points on a line, with corresponding coordinates $\,a\,,$ $\,b\,$ and $\,c\,.$

We say: $$ \begin{gather} \cssId{s47}{B \text{ is between } A \text{ and } C}\cr \cssId{s48}{\text{if and only if}}\cr \cssId{s49}{a\lt b\lt c \ \ \text{ or }\ \ a\gt b\gt c} \end{gather} $$

The sentence ‘$B\,$ is between $\,A\,$ and $\,C\,$’ will be notated by either $\,A{-}B{-}C\,$ or $\,C{-}B{-}A\,.$ Notice that whenever $\,B\,$ is between $\,A\,$ and $\,C\,,$ then $\,BA + BC = AC\,.$

Recall that $\,\angle AVB\,$ denotes the angle with vertex $\,V\,,$ with point $\,A\,$ on one side, and point $\,B\,$ on the other side.

As discussed earlier,
angles can be measured with a
protractor.
Often, angles are measured in *degrees*.

The notation
‘$\,m\angle AVB\,$’
is used to denote
*the measure of*
$\,\angle AVB\,.$
Thus,
$\,\angle AVB\,$ is a geometric figure
(an angle),
but $\,m\angle AVB\,$
is a *number* that measures
how ‘wide’ the angle is.

Angles are classified according to their size:

An angle is *acute*
if and only if
its measure is strictly between $\,0^{\circ}\,$ and $\,90^{\circ}\,.$

An angle is *a right angle*
if and only if
its measure is $\,90^{\circ}\,.$

An angle is *obtuse*
if and only if
its measure is strictly between $\,90^{\circ}\,$ and $\,180^{\circ}\,.$

An angle is *a straight angle*
if and only if
its measure is $\,180^{\circ}\,.$

Two geometric figures are
said to be *congruent*
if they coincide exactly
when superimposed, one on the other.
Congruent figures have exactly
the same size, and exactly the same shape.

For segments and angles, we can decide if they are congruent by looking at their measures: Line segments are congruent if and only if they have the same lengths. Angles are congruent if and only if they have the same measures.