Elements of Geometry
Point
Definition Point
A point shows an exact position in space. We draw a point as a small dot.
Definition Point Notation
We usually name a point with a capital letter, such as \(A\).
In mathematics, we imagine that a point has no size or shape. It only marks a position.
Example
The diagram below shows three points labeled \(A\), \(B\), and \(C\):
Lines, Segments and Rays
Definition Line
A line is a straight path that goes on forever in both directions.
Definition Line Notation
- A line can be named with a lowercase letter, written as \(\Line{l}\).
- A line is named using two points on it, written as \(\Line{AB}\).
Example
Name the line shown below:
The line is \(\Line{EF}\).
Definition Line Segment
A line segment is the part of a line between two endpoints. It has a fixed length.
Definition Line Segment Notation
We name a line segment by its endpoints, written as \(\Segment{AB}\). We read this as “segment \(AB\)”.
Example
Name the segment shown below:
The segment is \(\Segment{EF}\).
Definition Ray
A ray is a part of a line that starts at one endpoint and goes on forever in one direction.
Definition Ray Notation
We name a ray by its endpoint first and another point on it, written as \(\Ray{AB}\). We read this as “ray \(AB\)”.
Example
Name the ray shown below:
The ray is \(\Ray{EF}\).
Definition Collinear Points
Collinear points are points that all lie on the same straight line.
Example
The points \(A\), \(B\) and \(C\) are collinear points.
Element Relation
Definition Element Relation
The relation is a point of (or “belongs to”) is used to show that a point lies on a geometric figure, such as a line or a segment. We write this relation using the symbol \(\in\).
Example
\(C \in \Line{AB}\) and \(C \notin \Segment{AB}\)Length
Definition Length of a Line Segment
The length of a line segment is the distance between its two endpoints, measured in units such as centimeters (cm) or meters (m).
Definition Length Notation
If \(\Segment{AB}\) is a segment, its length is denoted by \(AB\) (without the bar). In diagrams, we may also write \(\LengthSegment{AB}\) for the length of segment \(AB\).
Definition Equal Lengths
Line segments are equal in length if they have the same length. We use tick marks on the segments to show that they are equal: segments with the same number of tick marks have the same length.
Example
Identify two segments that have the same length.
Segments \(\Segment{AB}\) and \(\Segment{AC}\) have the same length, as shown by the identical tick marks on each of them. Therefore, \(AB = AC\).
Method Measuring Length
We measure the length of a segment with a ruler. Place one endpoint on the 0 mark, then read the number at the other endpoint: that number is the length of the segment.
Example
Measure the length of segment \(\Segment{AB}\).
By aligning a ruler with segment \(\Segment{AB}\), we measure the length as \(AB = 4 \,\text{cm}\). So the length of segment \(AB\) is \(4\) cm.
Intersection Point
Definition Intersection Point
An intersection point is a point where two or more lines, segments, or rays cross each other.
Example
Find the intersection point of the lines \(\Line{AB}\) and \(\Line{CD}\).
The intersection point is \(I\).
Parallel Lines
Definition Parallel Lines
Two parallel lines are lines that are always the same distance apart and never meet, even if you extend them.
Definition Parallel Line Notation
On a diagram, parallel lines are shown using matching little arrows on each line.
Perpendicular Lines
Definition Perpendicular Lines
Two perpendicular lines are lines that intersect at a right angle (90 degrees). We write \(\Line{AB} \perp \Line{CD}\) to show that line \(\Line{AB}\) is perpendicular to line \(\Line{CD}\).
Example
Identify the pair of perpendicular lines in the figure below.
The lines \(\Line{AB}\) and \(\Line{CD}\) are perpendicular, as they intersect, forming a right angle, indicated by the right-angle mark.
Midpoint and Perpendicular Bisector
Definition Midpoint of a Line Segment
The midpoint of a line segment is a point that lies on the segment and divides it into two segments of equal length.For example, if \(I\) is the midpoint of segment \(\Segment{AB}\), then \(I \in \Segment{AB}\) and \(AI = IB\).
Proposition Midpoint Length Property
If point \(I\) is the midpoint of segment \(\Segment{AB}\), then \(AB = 2 \times AI\) and \(AI = \frac{AB}{2}\).
$$\begin{aligned}AB &= AI + IB &&(I \text{ is the midpoint of } \Segment{AB})\\
&= AI + AI\\
&= 2 \times AI\end{aligned}$$Thus, \(AB = 2 \times AI\). To find \(AI\) in terms of \(AB\), we rearrange the equation:$$AI = \frac{AB}{2}.$$
Definition Perpendicular bisector
The perpendicular bisector of a line segment is the line that passes through the midpoint of the segment and is perpendicular to the segment.
Method Constructing the Perpendicular Bisector of \(\Segment{AB}\)
- Using a compass, construct two arcs with the same radius and with centers at \(A\) and \(B\).
- The arcs intersect at points \(E\) and \(F\).
- The perpendicular bisector of \(\Segment{AB}\) is the line \(\Line{EF}\).
Properties of Parallel Lines
Proposition Properties of Parallel Lines
These properties help us decide when two lines are parallel or perpendicular.
- If line \(\Line{l_1}\) is parallel to line \(\Line{l_2}\) and line \(\Line{l_2}\) is parallel to line \(\Line{l_3}\), then line \(\Line{l_1}\) is parallel to line \(\Line{l_3}\).
If 
, then 
. - If line \(\Line{l_1}\) is perpendicular to line \(\Line{l_3}\) and line \(\Line{l_2}\) is perpendicular to line \(\Line{l_3}\), then line \(\Line{l_1}\) is parallel to line \(\Line{l_2}\).
If 
, then 
. - If line \(\Line{l_1}\) is parallel to line \(\Line{l_2}\) and line \(\Line{l_1}\) is perpendicular to line \(\Line{l_3}\), then line \(\Line{l_2}\) is perpendicular to line \(\Line{l_3}\).
If 
, then 
.