The concept of lines is straightforward, but much of geometry is concerned with portions of lines. Some of those portions are so special that they have their own names and symbols.

**Line segment**

**A line segment** is a connected piece of a line. It has two endpoints and is named by its endpoints. Sometimes, the symbol – written on top of two letters is used to denote the segment. This is line segment *CD* (Figure 1).

**Figure 1 **Line segment.

It is written *CD* (Technically, *CD* refers to the points *C* and *D* and all the points between them, and *CD* without the *C* to *D*.) Note that *CD* is a piece of

*Postulate 7* (Ruler Postulate): Each point on a line can be paired with exactly one real number called its **coordinate**. The distance between two points is the positive difference of their coordinates (Figure 2).

**Figure 2**Distance between two points.

**Example 1**: In Figure 3, find the length of *QU*.

**Figure 3**Length of a line segment.

*Postulate 8 (Segment Addition Postulate):* If *B* lies between *A* and *C* on a line, then *AB + BC = AC* (Figure 4).

**Figure 4**Addition of lengths of line segments.

**Example 2**: In Figure 5, *A* lies between *C* and *T*. Find *CT* if *CA* = 5 and *AT* = 8.

**Figure 5**Addition of lengths of line segments.

Because *A* lies between *C* and *T*, Postulate 8 tells you

**Midpoint**

A **midpoint** of a line segment is the halfway point, or the point equidistant from the endpoints (Figure 6).

**Figure 6 **Midpoint of a line segment.

*R* is the midpoint of *QS* because *QR* = *RS* or because *QR* = ½ *QS* or *RS* = ½ *QS*

**Example 3:** In Figure 7, find the midpoint of *KR* .

**Figure 7**Midpoint of a line segment.

**
**

The midpoint of *KR* would be ½(24), or 12 spaces from either *K* or *R*. Because the coordinate of *K* is 5, and it is smaller than the coordinate of R (which is 29), to get the coordinate of the midpoint you could either add 12 to 5 or subtract 12 from 29. In either case, you determine that the coordinate of the midpoint is 17. That means that point *O* is the midpoint of *KR* because *KO* = *OR*.

Another way to get the coordinate of the midpoint would be to find the average of the endpoint coordinates. To find the average of two numbers, you find their sum and divide by two. (5 + 29) ÷ 2 = 17. The coordinate of the midpoint is 17, so the midpoint is point *O*.

*Theorem 4:* A line segment has exactly one midpoint.

**Ray**

A **ray** is also a piece of a line, except that it has only one endpoint and continues forever in one direction. It could be thought of as a half‐line with an endpoint. It is named by the letter of its endpoint and any other point on the ray. The symbol → written on top of the two letters is used to denote that ray. This is ray *AB* (Figure 8).

**Figure 8 **Ray *AB*.

It is written as

This is ray *CD* (Figure 9).

**Figure 9 **Ray *CD*.

It is written as

Note that the nonarrow part of the ray symbol is over the endpoint.